seaway theory manual
TRANSCRIPT
Theoretical Manual ofStrip Theory Program
“SEAWAY for Windows”
J.M.J. Journée and L.J.M. Adegeest
Report 1370 September 2003
TU DELFT Ship Hydromechanics Laboratory
Delft University of Technology
AMARCON Advanced Maritime Consulting
www.amarcon.com
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
2
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
3
Summary
This report aims to be a guide and help for those people who want to study the theoreticalbackgrounds and the algorithms of a ship motions computer code based on the strip theory.The underlying report describes in detail the theoretical backgrounds and algorithms used bythe first author during the development of his six-degrees-of-freedom ship motions computercode, called SEAWAY.
The six ship motions of and about the centre of gravity G of the vessel have been defined inthe next figure.
Definition of ship motions
According to Newton’s second law, the equations of motion for six degrees of freedom of anoscillating ship in waves in a earth-bounded axes system have to be written as follows:
ixMj
iij direction in momentsor forces all of sum 6
1
=⋅∑=
&& for: 6,...1=i
Because a linear system has been considered here, the forces and moments in the right handside of these equations consist of a superposition of:• so-called hydromechanic forces and moments, caused by a harmonic oscillation of the
rigid body in the undisturbed surface of a fluid being previously at rest, and• so-called exciting wave forces and moments on the restrained body, caused by the
incoming harmonic waves.With this, the system of a with six degrees of freedom moving ship in waves can considered tobe a linear mass-damper-spring system with frequency-dependent coefficients and linearexciting forces and moments:
( ) ij
iijiijiijij FxcxbxaM =⋅+⋅+⋅+∑=
6
1
&&& for: 6,...1=i
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
4
In here, ix with indices 3,2,1=i are the displacements of G (surge, sway and heave) and ixwith indices 6,5,4=i are the rotations about the axes through G (roll, pitch and yaw). Theindices ij present at motion i the coupling with motion j .
The masses in the equations of motion above consist of solid masses or solid mass momentsof inertia of the ship ( ijM ) and “added” masses or “added” mass moments of inertia caused
by the disturbed water, the hydrodynamic masses or mass moments of inertia ( ija ). Anoscillating ship generates waves it self too; energy will be radiated from the ship. Thehydrodynamic damping-terms ( iij xb &⋅ ) account for this. For the heave, roll and pitch motions,
hydrostatic spring-terms ( iij xc ⋅ ) have to be added. The right hand sides of the equations of
motion consist of exciting wave forces and moments ( iF ).
In the so-called strip theory, the ship will be divided in 20 tot 30 cross sections, of which thetwo-dimensional hydromechanic coefficients and exciting wave loads will be calculated. Toobtain the three-dimensional values, these values will be integrated over the ship lengthnumerically. Finally, the differential equations will be solved to obtain the motions. Thesecalculations will be performed in the frequency domain.
It was in 1949 that Ursell published his potential theory for determining the hydrodynamiccoefficients of semicircular cross sections, oscillating in deep water in the frequency domain.Using this, for the first time a rough estimation could be made of the motions of a ship inregular waves at zero forward speed.Shortly after that Tasai, Grim, Gerritsma and many other scientists used various alreadyexisting conformal mapping techniques (to transform ship-like cross sections to a semicircle)together with Ursell’s theory, in such a way that the motions in regular waves of more realistichull forms could be calculated too. Most popular was the 2-parameter Lewis conformalmapping technique. The exciting wave loads were found from the loads in undisturbed waves– the so-called Froude-Krilov forces or moments – completed with diffraction termsaccounting for the presence of the ship in these waves.Borrowed from the broadcasting technology, Denis en Pierson published in 1953 asuperposition method to describe the irregular waves too. The sea was considered to be thesum of many simple harmonic waves; each wave with its own frequency, amplitude, directionand random phase lag. By calculating the responses of the ship on each of these individualharmonic waves and adding up the responses of the ship, the energy distribution of the ship’sbehaviour in irregular waves could be found. These irregular motions are characterised bysignificant amplitude and average period.However, these theories provided the motions at zero forward speed only. In 1957, Korvin-Kroukovski en Jacobs published a method - which was further improved in the sixties - toaccount for the effect of forward ship speed.So at the end of the fifties, all components for an elementary ship motions computer programfor deep water were already available.
Fukuda published in 1962 a calculation technique for the internal sheer forces and bendingmoments in a cross section of a ship.Frank published in 1967 his pulsating source theory to calculate the hydrodynamiccoefficients of a cross section of a ship in deep water directly, without using conformalmapping. The potential coefficients of a fully submerged cross section (bulbous bow) and
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
5
sections with a very low area coefficient (often present in the aft body) could be calculatednow too.Using Lewis conformal mapping, Keil published in 1974 his theory for obtaining the potentialcoefficients in very shallow water.Useful theories to calculate the added resistance of a ship due to waves were given by Boese(integrated pressure method) in 1970 and Gerritsma and Beukelman (radiated energy method)in 1972.So far, all hydrodynamic coefficients had been determined with the potential theory. However,roll requires a viscous correction on that. Ikeda, Himeno and Tanaka published in 1978 a veryuseful semi-empirical method for determining the viscous roll damping components.
The introduction of personal computers in the early eighties increased the accessibility forcarrying out ship motion calculations considerably; even non-specialists could become userstoo. From then on the computer capacity and the computing speed increased very fast, so thatthree-dimensional theories could be developed much easier and cheaper now.Because of the complex problem of forward speed in 3-D theories however, the 2-D approach(strip theory) is still very favourable for calculating the behaviour of a ship at forward speed.The many advantages and just a few disadvantages, when comparing 2-D with 3-D, had beendiscussed very clearly by Faltinsen and Svensen in 1990.
Among others as a consequence of the work of the researchers mentioned above, a DOSpersonal computer strip theory program - called SEAWAY - had been completed by the DelftUniversity of Technology at the end of the eighties. Recently, a Windows version has beencompleted too, see web site www.shipmotions.nl or www.amarcon.com.Based on the linear strip theory, this program calculates for 6 degrees of freedom in thefrequency domain the hydromechanic loads, wave loads, absolute and relative motions, addedresistance and internal loads of displacement ships, barges and yachts in regular and irregularwaves. When ignoring interaction effects between the two individual hulls, the behaviour ofcatamarans and semi-submersibles can be calculated too. The program is suitable for deepwater as well as for very shallow water. Viscous roll damping, bilge keels, free-surface anti-roll tanks, external moments and (linear) mooring springs can be added.The computer code has been verified and validated extensively by the authors, many studentsand a large number of industrial users.
Error messages, advises and all type of comments on this technical report are very welcomeby e-mail to [email protected] or [email protected].
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
6
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
7
Table of Contents:
1 Introduction........................................................................................................................111.1 About the Authors..........................................................................................................111.2 About this Manual........................................................................................................ 12
2 Strip Theory Methods ....................................................................................................... 152.1 Definitions .................................................................................................................... 172.2 Incident Wave Potential................................................................................................ 20
2.2.1 Continuity Condition............................................................................................ 212.2.2 Laplace Equation.................................................................................................. 212.2.3 Seabed Boundary Condition................................................................................. 222.2.4 Free Surface Dynamic Boundary Condition......................................................... 222.2.5 Free Surface Kinematic Boundary Condition....................................................... 242.2.6 Dispersion Relationship ........................................................................................ 252.2.7 Relationships in Regular Waves........................................................................... 26
2.3 Floating Rigid Body in Waves...................................................................................... 282.3.1 Fluid Requirements............................................................................................... 282.3.2 Forces and Moments............................................................................................. 302.3.3 Hydrodynamic Loads............................................................................................ 312.3.4 Wave and Diffraction Loads ................................................................................. 362.3.5 Hydrostatic Loads................................................................................................. 38
2.4 Equations of Motion..................................................................................................... 392.5 Strip Theory Approaches.............................................................................................. 43
2.5.1 Zero Forward Ship Speed ..................................................................................... 432.5.2 Forward Ship Speed.............................................................................................. 442.5.3 End-Terms............................................................................................................. 46
2.6 Hydrodynamic Coefficients.......................................................................................... 483 2-D Potential Coefficients ................................................................................................ 51
3.1 Conformal Mapping Methods....................................................................................... 533.1.1 Lewis Conformal Mapping................................................................................... 543.1.2 Extended Lewis Conformal Mapping................................................................... 583.1.3 Close-Fit Conformal Mapping.............................................................................. 593.1.4 Mapping Comparisons .......................................................................................... 63
3.2 Potential Theory of Tasai.............................................................................................. 653.2.1 Heave Motions ...................................................................................................... 663.2.2 Sway Motions ....................................................................................................... 763.2.3 Roll Motions ......................................................................................................... 883.2.4 Low and High Frequencies................................................................................. 100
3.3 Potentia l Theory of Keil.............................................................................................. 1033.3.1 Notations of Keil................................................................................................. 1033.3.2 Basic Assumptions.............................................................................................. 1043.3.3 Vertical Motions.................................................................................................. 1063.3.4 Horizontal Motions ............................................................................................. 1333.3.5 Appendices ......................................................................................................... 144
3.4 Potential Theory of Frank ........................................................................................... 1533.4.1 Notations of Frank .............................................................................................. 1533.4.2 Formulation of the Problem................................................................................ 1553.4.3 Solution of the Problem...................................................................................... 1573.4.4 Low and High Frequencies................................................................................. 1613.4.5 Irregular Frequencies .......................................................................................... 162
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
8
3.4.6 Appendices ......................................................................................................... 1653.5 Comparisons between Calculated Potential Data ....................................................... 1733.6 Estimated Potential Surge Coefficients ...................................................................... 175
4 Viscous Damping............................................................................................................ 1774.1 Surge Damping ........................................................................................................... 178
4.1.1 Total Surge Damping.......................................................................................... 1784.1.2 Viscous Surge Damping...................................................................................... 179
4.2 Roll Damping.............................................................................................................. 1804.2.1 Experimental Determination............................................................................... 1814.2.2 Empirical Formula for Barges ............................................................................ 1834.2.3 Empirical Method of Miller................................................................................ 1834.2.4 Semi-Empirical Method of Ikeda ....................................................................... 184
5 Hydromechanical Loads ................................................................................................. 1975.1 Hydromechanical Forces for Surge ............................................................................ 1985.2 Hydromechanical Forces for Sway............................................................................. 2015.3 Hydromechanical Forces for Heave ........................................................................... 2045.4 Hydromechanical Moments for Roll .......................................................................... 2075.5 Hydromechanical Moments for Pitch......................................................................... 2105.6 Hydromechanical Moments for Yaw.......................................................................... 213
6 Exciting Wave Loads ...................................................................................................... 2176.1 Wave Potential............................................................................................................ 2176.2 Classical Approach..................................................................................................... 219
6.2.1 Exciting Wave Forces for Surge ......................................................................... 2196.2.2 Exciting Wave Forces for Sway.......................................................................... 2216.2.3 Exciting Wave Forces for Heave ........................................................................ 2236.2.4 Exciting Wave Moments for Roll ....................................................................... 2256.2.5 Exciting Wave Moments for Pitch...................................................................... 2276.2.6 Exciting Wave Moments for Yaw....................................................................... 228
6.3 Approximating 2-D Diffraction Approach................................................................. 2296.3.1 Hydromechanical Loads ..................................................................................... 2296.3.2 Energy Considerations ........................................................................................ 2316.3.3 Wave Loads......................................................................................................... 232
6.4 Numerical Comparisons ............................................................................................. 2387 Transfer Functions of Motions ....................................................................................... 239
7.1 Centre of Gravity Motions .......................................................................................... 2407.2 Local Absolute Displacements ................................................................................... 2437.3 Local Absolute Velocities ........................................................................................... 2447.4 Local Absolute Accelerations ..................................................................................... 245
7.4.1 Accelerations in the Earth-Bound Axes System................................................. 2457.4.2 Accelerations in the Ship-Bound Axes System.................................................. 245
7.5 Local Vertical Relative Displacements....................................................................... 2477.6 Local Vertical Relative Velocities............................................................................... 248
8 Anti-Rolling Devices ...................................................................................................... 2498.1 Bilge Keels.................................................................................................................. 2508.2 Passive Free-Surface Tanks........................................................................................ 251
8.2.1 Theoretical Approach......................................................................................... 2518.2.2 Experimental Approach...................................................................................... 2558.2.3 Effect of Free-Surface Tanks .............................................................................. 257
8.3 Active Fin Stabilisers.................................................................................................. 2588.4 Active Rudder Stabilisers ........................................................................................... 261
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
9
9 External Linear Springs .................................................................................................. 2639.1 External Loads ............................................................................................................ 2639.2 Additional Coefficients............................................................................................... 2649.3 Linearised Mooring Coefficients................................................................................ 266
10 Added Resistance due to Waves ..................................................................................... 26710.1 Radiated Energy Method ........................................................................................ 26910.2 Integrated Pressure Method .................................................................................... 27110.3 Comparison of Results............................................................................................ 273
11 Bending and Torsion Moments....................................................................................... 27511.1 Still Water Loads .................................................................................................... 28111.2 Dynamical Lateral Loads........................................................................................ 28211.3 Dynamical Vertical Loads....................................................................................... 28411.4 Dynamical Torsion Loads....................................................................................... 287
12 Statistics in Irregular Waves........................................................................................... 28912.1 Normalised Wave Energy Spectra .......................................................................... 290
12.1.1 Neumann Wave Spectrum................................................................................... 29012.1.2 Bretschneider Wave Spectrum............................................................................ 29012.1.3 Mean JONSWAP Wave Spectrum...................................................................... 29112.1.4 Definition of Parameters..................................................................................... 291
12.2 Response Spectra and Statistics.............................................................................. 29512.3 Shipping Green Water............................................................................................. 30012.4 Bow Slamming........................................................................................................ 302
12.4.1 Criterium of Ochi................................................................................................ 30212.4.2 Criterium of Conolly........................................................................................... 303
13 Twin-Hull Ships.............................................................................................................. 30713.1 Hydromechanical Coefficients ............................................................................... 30713.2 Equations of Motion............................................................................................... 30813.3 Hydromechanical Forces and Moments ................................................................. 30913.4 Exciting Wave Forces and Moments...................................................................... 31013.5 Added Resistance due to Waves ............................................................................. 314
13.5.1 Radiated Energy Method .................................................................................... 31413.5.2 Integrated Pressure Method ................................................................................ 314
13.6 Bending and Torsion Moments............................................................................... 31514 Numerical Recipes.......................................................................................................... 317
14.1 Polynomials ............................................................................................................ 31714.1.1 First Degree Polynomial..................................................................................... 31714.1.2 Second Degree Polynomial................................................................................. 318
14.2 Integrations ............................................................................................................. 31914.2.1 First Degree Integration...................................................................................... 31914.2.2 Second Degree Integration................................................................................. 31914.2.3 Integration of Wave Loads.................................................................................. 320
14.3 Derivatives.............................................................................................................. 32314.3.1 First Degree Derivative....................................................................................... 32314.3.2 Second Degree Derivative .................................................................................. 323
14.4 Curve Lengths......................................................................................................... 32614.4.1 First Degree Curve.............................................................................................. 32614.4.2 Second Degree Curve ......................................................................................... 326
15 References....................................................................................................................... 329
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
10
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
11
1 Introduction
SEAWAY is a frequency-domain ship motions PC program, based on the linear strip theoryfor calculating the hydromechanic loads, wave-induced loads, motions, added resistance andinternal loads for six degrees of freedom of displacement ships and yachts, barges, semi-submersibles or catamarans, sailing in regular and irregular waves. The program is suitable fordeep water as well as for shallow water. Viscous roll damping, bilge keels, anti-roll tanks, freesurface effects and (linear springs) can be added.This computer code has been developed in the late eighties and early nineties under DOS bythe first author. His last “SEAWAY for DOS” version was released in 2002.In 2003, the second author took over the software implementation and distribution part of thejob and developed the new “SEAWAY for Windows” release.Information can be found at web site www.shipmotions.nl or www.amarcon.com.
1.1 About the Authors
Johan Journée had obtained his Polytechnical Degree in 1964 at the Avond-HTS Rotterdamand in 1975 his MSc degree at the Delft University of Technology. Both degrees in NavalArchitecture were obtained, alongside a full-time job, by studying in the evening hours.He started his working career in 1958 at the Rotterdam Dockyard Company, first withconstruction work in the shipbuilding factory and two years later with technical ship designwork in the drawing office of this yard. In 1963, he became Technical Officer at the ShipHydromechanics Laboratory of the Delft University of Technology. After obtaining his MScdegree in 1975, he became Scientific Officer there, some years later Assistant Professor and in1992 Associate Professor.During the years 1985 through 1990, Johan had developed - as a more or less derailed hobby -this 2-D ship motions computer code SEAWAY. This development was a very useful exercisefor him to understand in a very detailed way the theory and the practice of the behaviour of aship in waves, see Journée [1992]. Parts of this study were basis for comprehensive lecturenotes on Offshore Hydromechanics, see Journée and Massie [2001].Since 1984, Johan Journée is teaching Ship and Offshore Hydromechanics to students of theMechanical and Civil Engineering Departments and since 1990 also to students of theMaritime Technology Department.
Leon Adegeest was one of the founders of AMARCON in January 2001. AMARCON’s majoractivities are developing software for decision support onboard using seakeeping theory andrelated consultancy and engineering work.Before AMARCON, Leon has worked at Det Norske Veritas (DNV) in Norway and atMARIN in the Netherlands. At DNV (1997 – 2001) he developed methods for the predictionof extreme non-linear ship responses in irregular seas. As Group Leader Hydrodynamics, hewas heavily involved in the development and implementation of DNV's non-linear 3-Dseakeeping software package WASIM. Practical experience in the use of seakeeping codeswas gained during commercial projects, which include non-linear wave load analyses for largecontainer carriers, fatigue analyses for a RoRo-carrier and water on deck evaluations forproduction vessels. From 1994 to 1997, he worked at the Maritime Research InstituteNetherlands (MARIN) as project manager in the Trials and Monitoring group.Leon Adegeest holds a MSc degree in Naval Architecture from the Delft University ofTechnology. In 1994, he finished his PhD study at this university. The title of the thesis was“Non-linear Hull Girder Loads in Ships”, see Adegeest [1994].
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
12
1.2 About this Manual
This manual aim at being a guide and aid for those who want to study the theoreticalbackgrounds and the algorithms of a ship motions computer code, like SEAWAY, based on thestrip theory. The theoretical backgrounds and the algorithms of this program have beendescribed here in detail.
Chapter 1, this introduction, gives a short survey of the contents of all chapters in this report.
Chapter 2 gives a general description of the various strip theory approaches. A generaldescription of the potential flow theory is given. The derivations of the hydromechanic forcesand moments, the wave potential and the wave and diffraction forces and moments have beendescribed.The equations of motion are given with solid mass and inertia terms and hydromechanicforces and moments in the left hand side and the wave exciting forces and moments in theright hand side. The principal assumptions are a linear relation between forces and motionsand the validity of obtaining the total forces by a simple integration over the ship length of thetwo-dimensional cross sectional forces.This includes for all motions a forward speed effect caused by the potential mass, as has beendefined by Korvin-Kroukovsky and Jacobs [1957] for the heave and pitch motions. Thisapproach is called the ''Ordinary Strip Theory Method''. Also an inclusion of the forwardspeed effect caused by the potential damping, as for instance given by Tasai [1969]. Thisapproach is called the ''Modified Strip Theory Method''.The inclusion of so-called ''End-Terms'' has been described too.
Chapter 3 describes the determination of the two-dimensional potential mass and dampingcoefficients for the six modes of motions at infinite and finite water depths.Firstly, it describes several conformal mapping methods. For the determination of the two-dimensional hydrodynamic potential coefficients for sway, heave and roll motions of ship-likecross sections, these sections are conformal mapped to the unit circle. The advantage ofconformal mapping is that the velocity potential of the fluid around an arbitrary shape of across section in a complex plane can be derived from the more convenient circular section inanother complex plane. In this manner hydrodynamic problems can be solved directly withthe coefficients of the mapping function.The close-fit multi-parameter conformal mapping method is given. A very simple and straighton iterative least squares method, used to determine the conformal mapping coefficients, hasbeen described. Two special cases of multi-parameter conformal mapping have beendescribed too: the well known classic transformation of Lewis [1929] with two parametersand an Extended-Lewis transformation with three parameters, as given by Athanassoulis andLoukakis [1985].Then, it describes 3 methods for the determination of the two-dimensional potential mass anddamping coefficients for the six modes of motions at infinite and finite water depths. Atinfinite water depths, the principle of the calculation of these potential coefficients is based onwork of Ursell [1949] for circular cylinders and Frank [1967] for any arbitrary symmetriccross section.Starting from the velocity potentials and the conjugate stream functions of the fluid with aninfinite depth as have been given by Tasai [1959], Tasai [1960], Tasai [1961] and de Jong[1973] and using the multi-parameter conformal mapping technique, the calculation routines
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
13
of the two-dimensional hydrodynamic potential coefficients of ship-like cross sections aregiven for the sway, heave and roll motions.For any arbitrary water depth (deep to very shallow water), the method of Keil [1974] - basedon a variation of the theory of Ursell [1949] with Lewis conformal mapping - has been given.Finally, the pulsating source method of Frank [1967] for deep water has been described.Because of using the strip theory approach here, the pitch and yaw coefficients follow fromthe moments about the ship's centre of gravity of the heave and sway coefficients,respectively.Approximations are given for the surge coefficients.
Chapter 4 gives some corrections on the hydrodynamic damping due to viscous effects. Thesurge-damping coefficient is corrected for viscous effects by an empirical method, based on asimple still water resistance curve as published by Troost [1955].The analysis of free-rolling model experiments and two (semi) empirical methods publishedby Miller [1974] and Ikeda [1978], to determine a viscous correction of the roll-dampingcoefficients, are described in detail.
Chapter 5 describes the determination of the hydromechanic forces and moments in the left-hand side of the six equations of motion of a sailing ship, obtained with the hydromechaniccoefficients as determined in Chapter 3 and 4, for both the ordinary and the modified striptheory method.
Chapter 6 describes the wave exciting forces and moments in the right hand side of the sixequations of motion of a sailing ship in water with an arbitrarily depth, using the relativemotion concept for both the ordinary and the modified strip theory method.First, the classical approach has been described, using equivalent accelerations and velocitiesof the water particles. Then, an alternative approach, based on diffraction of waves, has beendescribed.
Chapter 7 describes the solution of the equations of motion and the determination of thefrequency characteristics of the absolute displacements, rotations, velocities and accelerationsand the vertical relative displacements. The use of a wave potential valid for any arbitrarywater depth makes a calculation method with deep water coefficients, suitable for shipssailing with keel clearances down to about 50 percent of the ship's draft. At lower waterdepths, Keil’s method should be used.
Chapter 8 describes some anti-rolling devices. A description is given of an inclusion ofpassive free-surface tanks as defined by the experiments of van den Bosch and Vugts [1966]and by the theory of Verhagen and van Wijngaarden [1965]. Active fin and rudder stabilisershave been described too.
Chapter 9 describes the inclusion of linear spring terms, to simulate the behavior of anchoredor moored ships.
Chapter 10 describes two methods to determine the transfer functions of the added resistancedue to waves. The first method is a radiated wave energy method, as published by Gerritsmaand Beukelman [1972]. The second method is an integrated pressure method, as published byBoese [1970].
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
14
Chapter 11 describes the determination of the frequency characteristics of the lateral andvertical shear forces and bending moments and the torsion moments in a way as had beenpresented by Fukuda [1962] for the vertical mode. Still water phenomena are described too.
Chapter 12 describes the statistics in irregular waves, by using the superposition principle.Three examples of normalized wave spectra are given: the somewhat wide wave spectrum ofNeumann, an average wave spectrum of Bretschneider and the more narrow Mean JONSWAPwave spectrum.A description is given of the calculation procedure of the energy spectra and the statistics ofthe ship motions for six degrees of freedom, the added resistance, the vertical relative motionsand the mechanic loads on the ship in waves coming from any direction.For the calculation of the probability of exceeding a threshold value by the motions, theRayleigh probability density function has been used.The static and dynamic swell-up of the waves, of importance when calculating the probabilityof shipping green water, are defined according to Tasaki [1963]. A theoretically determineddynamic swell-up had been given too.Bow slamming phenomena are defined by both the relative bow velocity criterion of Ochi[1964] and by the peak bottom-impact-pressure criteria of Conolly [1974].
Chapter 13 describes the additions to all algorithms in case of twin- hull ships, such as semi-submersibles and catamarans. However, for interaction effects between the two individualhulls will not be accounted here.
Chapter 14 shows some typical numerical recipes, as has been used in program SEAWAY.
Finally, Chapter 15 gives a survey of all literature used during the development of thiscomputer code.
Error messages, advises and all type of comments on this technical report are very welcomeby e-mail: [email protected].
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
15
2 Strip Theory Methods
The ship is considered to be a rigid body, floating in the surface of an ideal fluid, which ishomogeneous, incompressible, free of surface tension, irrotational and without viscosity. It isassumed that the problem of the motions of this floating body in waves is linear or can belinearised. Consequently, only the external loads on the underwater part of the ship areconsidered here and the effect of the above water part will be fully neglected.
Faltinsen and Svensen [1990] have discussed the incorporation of seakeeping theories in shipdesign clearly. An overview of seakeeping theories for ships were presented and it wasconcluded that - nevertheless some limitations - strip theories are the most successful andpractical tools for the calculation of the wave induced motions of the ship, at least in an earlydesign stage of a ship.The strip theory solves the three-dimensional problem of the hydromechanical and excitingwave forces and moments on the ship by integrating the two-dimensional potential solutionsover the ship's length. Interactions between the cross sections are ignored for the zero-speedcase. So, each cross section of the ship is considered to be part of an infinitely long cylinder.
The strip theory is a slender body theory, so one should expect less accurate predictions forships with low length to breadth ratios. However, experiments showed that the strip theoryappears to be remarkably effective for predicting the motions of ships with length to breadthratios down to about 3.0, or even sometimes lower.The strip theory is based on the potential flow theory. This holds that viscous effects areneglected, which can deliver serious problems when predicting roll motions at resonancefrequencies. In practice, for viscous roll damping effects can be accounted fairly by empiricalmethods.Because of the way that the forced motion problems are solved, generally in the strip theory,substantial disagreements can be found between the calculated results and the experimentaldata of the wave loads at low frequencies of encounter in following waves. In practicehowever, these ''near zero frequency of encounter problems'' can be solved by forcing thewave loads going to zero, artificially.
For high-speed vessels and for large ship motions, as appear in extreme sea states, the striptheory can deliver less accurate results. Then the so-called ''end-terms'' can become veryimportant.The strip theory accounts for the interaction with the forward speed in a very simple way. Theeffect of the steady wave system around the ship is neglected and the free surface conditionsare simplified, so that the unsteady waves generated by the ship are propagating in directionsperpendicular to the centre plane of the ship. In reality the wave systems around the ship arefar more complex. For high-speed vessels, unsteady divergent wave systems becomeimportant. This effect is neglected in the strip theory.The strip theory is based on linearity. This means that the ship motions are supposed to besmall, relative to the cross sectional dimensions of the ship. Only hydrodynamic effects of thehull below the still water level are accounted for. So, when parts of the ship go out of or in tothe water or when green water is shipped, inaccuracies can be expected. Also, the strip theorydoes not distinguish between alternative above water hull forms.Because of the added resistance of a ship due to the waves is proportional to the relativemotions squared, its inaccuracy will be gained strongly by inaccuracies in the predictedmotions.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
16
Nevertheless these limitations, seakeeping prediction methods based upon the strip theoryprovide a sufficiently good basis for optimisation studies at an early design stage of the ship.At a more detailed design stage, it can be considered to carry out additional modelexperiments to investigate for instance added resistance or extreme event phenomena, such asshipping green water and slamming.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
17
2.1 Definitions
Figure 2.1–1 shows a harmonic wave as seen from two different perspectives. Figure 2.1–1-ashows what one would observe in a snapshot photo made looking at the side of a (transparent)wave flume; the wave profile is shown as a function of distance x along the flume at a fixedinstant in time. Figure 2.1–1-b shows a time record of the water level observed at one locationalong the flume; it looks similar in many ways to the other figure, but time t has replaced xon the horizontal axis.
Figure 2.1–1: Harmonic wave definitions
Notice that the origin of the co-ordinate system is at the still water level with the positive z -axis directed upwards; most relevant values of z will be negative.The still water level is the average water level or the level of the water if no waves werepresent. The x -axis is positive in the direction of wave propagation. The water depth, h , (apositive value) is measured between the seabed ( hz −= ) and the still water level ( 0=z ).The highest point of the wave is called its crest and the lowest point on its surface is thetrough. If the wave is described by a harmonic wave, then its amplitude aζ is the distancefrom the still water level to the crest, or to the trough for that matter. The subscript a denotesthe amplitude, here.The horizontal distance (measured in the direction of wave propagation) between any twosuccessive wave crests is the wavelength, λ . The distance along the time axis is the waveperiod, T . The ratio of wave height to wavelength is often referred to as the dimensionlesswave steepness: λζ /2 a⋅ .Since the distance between any two corresponding points on successive harmonic waves is thesame, wave lengths and periods are usually actually measured between two consecutiveupward (or downward) crossings of the still water level. Such points are also called zero-crossings, and are easier to detect in a wave record.Since sine or cosine waves are expressed in terms of angular arguments, the wavelength andwave period are converted to angles using:
πωπλ
⋅=⋅⋅=⋅
2
2
T
k or
T
k
πω
λπ
⋅=
⋅=
2
2
Equation 2.1–1
in which k is the wave number (rad/m) and ω is the circular wave frequency (rad/s).
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
18
Obviously, the wave form moves one wave length during one period, so that its speed orphase velocity, c , is given by:
kTc
ωλ==
Equation 2.1–2
Suppose now a sailing ship in waves, with co-ordinate systems as given in Figure 2.1–2.
Figure 2.1–2: Co-ordinate systems
A right-handed co-ordinate system ( )000 ,, zyxS is fixed in space. The ( )00 , yx -plane lies in the
still water surface, 0x is directed as the wave propagation and 0z is directed upwards.
Another right-handed co-ordinate system ( )zyxO ,, is moving forward with a constant shipspeed V . The directions of the axes are: x in the direction of the forward ship speed V , y inthe lateral port side direction and z vertically upwards. The ship is supposed to carry outoscillations around this moving ( )zyxO ,, co-ordinate system. The origin O lies verticallyabove or under the time-averaged position of the centre of gravity G . The ( )yx, -plane lies inthe still water surface.
A third right-handed co-ordinate system ( )bbb zyxG ,, is connected to the ship with its origin at
G , the ship's centre of gravity. The directions of the axes are: bx in the longitudinal forward
direction, by in the lateral port side direction and bz upwards. In still water, the ( )bb yx , -planeis parallel to the still water surface.
If the wave moves in the positive 0x -direction (defined in a direction with an angle µ relativeto the ship's speed vector, V ), the wave profile - the form of the water surface - can now beexpressed as a function of both 0x and t as follows:
( )txka ⋅−⋅⋅= ωζζ 0cos or ( )0cos xkta ⋅−⋅⋅= ωζζ
Equation 2.1–3
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
19
The right-handed co-ordinate system ( )zyxO ,, is moving with the ship's speed V , whichyields:
µµµ sincoscos0 ⋅+⋅+⋅⋅= yxtVx
Equation 2.1–4
From the relation between the frequency of encounter eω and the wave frequency ω:
µωω cos⋅⋅−= Vke
Equation 2.1–5
follows:
( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= ykxktea
Equation 2.1–6
The resulting six ship motions in the ( )zyxO ,, system are defined by three translations of theship's centre of gravity in the direction of the x -, y - and z -axes and three rotations aboutthem:
( )( )( )( )( )( )ψζ
θζ
φζ
ζ
ζ
ζ
εωψψ
εωθθ
εωφφ
εω
εω
εω
+⋅⋅=
+⋅⋅=
+⋅⋅=
+⋅⋅=
+⋅⋅=
+⋅⋅=
t
t
t
tz
ty
txx
ea
ea
ea
zea
yea
xea
cos :yaw
cos :pitch
cos :roll
cosz :heave
cosy :sway
cos :surge
Equation 2.1–7
The phase shifts of these motions are related to the harmonic wave elevation at the origin ofthe ( )zyxO ,, system, i.e. the average position of the ship's centre of gravity:
( )tea ⋅⋅= ωζζ cos :wave
Equation 2.1–8
The harmonic velocities and accelerations in the ( )zyxO ,, system are found now by takingthe derivatives of the displacements, for instance for surge:
( )( )
( )ζ
ζ
ζ
εωω
εωω
εω
xeae
xeae
xea
txx
txx
txx
+⋅⋅⋅−=
+⋅⋅⋅−=
+⋅⋅=
cos :onaccelerati surge
sin : velocitysurge
cos :ntdisplaceme surge
2&&
&
Equation 2.1–9
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
20
2.2 Incident Wave Potential
In order to use the linear potential theory for waves, it will be necessary to assume that thewater surface slope is very small. This means that the wave steepness is so small that terms inthe equations of motion of the waves with a magnitude in the order of the steepness-squaredcan be ignored.Suppose a wave moving in the ( )zx, -plane. The profile of that simple wave with a smallsteepness looks like a sine or a cosine and the motion of a water particle in a wave depends onthe distance below the still water level. This is reason why the wave potential can be writtenas:
( ) ( ) ( )txkzPtzxw ⋅−⋅⋅=Φ ωsin,,
Equation 2.2–1
in which ( )zP is an (as yet) unknown function of z .
This velocity potential ( )tzxw ,,Φ of the harmonic waves has to fulfil four requirements:• Continuity condition or Laplace equation• Seabed boundary condition• Free surface dynamic boundary condition• Free surface kinematic boundary conditionThese requirements lead to a more complete expression for the velocity potential as will beexplained in the following subsections.
The relationships presented in these subsections are valid for all water depths, but the fact thatthey contain so many hyperbolic functions makes them cumbersome to use. Engineers - asopposed to (some) scientists - often look for ways to simplify the theory. The simplificationsstem from the following approximations for large and very small arguments, s , as shown inFigure 2.2–1:
For large arguments s : [ ] [ ][ ] 1 tan
coshsinh
≈>>≈
sh
sss
For small arguments s : [ ] [ ][ ] 1cos
tanhsinh
≈≈≈
sh
sss
Equation 2.2–2
Figure 2.2–1: Hyperbolic function limits
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
21
2.2.1 Continuity Condition
The velocity of the water particles ( )wvu ,, in the three translation directions, or alternatively
( )zyx vvv ,, , follow from the definition of the velocity potential, wΦ :
xvu w
x ∂Φ∂
== y
vv wy ∂
Φ∂==
zvw w
z ∂Φ∂
==
Equation 2.2–3
Since the fluid is homogeneous and incompressible, the continuity condition becomes:
0=∂∂
+∂∂
+∂∂
zw
yv
xu
Equation 2.2–4
2.2.2 Laplace Equation
The continuity condition in Equation 2.2–4 results in the Laplace equation for potential flows:
02
2
2
2
2
22 =
∂Φ∂
+∂Φ∂
+∂Φ∂
=Φ∇zyx
wwww
Equation 2.2–5
Water particles move here in the ( )zx, -plane only, so in the equations above:
0=∂Φ∂
=y
v w and 02
2
=∂Φ∂
=∂∂
yyv w
Equation 2.2–6
Taking this into account, a substitution of Equation 2.2–1 in Equation 2.2–5 yields ahomogeneous solution of this equation:
( ) ( ) 022
2
=⋅− zPkdz
zPd
Equation 2.2–7
with as a homogeneous solution for ( )zP :
( ) zkzk eCeCzP ⋅−⋅+ ⋅+⋅= 21
Equation 2.2–8
Using this result from the continuity condition and the Laplace equation, the wave potentialcan be written now with two unknown coefficients as:
( ) ( ) ( )txkeCeCtzx zkzkw ⋅−⋅⋅⋅+⋅=Φ ⋅−⋅+ ωsin,, 21
Equation 2.2–9
in which:( )tzxw ,,Φ wave potential (m2/s)
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
22
e base of natural logarithms (-)
21 ,CC as yet undetermined constants (m2/s)k wave number (1/m)t time (s)x horizontal distance (m)z vertical distance, positive upwards (m)ω wave frequency (1/s)
2.2.3 Seabed Boundary Condition
The vertical velocity of water particles at the seabed is zero (no-leak condition):
0=∂Φ∂z
w for: hz −=
Equation 2.2–10
Substituting this boundary condition in Equation 2.2–9 provides:021 =⋅⋅−⋅⋅ ⋅+⋅− hkhk eCkeCk
Equation 2.2–11
By defining:hkhk eCeCC ⋅+⋅− ⋅⋅=⋅⋅= 21 22
or:hke
CC ⋅+⋅=
21 and hkeC
C ⋅−⋅=22
it follows that ( )zP in Equation 2.2–8 can be worked out to:
( ) ( ) ( )( )( )[ ]zhkC
eeC
zP zhkzhk
+⋅⋅=
+⋅= +⋅−+⋅+
cosh2
Equation 2.2–12
and the wave potential Equation 2.2–1 becomes:( ) ( )[ ] ( )txkzhkCtzxw ⋅−⋅⋅+⋅⋅=Φ ωsincosh,,
Equation 2.2–13
in which C is an (as yet) unknown constant.
2.2.4 Free Surface Dynamic Boundary Condition
The pressure, p , at the free surface of the fluid, ζ=z , is equal to the atmospheric pressure,
0p . This requirement for the pressure is called the dynamic boundary condition at the freesurface.The Bernoulli equation for an instationary irrotational flow (with the velocity given in termsof its three components) is in its general form:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
23
( ) 021 222 =⋅++++⋅+
∂Φ∂
zgp
wvut
w
ρ
Equation 2.2–14
In two dimensions, 0=v , and since the waves have a small steepness (u and w are small),this equation becomes in a linearised format:
0=⋅++∂Φ∂
zgp
tw
ρ
Equation 2.2–15
At the free surface this condition becomes:
0=⋅++∂Φ∂ ζ
ρg
pt
w for: ζ=z
Equation 2.2–16
The constant value ρ0p can be included in tw ∂Φ∂ ; this will not influence the velocities
being obtained from the potential wΦ .With this the equation becomes:
0=⋅+∂Φ∂
ζgt
w for: ζ=z
Equation 2.2–17
The potential at the free surface can be expanded in a Taylor series, keeping in mind that thevertical displacement of the wave surface ζ is relatively small:
( ) ( ) ( ).........
,,,,,,
00 +
∂Φ∂
⋅+Φ=Φ=
==z
wzwzw z
tzxtzxtzx ζζ
or:( ) ( ) ( )2
0
,,,, εζ
Ot
tzxt
tzx
z
w
z
w +
∂Φ∂
=
∂Φ∂
==
Equation 2.2–18
which yields for the linearised form of the free surface dynamic boundary condition inEquation 2.2–17:
0=⋅+∂Φ∂
ζgt
w for: 0=z
Equation 2.2–19
With this, the wave surface profile becomes:
tgw
∂Φ∂
⋅−=1
ζ for: 0=z
Equation 2.2–20
A substitution of Equation 2.2–13 in Equation 2.2–20 yields the wave surface profile:
[ ] ( )txkhkgC
⋅−⋅⋅⋅⋅⋅
= ωωζ coscosh
or:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
24
( )txka ⋅−⋅⋅= ωζζ cos with: [ ]hkgC
a ⋅⋅⋅
= coshωζ
Equation 2.2–21
With this, depending on the water depth h , the wave potential in Equation 2.2–13 willbecome:
( )[ ][ ] ( )txk
hkzhkga
w ⋅−⋅⋅⋅+⋅
⋅⋅
=Φ ωω
ζsin
coshcosh
Equation 2.2–22
or when ω is the first of the sine function arguments, as generally will be used in ship motionequations:
( )[ ][ ] ( )xkt
hkzhkga
w ⋅−⋅⋅⋅+⋅
⋅⋅−
=Φ ωωζ
sincosh
cosh
Equation 2.2–23
In deep water, the expression for the wave potential reduces to:
( )xkteg zka
w ⋅−⋅⋅⋅⋅−
=Φ ⋅ ωω
ζsin (deep water)
Equation 2.2–24
2.2.5 Free Surface Kinematic Boundary Condition
So far the relation between the wave period T and the wavelength, λ , is still unknown. Thisrelation between T and λ (or equivalently ω and k ) follows from the boundary conditionthat the vertical velocity of a water particle in the free surface of the fluid is identical to thevertical velocity of that free surface itself (no-leak condition); this is a kinematic boundarycondition.Using Equation 2.2–21 of the free surface yields:
xu
t
dtdx
xtdtdz
∂∂
⋅+∂∂
=
⋅∂∂
+∂∂
=
ζζ
ζζ
for the wave surface: ζ=z
The second term in this expression is a product of two values, which are both small because ofthe assumed small wave steepness. This product becomes even smaller (second order) and canbe ignored, see Figure 2.2–2.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
25
Figure 2.2–2: Kinematic boundary condition
This linearisation provides the vertical velocity of the wave surface:
tdtdz
∂∂
=ζ
for the wave surface: ζ=z
Equation 2.2–25
The vertical velocity of a water particle in the free surface is then:
tzw
∂∂
=∂Φ∂ ζ
for: ζ=z
The vertical velocity of a water particle in the free surface is then:Analogous to Equation 2.2–19 this condition is valid for 0=z too, instead of for ζ=z only:
tzw
∂∂
=∂Φ∂ ζ
for: 0=z
Equation 2.2–26
A differentiation of the free surface dynamic boundary condition (Equation 2.2–19) withrespect to t provides:
02
2
=∂∂
⋅+∂Φ∂
tg
tw ζ
for: 0=z
or after re-arranging terms:
01
2
2
=∂Φ∂
⋅+∂∂
tgtwζ
for: 0=z
Equation 2.2–27
Together with Equation 2.2–25 this delivers the free surface kinematic boundary condition orthe so-called Cauchy-Poisson condition:
01
2
2
=∂Φ∂
⋅+tgdt
dz w for: 0=z
Equation 2.2–28
2.2.6 Dispersion Relationship
The information is now available to establish the relationship between ω and k (orequivalently T and λ ), referred to above. A substitution of the expression for the wave
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
26
potential (Equation 2.2–22) in Equation 2.2–28 gives the dispersion relation for any arbitrarywater depth h :
[ ]hkgk ⋅⋅⋅= tanh2ω
Equation 2.2–29
In many situations, ω or T will be known; one must determine k or λ . This equation willgenerally has to be solved iteratively, since k appears in a nonlinear way in Equation 2.2–29.
In deep water ( [ ] 1tanh =⋅hk ), Equation 2.2–29 degenerates to a quite simple form which canbe used without difficulty:
gk ⋅=2ω (deep water)
Equation 2.2–30
When calculating the hydromechanical forces and the wave exciting forces on a ship, it isassumed that bxx ≈ , byy ≈ and bzz ≈ . In case of forward ship speed, the wave frequency
ω has to be replaced by the frequency of encounter of the waves eω . This leads to the
following expressions for the wave surface in the ( )bbb zyxG ,, system:
( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbea ykxkt
Equation 2.2–31
and the expression for the velocity potential of the regular waves, wΦ , becomes:( )[ ][ ] ( )µµω
ωζ
sincossincosh
cosh⋅⋅−⋅⋅−⋅⋅
⋅+⋅
⋅⋅−
=Φ bbeba
w ykxkthk
zhkg
Equation 2.2–32
2.2.7 Relationships in Regular Waves
Figure 2.2–3 shows the relation between λ , T , c and h for a wide variety of conditions.Notice the boundaries 2≈hλ and 20≈hλ in this figure between short (deep water) andlong (shallow water) waves.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
27
Figure 2.2–3: Relationships between λ , T , c and h
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
28
2.3 Floating Rigid Body in Waves
Consider a rigid body, floating in an ideal fluid with harmonic waves. The water depth isassumed to be finite. The time-averaged speed of the body is zero in all directions. For thesake of simple notation, it is assumed here that the ( )zyxO ,, system is identical to the
( )000 ,, zyxS system. The x -axis is coincident with the undisturbed still water free surface
and the z -axis and 0z -axis are positive upwards.The linear fluid velocity potential can be split into three parts:
( ) dwrtzyx Φ+Φ+Φ=Φ ,,,
Equation 2.3–1
in which:rΦ radiation potential for the oscillatory motion of the body in still water
wΦ incident undisturbed wave potential
dΦ diffraction potential of the waves about the restrained body
2.3.1 Fluid Requirements
From the definition of a velocity potential Φ follows the velocity of the water particles in thethree translation directions:
xvx ∂
Φ∂=
yv y ∂
Φ∂=
zvz ∂
Φ∂=
Equation 2.3–2
The velocity potentials, dwr Φ+Φ+Φ=Φ , have to fulfil a number of requirements andboundary conditions in the fluid. Of these, the first three are identical to those in the incidentundisturbed waves. Additional boundary conditions are associated with the oscillating floatingbody.
1. Continuity Condition or Laplace EquationAs the fluid is homogeneous and incompressible, the continuity condition:
0=∂∂
+∂
∂+
∂∂
zv
y
v
x
v zyx
Equation 2.3–3
results into the equation of Laplace:
02
2
2
2
2
22 =
∂Φ∂
+∂
Φ∂+
∂Φ∂
=Φ∇zyx
Equation 2.3–4
2. Seabed Boundary ConditionThe boundary condition on the seabed (no-leak condition), following from the definition ofthe velocity potential, is given by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
29
0=∂Φ∂z
for: hz −=
Equation 2.3–5
3. Dynamic Boundary Condition at the Free SurfaceThe pressure in a point ( )zyxP ,, is given by the linearised Bernoulli equation:
zgt
p ⋅⋅−∂Φ∂
⋅−= ρρ or ρp
zgt
−=⋅+
∂Φ∂
Equation 2.3–6
At the free surface of the fluid, so for ( )tzyxz ,,,ζ= , the pressure p is constant.Because of the linearisation, the vertical velocity of a water particle in the free surfacebecomes:
tzdtdz
∂∂
≈∂Φ∂
=ζ
Equation 2.3–7
Combining these two conditions provides the boundary condition at the free surface:
02
2
=∂Φ∂
⋅+∂
Φ∂z
gt
for: 0=z
Equation 2.3–8
4. Kinematic Boundary Condition on the Oscillating Body SurfaceIt is obvious that the boundary condition at the surface of the rigid body plays a veryimportant role. The velocity of a water particle at a point at the surface of the body is equal tothe velocity of this (watertight) body point itself. The outward normal velocity, nv , at a point
( )zyxP ,, at the surface of the body (positive in the direction of the fluid) is given by:
( )tzyxvn n ,,,=
∂Φ∂
Equation 2.3–9
Because the solution is linearised, this can be written as:
( ) ∑=
⋅==∂Φ∂ 6
1
,,,j
jjn fvtzyxvn
Equation 2.3–10
in terms of oscillatory velocities, jv , and generalised direction-cosines, jf , on the surface of
the body, S , given by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
30
( )( )( )
( ) ( )( ) ( )( ) ( ) 126
315
234
3
2
1
,cos,cos
,cos,cos
,cos,cos
,cos
,cos
,cos
fyfxxnyynxf
fxfzznxxnzf
fzfyynzznyf
znf
ynf
xnf
⋅−⋅=⋅−⋅=⋅−⋅=⋅−⋅=
⋅−⋅=⋅−⋅===
=
Equation 2.3–11
The direction cosines are called generalised, because 1f , 2f and 3f have been normalised
(the sum of their squares is equal to 1) and used to obtain 4f , 5f and 6f .Note: The subscripts 1,2,...6 are used here to indicate the mode of the motion. Alsodisplacements are often indicated in literature in the same way: 1x , 2x ,... 6x , as used here inthe summary.
5. Radiation ConditionThe radiation condition states that when the distance R of a water particle to the oscillatingbody tends to infinity, the potential value tends to zero:
0lim =Φ∞→R
Equation 2.3–12
6. Symmetric or Anti-symmetric ConditionSince ships and many floating bodies are symmetric with respect to its middle line plane, onecan make use of this to simplify the potential equations:
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )yxyx
yxyx
yxyx
,, :rollfor
,, :heavefor
,, :swayfor
44
33
22
+Φ−=−Φ
+Φ+=−Φ
+Φ−=−Φ
Equation 2.3–13
in which ( )iΦ is the velocity potential for the given direction i .This indicates that for sway and roll oscillations, the horizontal velocities of the waterparticles, thus the derivative x∂Φ∂ , at any time on both sides of the body must have the samedirection; these motions are anti-symmetric. For heave oscillations these velocities must be ofopposite sign; this is a symmetric motion. However, for all three modes of oscillations thevertical velocities, thus the derivative y∂Φ∂ , on both sides must have the same directions atany time.
2.3.2 Forces and Moments
The forces F and moments M follow from an integration of the pressure, p , over thesubmerged surface, S , of the body:
( )∫∫ ⋅⋅−=S
dSnpF and ( )∫∫ ⋅×⋅−=S
dSnrpM
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
31
Equation 2.3–14
in which n is the outward normal vector on surface dS and r is the position vector ofsurface dS in the ( )zyxO ,, co-ordinate system.The pressure p - via the linearised Bernoulli equation - is determined from the velocitypotentials by:
zgttt
zgt
p
dwr ⋅⋅−
∂Φ∂
+∂Φ∂
+∂Φ∂
⋅−=
⋅⋅−∂Φ∂
⋅−=
ρρ
ρρ
Equation 2.3–15
which can obviously be split into four separate parts, so that the hydromechanical forces Fand moments M can be split into four parts too:
( )∫∫
∫∫
⋅×⋅
⋅+
∂Φ∂
+∂Φ∂
+∂Φ∂
⋅=
⋅⋅
⋅+
∂Φ∂
+∂Φ∂
+∂Φ∂
⋅=
S
dwr
S
dwr
dSnrzgttt
M
dSnzgttt
F
ρ
ρ
Equation 2.3–16
or:
sdwr FFFFF +++= and sdwr MMMMM +++=
Equation 2.3–17
2.3.3 Hydrodynamic Loads
The hydrodynamic loads are the dynamic forces and moments caused by the fluid on anoscillating body in still water; waves are radiated from the body. The radiation potential,
rΦ , which is associated with this oscillation in still water, can be written in terms, jΦ , for 6degrees of freedom as:
( ) ( )
( ) ( )∑
∑
=
=
⋅=
Φ=Φ
6
1
6
1
,,
,,,,,,
jjj
jjr
tvzyx
tzyxtzyx
φ
Equation 2.3–18
in which the space and time dependent potential term, ( )tzyxj ,,.Φ in direction j , is now
written in terms of a separate space dependent potential, ( )zyxj ,,φ in direction j , multiplied
by an oscillatory velocity, ( )tv j in direction j .
This allows the normal velocity on the surface of the body to be written as:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
32
∑
∑
=
=
⋅∂
∂=
Φ∂∂
=∂Φ∂
6
1
6
1
jj
j
jj
r
vn
nn
φ
Equation 2.3–19
and the generalised direction cosines are given by:
nf j
j ∂
∂=
φ
Equation 2.3–20
With this the radiation terms in the hydrodynamic force and moment becomes:
( )
( )∫∫ ∑
∫∫
∫∫ ∑
∫∫
⋅×⋅
⋅
∂∂
=
⋅×⋅
∂Φ∂
=
⋅⋅
⋅
∂∂
=
⋅⋅
∂Φ∂
=
=
=
S jjj
S
r
S jjj
S
r
dSnrvt
dSnrt
M
dSnvt
dSnt
F
6
1
6
1
φρ
ρ
φρ
ρ
Equation 2.3–21
The components of these radiation forces and moments are defined by:
( )321 ,, rrrr XXXF = and ( )654 ,, rrrr XXXM =with:
∫∫ ∑
∫∫ ∑
⋅∂
⋅
⋅
∂∂
=
⋅⋅
⋅
∂∂
=
=
=
S
k
jjj
S
kj
jjrk
dSdn
vt
dSfvt
X
φφρ
φρ
6
1
6
1 for: 6,...1=k
Equation 2.3–22
Since jφ and kφ are not time-dependent in this expression, it reduces to:
∑=
=6
1jrkjrk XX for: 6,...1=k
with:
∫∫ ⋅∂∂
⋅⋅=S
kj
jrkj dS
ndt
dvX
φφρ
Equation 2.3–23
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
33
This radiation force or moment rkjX in the direction k is caused by a forced harmonicoscillation of the body in the direction j . This is generally true for all j and k in the rangefrom 1 to 6. When kj = , the force or moment is caused by a motion in that same direction.When kj ≠ , the force in one direction results from the motion in another direction. Thisintroduces what is called coupling between the forces and moments (or motions).Equation 2.3–23 expresses the force and moment components, rkjX , in terms of still unknown
potentials, jφ . But not everything is solved yet, a solution for this will be found later in this
Chapter.
2.3.3.1 Oscillatory Motion
Now an oscillatory motion is defined; suppose a motion (in a complex notation) given by:ti
ajj ess ω−⋅=
Equation 2.3–24
Then the velocity and acceleration of this oscillation are:
tiaj
jj
tiajjj
esdt
dvs
esivs
⋅⋅−
⋅⋅−
⋅⋅−==
⋅⋅⋅−==
ω
ω
ω
ω
2&&
&
Equation 2.3–25
The hydrodynamic forces and moments can be split into a load in-phase with the accelerationand a load in-phase with the velocity:
( )ti
S
kjaj
tikjajkjaj
jkjjkjrkj
edSn
s
eNsiMs
sNsMX
⋅⋅−
⋅⋅−
⋅
⋅
∂∂
⋅⋅⋅−=
⋅⋅⋅⋅+⋅⋅=
⋅−⋅−=
∫∫ ω
ω
φφρω
ωω
2
2
&&&
Equation 2.3–26
So in case of an oscillation of the body in the direction j with a velocity potential jφ , thehydrodynamic mass and damping (coupling) coefficients are defined by:
⋅∂∂
⋅−= ∫∫S
kjkj dS
nM
φφρRe and
⋅∂
∂⋅⋅−= ∫∫
S
kjkj dS
nN
φφωρIm
Equation 2.3–27
In case of an oscillation of the body in the direction k with a velocity potential kφ , thehydrodynamic mass and damping (coupling) coefficients are defined by:
⋅∂
∂⋅−= ∫∫
S
jkjk dS
nM
φφρRe and
⋅∂
∂⋅⋅−= ∫∫
S
jkjk dS
nN
φφωρIm
Equation 2.3–28
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
34
2.3.3.2 Green's Second Theorem
Green's second theorem transforms a large volume-integral into a much easier to handlesurface-integral. Its mathematical background is beyond the scope of this text. It is valid forany potential function, regardless the fact if it fulfils the Laplace condition or not.
Consider two separate velocity potentials jφ and kφ . Green's second theorem, applied tothese potentials, is then:
( ) ∫∫∫∫∫ ⋅
∂
∂⋅−
∂∂
⋅=⋅∇⋅−∇⋅**
**22
S
jk
kj
V
jkkj dSnn
dVφ
φφ
φφφφφ
Equation 2.3–29
As said before, this theorem is generally valid for all kinds of potentials; it is not necessarythat they fulfil the Laplace equation. In Green's theorem, *S is a closed surface with a volume
*V . This volume is bounded by the wall of an imaginary vertical circular cylinder with a verylarge radius R , the seabed at hz −= , the water surface at ζ=z and the wetted surface of thefloating body, S ; see Figure 2.3–1.
Figure 2.3–1: Boundary conditions
Both of the above radiation potentials jφ and kφ must fulfil 022 =∇=∇ kj φφ , the Laplace
equation. So the left-hand side of Equation 2.3–29 becomes zero which yields for the right-hand side of this equation:
**
**
dSn
dSn
S
jk
S
kj ⋅
∂
∂⋅=⋅
∂∂
⋅ ∫∫∫∫φ
φφ
φ
Equation 2.3–30
The boundary condition at the free surface becomes for tie ⋅⋅−⋅=Φ ωφ :
02 =∂∂
⋅+⋅−z
gφφω for: 0=z
Equation 2.3–31
or with the dispersion relation, [ ]hkgk ⋅⋅⋅= tanh2ω :
[ ]z
hkk∂∂
=⋅⋅⋅φφtanh for: 0=z
Equation 2.3–32
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
35
This implies that at the free surface of the fluid one can write:
[ ] [ ][ ] [ ]
∂∂
⋅⋅⋅
=→∂∂
=∂∂
=⋅⋅⋅
∂∂
⋅⋅⋅
=→∂∂
=∂∂
=⋅⋅⋅
nhkknzhkk
nhkknzhkk
jj
jjj
kk
kkk
φφ
φφφ
φφφφφ
tanh1tanh
tanh1
tanh
at the free surface
Equation 2.3–33
When taking also the boundary condition at the seabed and the radiation condition on the wallof the cylinder in Figure 2.3–1:
0=∂∂
nφ
for: hz −= and 0lim =∞→φ
R
Equation 2.3–34
into account, the integral equation over the surface *S reduces to:
∫∫∫∫ ⋅∂
∂⋅=⋅
∂∂
⋅S
jk
S
kj dS
ndS
n
φφ
φφ
Equation 2.3–35
in which S is the wetted surface of the oscillating body only.Notice that jφ and kφ still have to be evaluated.
2.3.3.3 Potential Coefficients
The previous subsection provides - for the zero forward ship speed case - symmetry in thecoefficients matrices with respect to their diagonals so that:
kjjk MM = and kjjk NN =
Equation 2.3–36
Because of the symmetry of a ship, some coefficients are zero and the two matrices withhydrodynamic coefficients for a ship become:
666462
555351
464442
353331
262422
151311
666462
555351
464442
353331
262422
151311
000
000
000
000
000
000
:matrix damping icHydrodynam
000
000
000
000
000
000
:matrix mass icHydrodynam
NNN
NNN
NNN
NNN
NNN
NNN
MMM
MMM
MMM
MMM
MMM
MMM
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
36
Equation 2.3–37
For clarity, the symmetry of terms about the diagonal in these matrices (for example that
3113 MM = for zero forward speed) has not been included here. The terms on the diagonals
( nnM ) are the primary coefficients relating properties such as hydrodynamic mass in one
direction to the inertia forces in that same direction. Off-diagonal terms (such as 13M )represent hydrodynamic mass only, which is associated with an inertia dependent force in onedirection caused by a motion component in another.
Forward speed has an effect on the velocity potentials itself, but is not discussed in thisSection. This effect is quite completely explained by Timman and Newman [1962].
2.3.4 Wave and Diffraction Loads
The wave and diffraction terms in the hydrodynamic force and moment are:
( )∫∫
∫∫
⋅×⋅
∂Φ∂
+∂Φ∂
=+
⋅⋅
∂Φ∂
+∂Φ∂
=+
S
dwdw
S
dwdw
dSnrtt
MM
dSntt
FF
ρ
ρ
Equation 2.3–38
The principle of linear superposition allows the determination of these forces on a restrainedbody with zero forward speed: 0=∂Φ∂ n . This simplifies the boundary condition on thesurface of the body to:
0=∂Φ∂
+∂Φ∂
=∂Φ∂
nnndw
Equation 2.3–39
The space and time dependent potentials, ( )tzyxw ,,,Φ and ( )tzyxd ,,,Φ , are written now in
terms of isolated space dependent potentials, ( )zyxw ,,φ and ( )zyxd ,,φ , multiplied by a
normalised oscillatory velocity, ( ) tietv ⋅⋅−⋅= ω1 :
( ) ( )( ) ( ) ti
dd
tiww
ezyxtzyx
ezyxtzyx⋅⋅−
⋅⋅−
⋅=Φ
⋅=Φω
ω
φ
φ
,,,,,
,,,,,
Equation 2.3–40
This results into:
nndw
∂∂
−=∂
∂ φφ
Equation 2.3–41
With this and the expressions for the generalised direction-cosines it is found for the waveforces and moments on the restrained body in waves:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
37
( )
( )∫∫
∫∫
⋅∂∂
⋅+⋅⋅−=
⋅⋅+⋅⋅−=
⋅⋅−
⋅⋅−
S
kdw
ti
S
kdwti
wk
dSn
ei
dSfeiX
φφφρ
φφρ
ω
ω
for: 6,...1=k
Equation 2.3–42
in which kφ is the radiation potential.
The potential of the incident waves, wφ , is known, but the diffraction potential, dφ , has to be
determined. Green's second theorem provides a relation between this diffraction potential, dφ ,
and a radiation potential, kφ :
∫∫∫∫ ⋅∂∂
⋅=⋅∂∂
⋅S
dk
S
kd dS
ndS
nφ
φφ
φ
Equation 2.3–43
and with nn dw ∂∂−=∂∂ φφ from Equation 2.3–41 one finds:
∫∫∫∫ ⋅∂∂
⋅−=⋅∂∂
⋅S
wk
S
kd dS
ndS
nφ
φφ
φ
Equation 2.3–44
This elimination of the diffraction potential results into the so-called Haskind relations:
∫∫ ⋅
∂∂
⋅+∂
∂⋅⋅⋅−= ⋅⋅−
S
wk
kw
tiwk dS
nneiX
φφ
φφρ ω for: 6,...1=k
Equation 2.3–45
This limits the problem of the diffraction potential because the expression for wkX depends
only on the undisturbed wave potential wφ and the radiation potential kφ .These relations, found by Haskind [1957], are very important; they underlie the relativemotion (displacement - velocity - acceleration) hypothesis, as used in strip theory. Theserelations are valid only for a floating body with a zero time-averaged speed in all directions.Newman [1962] however, has generalised the Haskind relations for a body with a constantforward speed. He derived equations, which differ only slightly from those found by Haskind.According to Newman's approach the wave potential has to be defined in the moving
( )zyxO ,, system. The radiation potential has to be determined for the constant forward speedcase, taking an opposite sign into account.
The corresponding wave potential for deep water - as given in a previous section - nowbecomes:
( )
( ) tiyxkizka
zkaw
eeegi
ykxkteg
⋅⋅−⋅+⋅⋅⋅
⋅
⋅⋅⋅⋅⋅−
=
⋅⋅−⋅⋅−⋅⋅⋅⋅−
=Φ
ωµµ
ωζ
µµωωζ
sincos
sincossin
Equation 2.3–46
so that the isolated space dependent term is given by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
38
( )µµ
ωζφ sincos ⋅+⋅⋅⋅ ⋅⋅
⋅⋅−= yxkizka
w eegi
Equation 2.3–47
In these equations is µ the wave direction, defined as given in Figure 2.1–2.The velocity of the water particles in the direction of the outward normal n on the surface ofthe body is:
( ) µµφ
µµφφ
sincos
sincos
213 ⋅+⋅⋅+⋅⋅=
⋅
∂∂
+⋅∂∂
⋅+∂∂
⋅⋅=∂∂
ffifk
ny
nx
inz
kn
w
ww
Equation 2.3–48
With this, the wave loads are given by:
( ) ∫∫
∫∫⋅⋅+⋅⋅+⋅⋅⋅⋅⋅⋅+
⋅⋅⋅⋅⋅−=
⋅⋅−
⋅⋅−
dSffifkei
dSfeiX
kwti
Skw
tiwk
µµφφρ
φρ
ω
ω
sincos 213
for: 6,...1=k
Equation 2.3–49
The first term in this expression for the wave forces and moments is the so-called Froude-Krilov force or moment, which is the wave load caused by the undisturbed incident wave. Thesecond term is related to the disturbance caused by the presence of the (restrained) body.
2.3.5 Hydrostatic Loads
In the notations used here, the buoyancy forces and moments are:
∫∫ ⋅⋅⋅=S
s dSnzgF ρ and ( )∫∫ ⋅×⋅⋅=S
s dSnrzgM ρ
Equation 2.3–50
or more generally:
∫∫ ⋅⋅⋅=S
ksk dSfzgX ρ for: 6,...1=k
Equation 2.3–51
in which the skX are the components of these hydrostatic forces and moments.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
39
2.4 Equations of Motion
The equations of motion are given here in a - with the ship speed V steadily moving - right-handed co-ordinate system ( )zyxG ,, , with the origin in the average position of the ship’scentre of gravity G .The total mass as well as its distribution over the body is considered to be constant with time.For ships and other floating structures, this assumption is normally valid during a time that islarge relative to the period of the motions. This holds that small effects - such as for instance adecreasing mass due to fuel consumption - can be ignored.The solid mass matrix of a floating structure is given below.
−
−∇⋅
∇⋅
∇⋅
=
zzzx
yy
xzxx
II
I
IIm
0000
00000
0000
00000
00000
00000
:matrix mass Solidρ
ρρ
Equation 2.4–1
The moments of inertia here are often expressed in terms of the radii of inertia and the solidmass of the structure. Since Archimedes’ law ( ∇⋅= ρm ) is valid for a free floating structure:
∇⋅⋅=
∇⋅⋅=
∇⋅⋅=
ρ
ρ
ρ
2
2
2
zzzz
yyyy
xxxx
kI
kI
kI
Equation 2.4–2
When the actual distribution of the solid mass of a ship is unknown, the radii of inertia can beapproximated by:
⋅⋅≈
⋅⋅≈
⋅⋅≈
LL
LL
BB
28.0 to22.0k
28.0 to22.0k
40.0 to30.0k
:shipsfor
zz
yy
xx
Equation 2.4–3
in which L is the length and B is the breadth of the ship.Often, the (generally small) coupling terms, zxxz II = , are simply neglected.Bureau Veritas proposes for the radius of inertia for roll of the ship's solid mass:
⋅+⋅⋅≈
22
0.1289.0BKG
Bk xx
Equation 2.4–4
in which KG is the height of the centre of gravity, G , above the keel.For many ships without cargo on board (ballast condition), the mass is concentrated at theends (engine room aft and ballast water forward to avoid a large trim), while for ships withcargo on board (full load condition) the - more or less amidships laden - cargo plays an
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
40
important role. Thus for normal ships, the radii of inertia, yyk and zzk , are usually smaller inthe full load condition than in the ballast condition.Notice here that the longitudinal radius of gyration of a long homogeneous rectangular beamwith a length L is equal to about L⋅121 or L⋅289.0 .
The equations of motions of a rigid body in a space fixed co-ordinate system follow fromNewton's second law. The vector equations for the translations of and the rotations about thecentre of gravity are given respectively by:
( )Umdtd
F ⋅= and ( )Hdtd
M =
Equation 2.4–5
in which:F resulting external force acting in the centre of gravitym mass of the rigid body
U instantaneous velocity of the centre of gravity
M resulting external moment acting about the centre of gravity
H instantaneous angular momentum about the centre of gravityt time
Two important assumptions are made for the loads in the right-hand side of these equations:
a) The so-called hydromechanic forces and moments are induced by the harmonicoscillations of the rigid body, moving in the undisturbed surface of the fluid.
b) The so-called wave exciting forces and moments are produced by waves coming in on therestrained body.
Since the system is linear, these loads are added up for obtaining the total loads. Thus, afterassuming small motions, symmetry of the body and that the x -, y - and z -axes are principalaxes, one can write the following six equations of motion for the ship:
( )
( )
( )
( )( )( ) 66
55
44
33
22
11
Yaw
Pitch
:Roll
:Heave
:Sway
:Surge
whzxzzzxzz
whxxyy
whxzxxxzxx
wh
wh
wh
XXIIIIdtd
XXIIdtd
XXIIIIdtd
XXzzdtd
XXyydtd
XXxxdtd
+=⋅−⋅=⋅−⋅
+=⋅=⋅
+=⋅−⋅=⋅−⋅
+=⋅∇⋅=⋅∇⋅
+=⋅∇⋅=⋅∇⋅
+=⋅∇⋅=⋅∇⋅
φψφψ
θθ
ψφψφ
ρρ
ρρ
ρρ
&&&&&&
&&&
&&&&&&
&&&
&&&
&&&
Equation 2.4–6
in which:ρ density of water
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
41
∇ volume of displacement of the ship
ijI solid mass moment of inertia of the ship
321 ,, hhh XXX hydromechanic forces in the x -, y - and z -directions
654 ,, hhh XXX hydromechanic moments about the x -, y - and z -axes
321 ,, www XXX exciting wave forces in the x -, y - and z -directions
654 ,, www XXX exciting wave moments about the x -, y - and z -axes
Generally, a ship has a vertical-longitudinal plane of symmetry, so that its motions can be splitinto symmetric and anti-symmetric components. Surge, heave and pitch motions aresymmetric motions, that is to say that a point to starboard has the same motion as the mirroredpoint to port side. It is obvious that the remaining motions sway, roll and yaw are anti-symmetric motions. Symmetric and anti-symmetric motions of a free-floating structure are notcoupled; they don't have any effect on each other. For instance, a vertical force acting at thecentre of gravity can cause surge, heave and pitch motions, but will not result in sway, roll oryaw motions.
Because of this symmetry and anti-symmetry, two sets of three coupled equations of motioncan be distinguished for ships:
motions symmetric-anti
:Yaw
:Roll
:Sway
motions symmetric
:Pitch
:Heave
:Surge
66
44
22
55
33
11
=−⋅−⋅
=−⋅−⋅
=−⋅∇⋅
=−⋅
=−⋅∇⋅
=−⋅∇⋅
whzxzz
whxzxx
wh
whxx
wh
wh
XXII
XXII
XXy
XXI
XXz
XXx
φψ
ψφ
ρ
θ
ρρ
&&&&
&&&&
&&
&&
&&
&&
Equation 2.4–7
Note that this distinction between symmetric and anti-symmetric motions disappears when theship is anchored. Then, for instance, the pitch motions can generate roll motions via theanchor lines.
The coupled surge, heave and pitch equations of symmetric motion are:
( )
( )
( ) (pitch)
(heave)
(surge)
5555555
535353
515151
3353535
333333
313131
1151515
131313
111111
wyy
w
w
XcbaI
zczbza
xcxbxa
Xcba
zczbza
xcxbxa
Xcba
zczbza
xcxbxa
=⋅+⋅+⋅++⋅+⋅+⋅+⋅+⋅+⋅
=⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+⋅+⋅+⋅
=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅
θθθ
θθθρ
θθθ
ρ
&&&&&&
&&&
&&&&&&
&&&
&&&&&&
&&&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
42
Equation 2.4–8
The coupled sway, roll and yaw equations of anti-symmetric motion are:
( )
( )( )
( )( ) (yaw)
(roll)
(sway)
6666666
646464
626262
4464646
444444
424242
2262626
242424
222222
wzz
zx
wxz
xx
w
XcbaI
cbaI
ycybya
XcbaI
cbaI
ycybya
Xcba
cba
ycybya
=⋅+⋅+⋅++⋅+⋅+⋅+−+⋅+⋅+⋅
=⋅+⋅+⋅+−+⋅+⋅+⋅++⋅+⋅+⋅
=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅
ψψψφφφ
ψψψφφφ
ψψψφφφ
ρ
&&&
&&&
&&&
&&&
&&&
&&&
&&&
&&&
&&&
Equation 2.4–9
In many applications, zxxz II = is not known or small; hence their terms are often omitted. Inprogram SEAWAY they have been introduced in the equations of motion if they can becalculated from an input of the mass distribution along the ship’s length, only.
After the determination of the in and out of phase terms of the hydromechanic and the waveloads, these equations can be solved with a numerical method.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
43
2.5 Strip Theory Approaches
Strip theory is a computational method by which the forces on and motions of a three-dimensional floating body can be determined using results from two-dimensional potentialtheory. Strip theory considers a ship to be made up of a finite number of transverse two-dimensional slices, which are rigidly connected to each other. Each of these slices will have aform that closely resembles the segment of the ship that it represents. Each slice is treatedhydrodynamically as if it is a segment of an infinitely long floating cylinder; see Figure 2.5–1.
Figure 2.5–1: Strip theory representation by cross sections
This means that all waves which are produced by the oscillating ship (hydromechanic loads)and the diffracted waves (wave loads) are assumed to travel perpendicular to the middle lineplane - thus parallel to the ( )zy, -plane - of the ship. This holds too that the strip theorysupposes that the fore and aft side of the body (such as a pontoon) does not produce waves inthe x -direction. For the zero forward speed case, interactions between the cross sections areignored as well.Fundamentally, strip theory is valid for long and slender bodies only. In spite of thisrestriction, experiments have shown that strip theory can be applied successfully for floatingbodies with a length to breadth ratio larger than three, 3≥BL , at least from a practical pointof view.
2.5.1 Zero Forward Ship Speed
When applying the strip theory, the loads on the body are found by an integration of the 2-Dloads:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
44
∫∫∫∫
∫∫∫∫∫∫∫∫
⋅⋅+=⋅⋅+=
⋅⋅−=⋅⋅−=
⋅=⋅=
⋅=⋅=
⋅=⋅=
⋅=⋅=
Lbbww
Lbbhh
Lbbww
Lbbhh
Lbww
Lbhh
Lbww
Lbhh
Lbww
Lbhh
Lbww
Lbhh
dxxXXdxxXX
dxxXXdxxXX
dxXXdxXX
dxXXdxXX
dxXXdxXX
dxXXdxXX
'26
'26
'35
'35
'44
'44
'33
'33
'22
'22
'11
'11
:Yaw
:Pitch
:Roll
:Heave
:Sway
:Surge
Equation 2.5–1
in which:'
hjX sectional hydromechanic force or moment in direction j per unit ship length'
wjX sectional exciting wave force or moment in direction j per unit ship length
The appearance of two-dimensional surge forces seems strange here. It is strange! A more orless empirical method has been used in SEAWAY for the surge motion, by defining anequivalent longitudinal cross section that is swaying. Then, the 2-D hydrodynamic swaycoefficients of this equivalent cross section are translated to 2-D hydrodynamic surgecoefficients by an empirical method based on theoretical results from three-dimensionalcalculations and these coefficients are used to determine 2-D loads. In this way, all sets of sixsurge loads can be treated in the same numerical way in SEAWAY for the determination of the3-D loads. Inaccuracies of the hydromechanic coefficients for surge of (slender) ships are ofminor importance, because these coefficients are relatively small.
Notice how in the strip theory the pitch and yaw moments are derived from the 2-D heave andsway forces, respectively, while the roll moments are obtained directly.
The equations of motions are defined in the moving axis system with the origin at the time-averaged position of the centre of gravity, G . All two-dimensional potential coefficients havebeen defined here in an axis system with the origin, O , in the water plane; the hydromechanicand exciting wave moments have to be corrected for the distance OG .
2.5.2 Forward Ship Speed
Relative to an oscillating ship moving forward in the undisturbed surface of the fluid, thedisplacements, *
hjζ , velocities, *hjζ& , and accelerations, *
hjζ&& , at forward ship speed V in oneof the 6 directions j of a water particle in a cross section are defined by:
*hjζ **
hjhj DtD ζζ =& **
hjhj DtD ζζ &&& =
Equation 2.5–2
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
45
∂∂
⋅−∂∂
=x
VtDt
D
Equation 2.5–3
is a mathematical operator which transforms the potentials ( )tzyx ,,, 000Φ , defined in the
earth bounded (fixed) co-ordinate system, to the potentials ( )tzyx ,,,Φ , defined in the ship'ssteadily translating co-ordinate system with speed V .
In waves the motions of the water particles are depending on its local vertical distance to themean or still water surface. At each cross section of the ship an average (or equivalent)constant value has to be found.Relative to a restrained ship, moving forward with speed V in waves, the equivalent j
constant components of water particle displacements ( *wjζ ), velocities ( *
wjζ& ) and
accelerations ( *wjζ&& ) in a cross section are defined in a similar way by:
*wjζ **
wjwj DtD ζζ =& **
wjwj DtD ζζ &&& =
Equation 2.5–4
The effect of the operator in Equation 2.5–3 can be understood easily when one realises that inthat earth-bound co-ordinate system the sailing ship penetrates through a ''virtual verticaldisk''. For instance, when a ship sails with speed V and constant trim angle θ through stillwater, the relative vertical velocity of a water particle with respect to the bottom of the sailingship becomes θθ ⋅≈⋅ VV sin .
Two different types of strip theory methods (as has been used in SEAWAY) are discussedhere:
1. Ordinary Strip Theory MethodAccording to this classic method, the uncoupled two-dimensional potentialhydromechanic loads and wave loads in an arbitrary direction j are defined by:
'*'*'*
'*'*'*
fkjwjjjwjjjwj
rsjhjjjhjjjhj
XNMDtD
X
XNMDtD
X
+⋅+⋅=
+⋅+⋅=
ζζ
ζζ
&&
&&
Equation 2.5–5
This is the first formulation of the strip theory that can be found in the literature. Itcontains a more or less intuitive approach to the forward speed problem, as published indetail by Korvin-Kroukovski and Jacobs [1957].
2. Modified Strip Theory MethodAccording to this modified method, these loads become:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
46
'*''*
'*''*
fkjwjjje
jjwj
rsjhjjje
jjhj
XNi
MDtD
X
XNi
MDtD
X
+
⋅
⋅−=
+
⋅
⋅−=
ζω
ζω
&
&
Equation 2.5–6
This formulation is a more fundamental approach of the forward speed problem, aspublished in detail by Tasai [1969] and many others.
In the equations above, 'jjM and '
jjN are the 2-D potential mass and damping coefficients.'
rsjX is the two-dimensional quasi-static restoring spring term, as generally present for heave,
roll and pitch only. 'fkjX is the two-dimensional Froude-Krilov force or moment which is
calculated by integration of the directional pressure gradient in the undisturbed wave over thecross sectional area of the hull.Equivalent directional components of the orbital acceleration and velocity - derived fromthese Froude-Krilov loads - are used here to calculate the diffraction parts of the total waveforces and moments.From a theoretical point of view, one should prefer the use of the modified strip theorymethod. However, it appeared from the authors’ experiences that for ships with moderateforward speed ( 30.0≤Fn ) the ordinary method could provide in some cases a better fit withexperimental data.
2.5.3 End-Terms
From the previous, it is obvious that in the equations of motion longitudinal derivatives of thetwo-dimensional potential mass '
jjM and damping 'jjN will appear. These derivatives have to
be determined numerically over the whole ship length in such a manner that the followingrelation is fulfilled:
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )( )
( )
( ) ( )( )
( )
( )
0
00
0
0
00
=
−⋅+=
⋅+⋅+⋅=⋅
∫
∫∫∫∫+
−
+
−
Lfdxdx
xdff
dxdx
xdfdx
dxxdf
dxdx
xdfdx
dxxdf
b
Lx
x b
b
b
Lx
Lx b
bb
Lx
x b
bb
x
x b
bb
Lx
x b
b
b
b
b
b
b
b
b
b
b
b
ε
ε
ε
ε
Equation 2.5–7
with L<<ε , while ( )bxf is equal to the local values of ( )bjj xN ' or ( )bjj xM ' ; see Figure 2.5–2.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
47
Figure 2.5–2: Integration of derivatives
The numerical integration of the derivatives will be carried out in the region( ) ( )Lxxx bbb ≤≤0 only. So, the additional so-called ''end terms'' are defined by ( )0f and
( )Lf .Because the integration of the derivatives should be carried out in the region just behind untiljust before the ship, so ( ) ( ) εε +≤≤− Lxxx bbb 0 , some can algebra provide the integral andthe first and second order moments (with respect to G ) over the whole ship length (slenderbody assumption):
( )( )
( )
( )( )
( )
( )( )
( )
( )( )
( )
( )( )
( )
bb
Lx
xbbb
Lx
x b
b
b
Lx
xbbb
Lx
x b
b
b
Lx
x b
b
dxxxfdxxdx
xdf
dxxfdxxdx
xdf
dxdx
xdf
b
b
b
b
b
b
b
b
b
b
⋅⋅⋅−=⋅⋅
⋅−=⋅⋅
=⋅
∫∫
∫∫
∫
+
−
+
−
+
−
0
2
0
00
0
2
0
ε
ε
ε
ε
ε
ε
Equation 2.5–8
Notice that these expressions are valid for the integration of the potential coefficients over thefull ship length only. They can not be used for calculating local hydromechanic loads. Also forthe wave loads, these expressions can not be used, because there these derivatives aremultiplied with equivalent bx -depending orbital motion amplitudes.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
48
2.6 Hydrodynamic Coefficients
In strip theory, the two-dimensional hydrodynamic sway, heave and roll coefficients can becalculated by several methods:
1. Methods based on Ursell's Theory and Conformal MappingUrsell [1949] derived an analytical solution for solving the problem of calculating thehydrodynamic coefficients of an oscillating circular cylinder in the surface of a fluid:
a) Deep Water Coefficients with Lewis Conformal MappingTasai [1959], Tasai [1961] and many others added the so-called Lewis transformation -which is a very simple and in a lot of cases also more or less realistic method totransform ship-like cross sections to this unit circle - to Ursell's solution. Thistransformation will be carried out using a scale factor and two mapping coefficients.Only the breadth, the draft and the area of the mapped cross section will be similar tothat of the actual cross section.
b) Deep Water Coefficients with Close-Fit Conformal MappingA more accurate way of mapping has been added by Tasai [1960] and others too, byusing more than only two mapping coefficients. The accuracy obtained depends on thenumber of mapping coefficients. Generally, a maximum number of 10 coefficients areused for defining the cross section. These coefficients are determined in such a waythat the Root Mean Square of the differences between the offsets of the mapped andthe actual cross section is minimal.
c) Shallow Water Coefficients with Lewis Conformal MappingFor shallow water, the theory of Keil [1974] - based on an expansion of Ursell'spotential theory for circular cylinders at deep water to shallow water - and Lewisconformal mapping can be used.
2. Frank's Pulsating Source Theory for Deep WaterMapping methods require an intersection of the cross section with the water plane and, asa consequence of this, they are not suitable for submerged cross sections, like at a bulbousbow. Also, conformal mapping can fail for cross sections with very low sectional areacoefficients, such as are sometimes present in the aft body of a ship.Frank [1967] considered a cylinder of constant cross sections with an arbitrarilysymmetrical shape, of which the cross sections are simply a region of connected lineelements. This vertical cross section can be fully or partly immersed in a previouslyundisturbed fluid of infinite depth. He developed an integral equation method utilising theGreen's function, which represents a complex potential of a pulsating point source of unitstrength at the midpoint of each line element. Wave systems were defined in such a waythat all required boundary conditions were fulfilled. The linearised Bernoulli equationprovides the pressures after which the potential coefficients were obtained from the in-phase and out-of-phase components of the resultant hydrodynamic loads.
The 2-D potential pitch and yaw (moment) coefficients follow from the previous heave andsway coefficients and the lever, i.e., the distance of the cross section to the centre of gravityG .
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
49
A more or less empirical procedure has followed by the author for the surge motion. Anequivalent longitudinal cross section has been defined. For each frequency, the two-dimensional potential hydrodynamic sway coefficient of this equivalent cross section istranslated to two-dimensional potential hydrodynamic surge coefficients, by an empiricalmethod based on theoretical results of three-dimensional calculations.
The 3-D coefficients follow from an integration of these 2-D coefficients over the ship'slength. Viscous terms have been be added for surge and roll.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
50
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
51
3 2-D Potential Coefficients
This Chapter described the various methods, used in the SEAWAY computer code, to obtainthe 2-D potential coefficients:• the theory of Tasai for deep water, based on Ursell's potential theory for circular cylinders
and Lewis and N-parameter conformal mapping• the theory of Keil for very shallow to deep water, based on a variation of Ursell's potential
theory for circular cylinders and Lewis conformal mapping• the theory of Frank for deep water, using pulsating sources on the cross sectional contour.
During the ship motions calculations different co-ordinate systems, as shown before, will beused. The two-dimensional hydrodynamic potential coefficients have been defined here withrespect to the ( )zyxO ,, co-ordinate system for the moving ship in still water.However, in this section deviating axes systems are used for the determination of the two-dimensional hydrodynamic potential coefficients for sway, heave and roll motions. This holdsfor the sway and roll coupling coefficients a change of sign. The signs of the uncoupled sway,heave and roll coefficients do not change.
For each cross section, the following two-dimensional hydrodynamic coefficients have to beobtained:• '
22M and '22N 2-D potential mass and damping coefficients of sway
• '24M and '
24N 2-D potential mass and damping coupling coefficients of roll into sway
• '33M and '
33N 2-D potential mass and damping coefficients of heave
• '44M and '
44N 2-D potential mass and damping coefficients of roll
• '
42M and '42N 2-D potential mass and damping coupling coefficients of sway into roll
The 2-D potential pitch and yaw (moment) coefficients, '55M , '
55N , '66M and '
66N , followfrom the previous heave and sway coefficients and the lever of the loads, i.e., the distance ofthe cross section to the centre of gravity G .
Finally, an approximation is given for the determination of the 2-D potential surge coefficients'
11M and '11N .
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
52
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
53
3.1 Conformal Mapping Methods
Ursell's derivation of potential coefficients is valid for semicircular cross sections only. Forthe determination of the two-dimensional added mass and damping in the sway, heave and rollmode of the motions of ship-like cross sections by Ursell's method, the cross sections have tobe mapped conformally to the unit semicircle. The advantage of conformal mapping is that thevelocity potential of the fluid around an arbitrarily shape of a cross section in a complex planecan be derived from the more convenient semicircular section in another complex plane. Inthis manner hydrodynamic problems can be solved directly with the coefficients of themapping function.The general transformation formula – see also Figure 3.1–1 - is given by:
( ) ∑ −−− ⋅⋅= 12
12n
ns aMz ζ
Equation 3.1–1
with:iyxz += plane of the ship's cross section
θαζ ieie −⋅= plane of the unit circle
sM scale factor
1−a 1+=
12 −na conformal mapping coefficients ( Nn ,...1= )N Maximum parameter index number
Figure 3.1–1: Mapping relation between two planes
From this follows the relation between the co-ordinates in the z -plane (the ship's crosssection) and the variables in the ζ -plane (the circular cross section):
( ) ( ) ( )( )
( ) ( ) ( )( ) ∑
∑
=
⋅−−−
=
⋅−−−
⋅−⋅⋅⋅−⋅+=
⋅−⋅⋅⋅−⋅−=
N
n
ann
ns
N
n
ann
ns
neaMy
neaMx
0
1212
0
1212
12cos1
12sin1
θ
θ
Equation 3.1–2
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
54
The contour of the - by conformal mapping approximated - ship's cross section follows fromputting 0=α in the previous relations in Equation 3.1–2:
( ) ( )( )
( ) ( )( ) ∑
∑
=−
=−
⋅−⋅⋅−⋅+=
⋅−⋅⋅−⋅−=
N
nn
ns
N
nn
ns
naMy
naMx
0120
0120
12cos1
12sin1
θ
θ
Equation 3.1–3
The breadth on the waterline of the approximated ship's cross section is defined by:
asMb λ⋅⋅= 20 with: ∑=
−=N
nna a
012λ and
as
bM
λ⋅=
20
Equation 3.1–4
The draught is defined by:
bsMd λ⋅=0 with: ( ) ∑=
−⋅−=N
nn
nb a
0121λ
Equation 3.1–5
3.1.1 Lewis Conformal Mapping
A very simple and in a lot of cases also a more or less realistic transformation of the crosssectional hull form will be obtained with 2=N in the transformation formula, the wellknown Lewis transformation, see reference Lewis [1929]. An extended and clear descriptionof the representation of ship hull forms by this Lewis two-parameter conformal mapping isgiven by von Kerczek and Tuck [1969].The two-parameter Lewis transformation of a cross section is defined by:
( )33
111
−−− ⋅+⋅+⋅⋅= ζζζ aaaMz s
Equation 3.1–6
In here 11 +=−a and the conformal mapping coefficients 1a and 3a are called Lewis
coefficients, while sM is the scale factor.Then:
( )( )θθθ
θθθααα
ααα
3coscoscos
3sinsinsin3
31
331
⋅⋅+⋅⋅−⋅⋅=
⋅⋅−⋅⋅+⋅⋅=−−
−−
eaeaeMy
eaeaeMx
s
s
Equation 3.1–7
By putting 0=α is the contour of this so-called Lewis form expressed as:( )( )( )( )θθ
θθ3coscos1
3sinsin1
310
310
⋅+⋅−⋅=⋅−⋅+⋅=
aaMy
aaMx
s
s
with scale factor:
3131 112
aaD
aaB
M sss +−
=++
=
Equation 3.1–8
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
55
in which:
sB sectional breadth on the water line
sD sectional draught
Now the coefficients 1a and 3a and the scale factor sM will be determined in such a mannerthat the sectional breadth, draught and area of the approximated cross section and of the actualcross section are identical.The half breadth to draught ratio 0H is given by:
31
310 1
12aaaa
DB
Hs
s
+−++
==
Equation 3.1–9
An integration of the Lewis form delivers the sectional area coefficient sσ :
( ) 21
23
23
21
1
314 aa
aaDB
A
ss
ss
−+
⋅−−⋅=
⋅=
πσ
Equation 3.1–10
in which sA is the area of the cross section.
Putting 1a , derived from the expression for 0H in Equation 3.1–9, into the expression for sσin Equation 3.1–10 yields a quadratic equation in 3a :
03322
31 =+⋅+⋅ cacacin which:
4
62
114
14
3
13
12
2
0
01
−=−⋅=
+−
⋅
⋅−+
⋅+=
cc
cc
HH
c ss
πσ
πσ
Equation 3.1–11
The (valid) solutions for 3a and 1a become:
( )111
293
30
01
1
113
+⋅+−
=
⋅−++−=
aHH
a
c
cca
Equation 3.1–12
Lewis forms with the other solution of 3a in the quadratic equation, with a minus sign beforethe square root expression:
1
113
293
c
cca
⋅−−+−=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
56
are looped; they intersect themselves at a point within the fourth quadrant. Since ships are''better behaved'', these solutions are not considered.
It is obvious that a transformation of a half-immersed circle with radius R will result inRM s = , 01 =a and 03 =a .
Some typical and realistic Lewis forms are presented in Figure 3.1–2.
Figure 3.1–2: Typical Lewis forms
3.1.1.1 Boundaries of Lewis Forms
In some cases the Lewis transformation can give more or less ridiculous results. Thefollowing typical Lewis hull forms, with the regions of the half breadth to draught ratio 0H
and the area coefficient sσ to match as presented before, can be distinguished:
• re-entrant forms, bounded by:
for 0.10 ≤H : ( )0232
3Hs −⋅
⋅<
πσ
for 0.10 ≥H :
−⋅
⋅<
0
12
323
Hs
πσ
Equation 3.1–13
• conventional forms, bounded by:
for 0.10 ≤H : ( )
+⋅
⋅<<−⋅
⋅4
332
32
323 0
0
HH s
πσπ
for 0.10 ≥H :
⋅
+⋅⋅
<<
−⋅
⋅
00 41
332
312
323
HH s
πσ
π
Equation 3.1–14
• bulbous and not-tunneled forms, bounded by:
0.10 ≤H and
⋅
+⋅⋅
<<
+⋅⋅
0
0
41
332
34
332
3H
Hs
πσ
π
Equation 3.1–15
• tunneled and not-bulbous forms, bounded by:
for: 0.10 ≥H and
+⋅
⋅<<
⋅
+⋅⋅
43
323
41
332
3 0
0
HH s
πσπ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
57
Equation 3.1–16
• combined bulbous and tunneled forms, bounded by:
for: 0.10 ≤H and
++⋅<<
⋅
+⋅⋅
00
0
110
3241
332
3H
HH s
πσ
π
for: 0.10 ≥H and
++⋅<<
+⋅⋅
00
0 110
3243
323
HH
Hs
πσ
π
Equation 3.1–17
• non-symmetric forms, bounded by:
∞<< 00 H and
++⋅>
00
110
32 HHs
πσ
Equation 3.1–18
These ranges of the half breadth to draught ratio 0H and the area coefficient sσ for thedifferent typical Lewis forms are shown in Figure 3.1–3.
Figure 3.1–3: Ranges of 0H and sσ of Lewis Forms
3.1.1.2 Acceptable Lewis Forms
Not-acceptable forms of ships are supposed to be the re-entrant forms and the asymmetricforms. So conventional forms, bulbous forms and tunneled forms are considered to be validforms here, see Figure 3.1–3. To obtain ship-like Lewis forms, the area coefficient sσ isbounded by a lower limit to omit re-entrant Lewis forms and by an upper limit to omit non-symmetric Lewis forms:
for 0.10 ≤H : ( )
++⋅<<−⋅
⋅
000
110
322
323
HHH s
πσπ
for 0.10 ≥H :
++⋅<<
−⋅
⋅
00
0
110
321
232
3H
HH s
πσ
π
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
58
Equation 3.1–19
If a value of sσ is outside of this range it has to be set (from a practical point of view) to thevalue of the nearest border of this range, to calculate the Lewis coefficients.Numerical problems, for instance with bulbous or aft cross sections of a ship, are avoidedwhen the following requirements are fulfilled:
ss D
B⋅> γ
2 and
2s
s
BD ⋅> γ with for instance 01.0=γ .
3.1.2 Extended Lewis Conformal Mapping
Somewhat better approximations will be obtained by taking into account also the first ordermoments of half the cross section about the 0x - and 0y -axes. These two additions to theLewis formulation were proposed by Reed and Nowacki [1974] and have been simplified byAthanassoulis and Loukakis [1985] by taking into account the vertical position of the centroidof the cross section. Extending the Lewis transformation from 2=N to 3=N in the generaltransformation formula has done this.
The three-parameter Extended-Lewis transformation is defined by:( )5
53
31
11−−−
− ⋅+⋅+⋅+⋅⋅= ζζζζ aaaaMz s
with 11 +=−a .
Equation 3.1–20
So:( )( )θθθθ
θθθθαααα
αααα
5cos3coscoscos
5sin3sinsinsin5
53
31
55
331
⋅⋅−⋅⋅+⋅⋅−⋅⋅=
⋅⋅+⋅⋅−⋅⋅+⋅⋅=−−−
−−−
eaeaeaeMy
eaeaeaeMx
s
s
Equation 3.1–21
By putting 0=α , the contour of this approximated form is expressed as:( )( )( )( )θθθ
θθθ5cos3coscos1
5sin3sinsin1
5310
5310
⋅−⋅+⋅−⋅=⋅+⋅−⋅+⋅=
aaaMy
aaaMx
s
s
with scale factor:
531531 112
aaaD
aaaB
M sss −+−
=+++
=
Equation 3.1–22
in which:
sB sectional breadth on the water line
sD sectional draught
Now the coefficients 1a , 3a and 5a and the scale factor sM will be determined such that,except the sectional breadth, draught and area, also the centroids of the approximated crosssection and of the actual cross section have a similar position.The half breadth to draught ratio 0H is given by:
531
5310 1
12aaaaaa
DB
Hs
s
−+−+++
==
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
59
Equation 3.1–23
An integration of the approximated form results into the sectional area coefficient sσ :
( ) ( )251
23
25
23
21
1
5314 aaa
aaaDB
A
ss
ss +−+
⋅−⋅−−⋅=
⋅=
πσ
Equation 3.1–24
A more complex expression has been obtained by Athanassoulis and Loukakis [1985] for therelative distance of the centroid to the keel point:
∑
∑∑∑
=−
=−−−
==
⋅⋅
⋅⋅⋅−== 3
0
3120
3
0121212
3
0
3
01
iis
kkjiijk
ji
s aH
aaaA
DKB
σκ
in which:
( ) ( ) ( ) ( )
++−⋅−⋅−
+−+⋅−
⋅−+
+−⋅−⋅−
−++⋅−
⋅−⋅=
kjik
kjik
kjik
kjik
Aijk 2121
2121
2121
2321
41
Equation 3.1–25
The following requirements should be fulfilled when also bulbous cross sections are allowed:• re-entrant forms are avoided when:
0531
0531
531
531
>⋅+⋅−+>⋅−⋅−−
aaa
aaa
Equation 3.1–26
• existence of a point of self-intersection is avoided when:
020101459
020101459
50532
52
3
50532
52
3
>⋅⋅−⋅⋅−⋅+⋅
>⋅⋅+⋅⋅+⋅+⋅
aHaaaa
aHaaaa
Equation 3.1–27
Taking these restrictions into account, the equations above can be solved in an iterativemanner.
3.1.3 Close-Fit Conformal Mapping
A more accurate transformation of the cross sectional hull form can be obtained by using agreater number of parameters N . A very simple and straight on iterative least squares methodof the first author to determine the Close-Fit conformal mapping coefficients will be describedhere shortly.
The scale factor sM and the conformal mapping coefficients 12 −na , with a maximum value ofn varying from 2=N until 10=N , have been determined successfully from the offsets ofvarious cross sections in such that the sum of the squares of the deviations of the actual crosssection from the approximate described cross section is minimised.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
60
The general transformation formula is again given by:
( ) ∑=
−−− ⋅⋅=
N
n
nns aMz
0
1212 ζ
with: 11 +=−a .
Equation 3.1–28
Then the contour of the approximated cross section is given by:
( ) ( )( )
( ) ( )( ) ∑
∑
=−
=−
⋅−⋅⋅−⋅+=
⋅−⋅⋅−⋅−=
N
nn
ns
N
nn
ns
naMy
naMx
0120
0120
12cos1
12sin1
θ
θ
with scale factor:
( ) ∑∑=
−=
− ⋅−== N
nn
n
sN
nn
ss
a
D
a
BM
012
012 1
2
Equation 3.1–29
The procedure starts with initial values for [ ]12 −⋅ ns aM . The initial values of sM , 1a and 3aare obtained with the Lewis method as has been described before, while the initial values of
5a through 12 −Na are set to zero. With these [ ]12 −⋅ ns aM values, a iθ -value is determined for
each offset in such a manner that the actual offset ( )ii yx , lies on the normal of the
approximated contour of the cross section in ( )ii yx 00 , .
Now iθ has to be determined. Therefore a function ( )iF θ , will be defined by the distance
from the offset ( )ii yx , to the normal of the contour to the actual cross section through
( )ii yx 00 , , see Figure 3.1–4.
Figure 3.1–4: Cose-Fit conformal mapping
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
61
These i offsets ( Ii ,...0= ) have to be selected at approximately equal mutual circumferential
lengths, eventually with somewhat more dense offsets near sharp corners. Then iα is definedby:
( ) ( )
( ) ( )211
211
11
211
211
11
sin
cos
−+−+
−+
−+−+
−+
−+−
+−=
−+−
−+=
iiii
iii
iiii
iii
yyxx
yy
yyxx
xx
α
α
Equation 3.1–30
With this iθ -value, the numerical value of the square of the deviation of ( )ii yx , from ( )ii yx 00 ,is calculated:
( ) ( )20
20 iiiii yyxxe −+−=
Equation 3.1–31
After doing this for all 1+I offsets, the numerical value of the sum of the squares ofdeviations is known:
∑=
=I
iieE
0
Equation 3.1–32
The sum of the squares of these deviations can also be expressed as:
( ) [ ] ( )( )
( ) [ ] ( )( ) ∑
∑
∑=
=−
=−
⋅−⋅⋅⋅−−+
⋅−⋅⋅⋅−++
=I
i N
nins
ni
N
nins
ni
naMy
naMx
E0
2
012
2
012
12cos1
12sin1
θ
θ
Equation 3.1–33
Then, new values of [ ]12 −⋅ ns aM have to be determined such that E is minimised. This means
that the derivative of this equation to each coefficient [ ]12 −⋅ ns aM is zero, so:
012
=⋅∂
∂
−js aME
for: Nj ,...0=
Equation 3.1–34
This provides 1+N equations:
( )( ) ( ) [ ] ( )( )
( )( ) ( ) [ ] ( )( )
( )( ) ( )( ) ∑
∑∑
∑
=
=
=−
=−
⋅−⋅−⋅−⋅=
=
⋅−⋅⋅⋅−⋅⋅−−
⋅−⋅⋅⋅−⋅⋅−−
I
iiiii
I
iN
nins
ni
N
nins
ni
jyjx
naMj
naMj
0
0
012
012
12cos12sin
12cos112cos
12sin112sin
θθ
θθ
θθ
for: Nj ,...0=
which are rewritten as:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
62
( ) [ ] ( )( )
( )( ) ( )( ) ∑
∑ ∑
=
= =−
⋅−⋅+⋅−⋅−
=
⋅−⋅⋅⋅−
I
iiiii
N
n
I
iins
n
jyjx
njaM
0
0 012
12cos12sin
22cos1
θθ
θ for: Nj ,...0=
Equation 3.1–35
To obtain the exact actual breadth and draught, the last two equations ( 1−= Nj and Nj = )in Equation 3.1–35 are replaced by the equations for the breadth at the water line and thedraught:
( ) [ ] ( )( )
( )( ) ( )( )
[ ]
( ) [ ] s
N
nns
n
s
N
nns
I
iiiii
N
n
I
iins
n
DaM
BaM
Njjyjx
njaM
=⋅⋅−
=⋅
−=⋅−⋅+⋅−⋅−
=
⋅−⋅⋅⋅−
∑
∑
∑
∑ ∑
=−
=−
=
= =−
012
012
0
0 012
1
2
2,...0 :for 12cos12sin
22cos1
θθ
θ
Equation 3.1–36
These 1+N equations can be solved numerically, so that new values for [ ]12 −⋅ ns aM will be
obtained. These new values are used instead of the initial values to obtain new iθ -values ofthe 1+I offsets again, etc. This procedure will be repeated several times and stops when thedifference between the numerical E -values of two subsequent calculations becomes less thana certain threshold value E∆ , depending on the dimensions of the cross section; for instance:
( )2
2max
2max00005.01
+⋅⋅+=∆ dbIE
Equation 3.1–37
in which:
maxb maximum half breadth of the cross section
maxd maximum draught of the cross section
Because 11 +=−a , the scale factor sM is equal to the final solution of the first coefficient
( 0=n ). The N other coefficients 12 −na can be found by dividing the final solutions of
[ ]12 −⋅ ns aM by this sM -value.
Reference is also given here to a report of de Jong [1973]. In that report several other, suitablebut more complex, methods are described to determine the scale factor sM and the conformal
mapping coefficients 12 −na from the offsets of a cross section.
Attention has been paid in SEAWAY to divergence in the calculation routines and re-entrantforms. In these cases the number N will be decreased until the divergence or re-entrancevanish. In the worse case the ''minimum'' value of N will be attained without success. One
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
63
can then switch to Lewis coefficients with an area coefficient of the cross section, eventuallyset to the nearest border of the valid Lewis form area.
3.1.4 Mapping Comparisons
A first example has been given here for the amidships cross section of a container vessel, witha breadth of 25.40 meter and a draught of 9.00 meter, with offsets as tabled below.
Table 3.1-1: Offsets of a cross section
For the least square method in the conformal mapping method, 33 new offsets at equidistantlength intervals on the contour of this cross section can be determined by a second degreeinterpolation routine. The calculated data of the two-parameter Lewis and the N -parameterClose-Fit conformal mapping of this amidships cross section are tabled below. The last linelists the RMS -values for the deviations of the 33 equidistant points on the approximatecontour of this cross section.
Table 3.1-2: Conformal mapping coefficients
Another example is given in Figure 3.1–5, which shows the differences between a Lewistransformation and a 10-parameter close-fit conformal mapping of a rectangular cross section.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
64
Figure 3.1–5: Lewis and Close-Fit conformal mapping of a rectangle
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
65
3.2 Potential Theory of Tasai
In this section, the determination of the hydrodynamic coefficients of a heaving, swaying androlling cross section of a ship in deep water at zero forward speed is based on work publishedby Ursell [1949], Tasai [1959], Tasai [1960], Tasai [1961] and de Jong [1973]. Tasai'snotations have been maintained here as far as possible.
The axes system of Tasai (and used here) is given in Figure 3.2–1.
Figure 3.2–1: Tasai’s axes system for heave, sway and roll oscillations
The figure shows a cross section of an infinite long cylinder in the surface of a fluid. Thiscylinder will carry out forced harmonic heave, sway and roll motions, respectively. Using theapproach of Tasai (and de Jong), the determination of the hydrodynamic loads will be showedin the following Sections.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
66
3.2.1 Heave Motions
The determination of the hydrodynamic coefficients of a heaving cross section of a ship indeep and still water at zero forward speed, as described here, is based on work published byUrsell [1949], Tasai [1959] and Tasai [1960]. Starting points for the derivation thesecoefficients here are the velocity potentials and the conjugate stream functions of the fluid, asthey have been derived by Tasai and also by de Jong [1973].
Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic vertical motion about itsinitial position with a frequency of oscillation ω and small amplitude of displacement ay :
( )δω +⋅⋅= tyy a cos
Equation 3.2–1
in which δ is a phase angle.Respectively, the vertical velocity and acceleration of the cylinder are:
( )δωω +⋅⋅⋅−= tyy a sin& and ( )δωω +⋅⋅⋅−= tyy a cos2&&
Equation 3.2–2
This forced vertical oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.
Two kinds of waves will be produced:• A standing wave system, denoted here by subscript A .
The amplitudes of these waves decrease strongly with the distance to the cylinder.• A regular progressive wave system, denoted here by subscript B .
These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder ay , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and thewavelength is not much smaller than the diameter of the cylinder.
The two-dimensional velocity potential of the fluid has to fulfil the following requirements:
1. The velocity potential must satisfy to the equation of Laplace:
02
2
2
22 =
∂Φ∂
+∂
Φ∂=Φ∇
yx
Equation 3.2–3
2. Because the heave motion of the fluid is symmetrical about the (vertical) y -axis, thisvelocity potential has the following relation:
( ) ( )yxyx ,, +Φ=−Φ
Equation 3.2–4
from which follows:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
67
0=∂Φ∂θ
for: 0=θ
Equation 3.2–5
3. The linearised free surface condition in deep water is expressed as follows:
02
=∂Φ∂
+Φ⋅yg
ω for:
2sB
x ≥ and 0=y
Equation 3.2–6
In consequence of the conformal mapping, the free surface condition in Equation 3.2–6 can bewritten as:
( ) ( ) 0120
1212 =
∂Φ∂
±⋅⋅−⋅Φ⋅ ∑=
⋅−−− θσ
ξ αN
n
nn
a
b ean for: 0≥α and 2πθ ±=
in which:
sa
b Mg
⋅=2ω
σξ
or gb
b ⋅⋅
=2
02ωξ (non-dimensional frequency squared)
Equation 3.2–7
From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder for 0=α :
( )ny
yn ∂
∂⋅=
∂Φ∂ 00 &
θ
Equation 3.2–8
in which n is the outward normal of the cylinder surface.Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:
( )
( ) ( ) ( )( ) ∑=
− ⋅−⋅⋅−⋅−⋅⋅−=
∂∂
⋅=∂Ψ∂−
N
nn
ns nanMy
yy
012
00
12cos121 θ
αθθ
&
&
Equation 3.2–9
Integration results into the following requirement for the stream function on the surface of thecylinder:
( ) ( ) ( )( ) ( )tCnaMyN
nn
ns +⋅−⋅⋅−⋅⋅=Ψ ∑
=−
0120 12sin1 θθ &
Equation 3.2–10
in which ( )tC is a function of time only.
When defining:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
68
( )
( ) ( )( ) ∑=
− ⋅−⋅⋅−⋅−=
⋅=
N
nn
n
a
na
bx
h
012
0
0
12sin11
2
θσ
θ
Equation 3.2–11
the stream function on the surface of the cylinder is given by:
( ) ( ) ( )tChb
y +⋅⋅−=Ψ θθ20
0 &
Equation 3.2–12
Because of the symmetry of the fluid about the y -axis, it is clear that ( ) 0=tC , so that:
( ) ( )θθ hb
y ⋅⋅−=Ψ20
0 &
Equation 3.2–13
For the standing wave system a velocity potential and a stream function satisfying theequation of Laplace, the symmetrical motion of the fluid and the free surface condition has tobe found.The following set of velocity potentials, as they are given by Tasai [1959], Tasai [1960] andde Jong [1973], fulfil these requirements:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Φ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαφωθαφ
ωπη
in which:( ) ( )
( ) ( ) ( )( )∑=
⋅−+−−
⋅−
⋅−+⋅⋅⋅
−+−
⋅−⋅−
⋅⋅=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
12212
22
122cos122
121
2cos,
θσξ
θθαφ
α
α
Equation 3.2–14
The set of conjugate stream functions is expressed as:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Ψ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαψωθαψ
ωπη
in which:( ) ( )
( ) ( ) ( )( )∑=
⋅−+−−
⋅−
⋅−+⋅⋅⋅
−+−
⋅−⋅−
⋅⋅=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
12212
22
122sin122
121
2sin,
θσξ
θθαψ
α
α
Equation 3.2–15
These sets of functions tend to zero as α tends to infinity.In these expressions the magnitudes of the mP2 and mQ2 series follow from the boundaryconditions as will be explained further on.
Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the free surface condition and the symmetry about the y -
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
69
axis, representing such a train of waves at infinity. For this, a function describing a source atthe origin O is chosen.Tasai [1959], Tasai [1960] and de Jong [1973] gave the velocity potential of the progressivewave system as:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωφωφωπη
sin,cos,
in which:( )
( ) ( ) ( )∫∞
⋅−⋅−
⋅−
⋅⋅+
⋅⋅−⋅⋅+⋅⋅⋅+=
⋅⋅⋅+=
022
cossinsin
cos
dkek
ykkykxe
xe
xkyBs
yBc
νννπφ
νπφ
ν
ν
while:
g
2ων = (wave number for deep water)
Equation 3.2–16
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωθαφωθαφωπη
sin,cos,
Equation 3.2–17
The conjugate stream function is given by:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωψωψωπη
sin,cos,
in which:( )
( ) ( ) ( )∫∞
⋅−⋅−
⋅−
⋅⋅+
⋅⋅+⋅⋅+⋅⋅⋅−=
⋅⋅⋅+=
022
sincoscos
sin
dkek
ykkykxe
xe
xkyBs
yBc
νννπψ
νπψ
ν
ν
Equation 3.2–18
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωθαψωθαψωπη
sin,cos,
Equation 3.2–19
When calculating the integrals in the expressions for Bsψ and Bcψ numerically, theconvergence is very slowly.Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k . Summations in these expansions converge much faster than the numeric integrationprocedure. This will be shown in the Section 3.2.2 for the sway case.
The total velocity potential and stream function to describe the waves generated by a heavingcylinder are:
BA
BA
Ψ+Ψ=ΨΦ+Φ=Φ
Equation 3.2–20
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
70
So the velocity potential and the conjugate stream function are expressed by:
( )( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑
∑
∑
∞
=
∞
=
∞
=
∞
=
tQ
tPg
tQ
tPg
mmAmBs
mmAmBc
a
mmAmBs
mmAmBc
a
ωθαψθαψ
ωθαψθαψ
ωπηθα
ωθαφθαφ
ωθαφθαφ
ωπηθα
sin,,
cos,,
,
sin,,
cos,,
,
122
122
122
122
Equation 3.2–21
When putting 0=α , the stream function is equal to the expression in Equation 3.2–13, foundfrom the boundary condition on the surface of the cylinder:
( )( ) ( ) ( )
( ) ( ) ( )
( )θ
ωθψθψ
ωθψθψ
ωπηθ
hb
y
tQ
tPg
mmAmsB
mmAmcB
a
⋅⋅−=
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
∑
∑∞
=
∞
=
2
sin
cos
0
10220
10220
0
&
in which:( ) ( )
( ) ( )( )∑=
−
⋅−+⋅⋅
−+−
⋅−⋅−
⋅=N
nn
n
a
b
mA
nmanm
n
m
012
02
122sin122
121
2sin
θσξ
θθψ
Equation 3.2–22
In this expression, ( )θψ cB 0 and ( )θψ sB 0 are the values of ( )θαψ ,Bc and ( )θαψ ,Bs at thesurface of the cylinder, so for 0=α .
So for each θ , the following equation has been obtained from Equation 3.2–22:
( ) ( ) ( )
( ) ( ) ( )( )θ
ηωπ
ωθψθψ
ωθψθψh
gb
y
tQ
tP
a
mmAmsB
mmAmcB
⋅⋅⋅⋅⋅
⋅−=
⋅⋅
⋅++
⋅⋅
⋅++
∑
∑∞
=
∞
=
2sin
cos0
10220
10220
&
Equation 3.2–23
The right hand side of this equation can be written as:
( ) ( ) ( )
( ) ( ) ( ) tBtAh
ty
hhg
by b
a
a
a
⋅⋅+⋅⋅⋅=
+⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅
⋅−
ωωθ
δωξπη
θθη
ωπ
sincos
sin2
00
0&
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
71
δξπη
sin0 ⋅⋅⋅= ba
ayA and δξπ
ηcos0 ⋅⋅⋅= b
a
ayB
Equation 3.2–24
This results for each θ into a set of two equations:
( ) ( ) ( )
( ) ( ) ( ) 01
0220
01
0220
BhQ
AhP
mmAmsB
mmAmcB
⋅=⋅+
⋅=⋅+
∑
∑∞
=
∞
=
θθψθψ
θθψθψ
Equation 3.2–25
When putting 2πθ = , so at the intersection of the surface of the cylinder with the freesurface of the fluid where ( ) 1=θh , we obtain the coefficients 0A and 0B :
( ) ( )
( ) ( ) ∑
∑∞
=
∞
=
⋅+=
⋅+=
102200
102200
22
22
mmAmsB
mmAmcB
QB
PA
πψπψ
πψπψ
in which:
( ) ( ) ∑=
−
⋅
−+−
⋅−⋅=N
nn
m
a
bmA a
nmn
01202 122
1212
σξ
πψ
Equation 3.2–26
A substitution of 0A and 0B into the set of two equations for each θ , results for each θ-value
less than 2π in a set of two equations with the yet unknown parameters mP2 and mQ2 , so:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ∑
∑∞
=
∞
=
⋅=⋅−
⋅=⋅−
12200
12200
2
2
mmmsBsB
mmmcBcB
Qfh
Pfh
θπψθθψ
θπψθθψ
in which:( ) ( ) ( ) ( )202022 πψθθψθ mAmAm hf ⋅+−=
Equation 3.2–27
The series in these two sets of equations converges uniformly with an increasing value of m .For practical reasons the maximum value of m is limited to M , for instance 10=M .
Each θ-value less than 2π will provide an equation for the mP2 and mQ2 series. For a lot of
θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of a
least square method. Notice that at least M values of θ , less than 2π , are required to solvethese equations.
Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration.Herewith, two sets of M equations have been obtained, one set for mP2 and one set for mQ2 :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
72
( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( )∫∑ ∫
∫∑ ∫
⋅⋅⋅−=
⋅⋅⋅
⋅⋅⋅−=
⋅⋅⋅
=
=
2
0200
1
2
0222
2
0200
1
2
0222
2
2
ππ
ππ
θθπψθθψθθθ
θθπψθθψθθθ
dfhdffQ
dfhdffP
nsBsB
M
mnmm
ncBcB
M
mnmm
for: Mn ,...1=
Equation 3.2–28
Now the mP2 and mQ2 series can be solved by a numerical method and with these values, the
coefficients 0A and 0B are known too.
From the definition of these coefficients in Equation 3.2–24 follows the amplitude ratio of theradiated waves and the forced heave oscillation:
20
20 BAy
b
a
a
+
⋅=
ξπη
Equation 3.2–29
With the solved mP2 and mQ2 values, the velocity potential on the surface of the cylinder( 0=α ) is known too:
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑∞
=
∞
=
tQ
tPg
mmAmsB
mmAmcB
a
ωθφθφ
ωθφθφ
ωπηθ
sin
cos
10220
10220
0
in which:( ) ( )
( ) ( )( )∑=
−
⋅−+⋅⋅
−+−
⋅−⋅−
⋅=N
nn
n
a
b
mA
nmanm
n
m
012
02
122cos122
121
2cos
θσξ
θθφ
Equation 3.2–30
In this expression, ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at thesurface of the cylinder, so for 0=α .
3.2.1.1 Pressure Distribution During Heave Motions
Now the hydrodynamic pressure on the surface of the cylinder can be obtained from thelinearised equation of Bernoulli:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
73
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅+−
⋅⋅
⋅++
⋅⋅⋅−
=
∂Φ∂
⋅−=
∑
∑
=
=
tP
tQg
tp
M
mmAmcB
M
mmAmsB
a
ωθφθφ
ωθφθφ
πηρ
θρθ
sin
cos
10220
10220
0
Equation 3.2–31
It is obvious that this pressure is symmetric in θ .
3.2.1.2 Heave Coefficients
The two-dimensional hydrodynamic vertical force, acting on the cylinder in the direction ofthe y -axis, can be found by integrating the vertical component of the hydrodynamic pressureon the surface of the cylinder:
( )
( ) θθ
θ
θ
π
π
π
dddx
p
dsdsdx
pFy
⋅⋅⋅−=
⋅⋅−=
∫
∫+
−
02
0
02
2
'
2
Equation 3.2–32
With this the two-dimensional hydrodynamic vertical force due to heave oscillations can bewritten as follows:
( ) ( )( )tNtMbg
F ay ⋅⋅−⋅⋅⋅
⋅⋅⋅= ωω
πηρ
sincos 000'
in which:
( ) ( ) ( ) ( )( )
( ) ( )( ) ( )
( ) ( )
⋅⋅−⋅⋅−+⋅⋅
+
⋅−−
−⋅⋅−⋅−
⋅⋅−⋅⋅−⋅−⋅⋅−=
∑ ∑
∑ ∑
∫ ∑
=
−
=−+−
= =−
=−
N
m
mN
nnmnm
m
a
b
M
m
N
nnm
m
a
N
nn
nsB
a
aanQQ
anm
nQ
dnanM
1 012212222
1 01222
2
2
2
0 01200
1214
12212
11
12cos1211
σξπ
σ
θθθφσ
π
and 0N as obtained from this expression above for 0M , by replacing there ( )θφ sB0 by
( )θφ cB0 and mQ2 by mP2 .
Equation 3.2–33
For the determination of 0M and 0N , it is required that NM ≥ . These expressions coincidewith those as given by Tasai [1960].
With Equation 3.2–33 in some other format:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
74
( ) ( )( )
0
0
000'
cos
sin
sincos
By
Ay
tNtMbg
F
ba
a
ba
a
ay
⋅⋅⋅
=
⋅⋅⋅
=
−+⋅⋅−−+⋅⋅⋅⋅⋅⋅
=
ξπη
δ
ξπηδ
δδωδδωπ
ηρ
Equation 3.2–34
the two-dimensional hydrodynamic vertical force can be resolved into components in phaseand out phase with the vertical displacement of the cylinder:
( ) ( ) ( ) ( ) δωδω
ξπηρ
+⋅⋅⋅−⋅++⋅⋅⋅+⋅
⋅⋅⋅⋅⋅⋅
=
tBNAMtANBM
ybg
Fab
ay
sincos 00000000
2
20'
Equation 3.2–35
This hydrodynamic vertical force can also be written as:
( ) ( )δωωδωω +⋅⋅⋅⋅++⋅⋅⋅⋅=
⋅−⋅−=
tyNtyM
yNyMF
aa
y
sincos '33
2'33
'33
'33
' &&&
Equation 3.2–36
in which:'
33M 2-D hydrodynamic mass coefficient of heave'
33N 2-D hydrodynamic damping coefficient of heave
When using also the amplitude ratio of the radiated waves and the forced heave oscillation,found before in Equation 3.2–29, the two-dimensional hydrodynamic mass and dampingcoefficients of heave are given by:
ωρ
ρ
⋅+
⋅−⋅⋅
⋅=
+
⋅+⋅⋅
⋅=
20
20
00002
0'33
20
20
00002
0'33
2
2
BA
BNAMbN
BA
ANBMbM
Equation 3.2–37
The signs of these two coefficients are proper in both, the axes system of Tasai and the shipmotions ( )zyxO ,, co-ordinate system.
The energy delivered by the exciting forces should be equal to the energy radiated by thewaves, so:
( ) ( )2
1 2
0
'33
cgdtyyN
Ta
T
osc
osc ⋅⋅⋅=⋅⋅⋅⋅ ∫
ηρ&&
Equation 3.2–38
in which oscT is the period of oscillation.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
75
With the relation for the wave speed ωgc = at deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced heave oscillation:
2
3
2'
33
⋅
⋅=
a
a
yg
Nη
ωρ
Equation 3.2–39
With this amplitude ratio the two-dimensional hydrodynamic damping coefficient of heave isalso given by:
ωπρ
⋅+
⋅⋅⋅
= 20
20
20
2'
33
14 BA
bN
Equation 3.2–40
When comparing this expression for '33N with the expression found before, the following
energy balance relation is found:
2
2
0000
π=⋅−⋅ BNAM
Equation 3.2–41
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
76
3.2.2 Sway Motions
The determination of the hydrodynamic coefficients of a swaying cross section of a ship indeep and still water at zero forward speed is based here on work published by Tasai [1961] forthe Lewis conformal mapping method. Starting points for the derivation these coefficientshere are the velocity potentials and the conjugate stream functions of the fluid as they havebeen derived by Tasai [1961] and also by de Jong [1973].Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic lateral motion about itsinitial position with a frequency of oscillation ω and small amplitude of displacement ax :
( )εω +⋅⋅= txx a cos
Equation 3.2–42
in which ε is a phase angle.Respectively, the lateral velocity and acceleration of the cylinder are:
( )εωω +⋅⋅⋅−= txx a sin& and ( )εωω +⋅⋅⋅−= txx a cos2&&
Equation 3.2–43
This forced lateral oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.
Two kinds of waves will be produced:• A standing wave system, denoted here by subscript A .
The amplitudes of these waves decrease strongly with the distance to the cylinder.• A regular progressive wave system, denoted here by subscript B .
These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder ax , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and thewavelength is not much smaller than the diameter of the cylinder.
The two-dimensional velocity potential of the fluid has to fulfil the following requirements:
1. The velocity potential must satisfy to the equation of Laplace:
02
2
2
22 =
∂Φ∂
+∂
Φ∂=Φ∇
yx
Equation 3.2–44
2. Because the sway motion of the fluid is not symmetrical about the y -axis, this velocitypotential has the following anti-symmetric relation:
( ) ( )yxyx ,, +Φ−=−Φ
Equation 3.2–45
3. The linearised free surface condition in deep water is expressed as follows:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
77
02
=∂Φ∂
+Φ⋅yg
ω for:
2sB
x ≥ and 0=y
Equation 3.2–46
In consequence of the conformal mapping, this free surface condition can be written as:
( ) ( ) 0120
1212 =
∂Φ∂
±⋅⋅−⋅Φ⋅ ∑=
⋅−−− θσ
ξ αN
n
nn
a
b ean for: 0≥α and 2πθ ±=
in which:
sa
b Mg
⋅=2ω
σξ
or gb
b ⋅⋅
=2
02ωξ (non-dimensional frequency squared)
Equation 3.2–47
From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder S for 0=α :
( )nx
xn ∂
∂⋅=
∂Φ∂ 00 &
θ
Equation 3.2–48
in which n is the outward normal of the cylinder surface S .
Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:
( )
( ) ( ) ( )( ) ∑=
− ⋅−⋅⋅−⋅−⋅⋅−=
∂∂
⋅−=∂
Ψ∂
N
nn
ns nanMx
xx
012
00
12sin121 θ
αθθ
&
&
Equation 3.2–49
Integration results into the following requirement for the stream function on the surface of thecylinder:
( ) ( ) ( )( ) ( )tCnaMxN
nn
ns +⋅−⋅⋅−⋅⋅=Ψ ∑
=−
0120 12cos1 θθ &
Equation 3.2–50
in which ( )tC is a function of time only.
When defining:
( )
( ) ( )( ) ∑=
− ⋅−⋅⋅−⋅=
⋅=
N
nn
n
a
na
by
g
012
0
0
12cos11
2
θσ
θ
Equation 3.2–51
the stream function on the surface of the cylinder is given by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
78
( ) ( ) ( )tCgb
x +⋅⋅−=Ψ θθ20
0 &
Equation 3.2–52
For the standing wave system a velocity potential and a stream function satisfying to theequation of Laplace, the non-symmetrical motion of the fluid and the free surface conditionhas to be found.
The following set of velocity potentials, as they are given by Tasai [1961] and de Jong [1973],fulfil these requirements:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Φ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαφωθαφ
ωπη
in which:( ) ( ) ( )( )
( ) ( ) ( )( )∑=
⋅+−−
⋅+−
⋅+⋅⋅⋅
+−
⋅−⋅−
⋅+⋅+=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
2212
122
22sin2212
1
12sin,
θσξ
θθαφ
α
α
Equation 3.2–53
The set of conjugate stream functions is expressed as:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Ψ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαψωθαψ
ωπη
in which:( ) ( ) ( )( )
( ) ( ) ( )( )∑=
⋅+−−
⋅+−
⋅+⋅⋅⋅
+−
⋅−⋅+
⋅+⋅−=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
2212
122
22cos2212
1
12cos,
θσξ
θθαψ
α
α
Equation 3.2–54
These sets of functions tend to zero as α tends to infinity.
In these expressions the magnitudes of the mP2 and mQ2 series follow from the boundaryconditions as will be explained further on.
Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the equation of Laplace, the non-symmetrical motion andthe free surface condition, representing such a train of waves at infinity. For this, a functiondescribing a two-dimensional horizontal doublet at the origin O is chosen.Tasai [1961] and de Jong [1973] gave the velocity potential of the progressive wave systemas:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωφωφωπη
sin,cos,
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
79
( )
( ) ( ) ( )
( )22
022
sincoscos
sin
yxx
dkek
ykkykxej
xej
xkyBs
yBc
+⋅+
⋅⋅+
⋅⋅+⋅⋅−⋅⋅⋅+=⋅
⋅⋅⋅−=⋅
∫∞
⋅−⋅−
⋅−
ν
νννπφ
νπφ
ν
ν
while:1+=j for: 0>x1−=j for: 0<x
g
2ων = (wave number for deep water)
Equation 3.2–55
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωθαφωθαφωπη
sin,cos,
Equation 3.2–56
The conjugate stream function is given by:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωψωψωπη
sin,cos,
in which:( )
( ) ( ) ( )
( )22
022
cossinsin
cos
yxy
dkek
ykkykxe
xe
xkyBs
yBc
+−
⋅⋅+
⋅⋅−⋅⋅+⋅⋅⋅+=
⋅⋅⋅+=
∫∞
⋅−⋅−
⋅−
ν
νννπψ
νπψ
ν
ν
Equation 3.2–57
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωθαψωθαψωπη
sin,cos,
Equation 3.2–58
When calculating the integrals in the expressions for Bsφ and Bcφ numerically, theconvergence is very slowly.Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k :
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) yxk
yxk
exSxQdkek
ykkyk
exSxQdkek
ykkyk
⋅−∞
⋅−
⋅−∞
⋅−
⋅⋅⋅−+⋅⋅=⋅⋅+
⋅⋅−⋅⋅
⋅⋅⋅−−⋅⋅=⋅⋅+
⋅⋅+⋅⋅
∫
∫
ν
ν
νπνν
ν
νπνν
ν
sincoscossin
cossinsincos
022
022
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
80
( ) ( )
( )
( )constant)(Euler .....57722.0
!
arctan
sin
cosln
22
1
1
22
=⋅+⋅
=
=
⋅⋅+=
⋅⋅++⋅+=
∑
∑∞
=
∞
=
γ
ν
β
ββ
βνγ
nnyx
p
yx
npS
npyxQ
n
n
nn
nn
Equation 3.2–59
The summations in these expansions converge much faster than the numeric integrationprocedure. A suitable maximum value for n should be chosen, Nn ,...1= .
The total velocity potential and stream function to describe the waves generated by a swayingcylinder are:
BA
BA
Ψ+Ψ=ΨΦ+Φ=Φ
Equation 3.2–60
So the velocity potential and the conjugate stream function are expressed by:
( )( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑
∑
∑
∞
=
∞
=
∞
=
∞
=
tQ
tPg
tQ
tPg
mmAmBs
mmAmBc
a
mmAmBs
mmAmBc
a
ωθαψθαψ
ωθαψθαψ
ωπηθα
ωθαφθαφ
ωθαφθαφ
ωπηθα
sin,,
cos,,
,
sin,,
cos,,
,
122
122
122
122
Equation 3.2–61
When putting 0=α , the stream function is equal to the expression found before in Equation3.2–52 from the boundary condition on the surface of the cylinder:
( )( ) ( ) ( )
( ) ( ) ( )
( ) ( )tCgb
x
tQ
tPg
mmAmsB
mmAmcB
a
+⋅⋅=
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
∑
∑∞
=
∞
=
θ
ωθψθψ
ωθψθψ
ωπηθ
2
sin
cos
0
10220
10220
0
&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
81
in which:( ) ( )( )
( ) ( )( )∑=
−
⋅+⋅⋅
+−
⋅−⋅+
⋅+−=N
nn
n
a
b
mA
nmanm
n
m
012
02
22cos2212
1
12cos
θσξ
θθψ
Equation 3.2–62
In this expression, ( )θψ cB 0 and ( )θψ sB 0 are the values of ( )θαψ ,Bc and ( )θαψ ,Bs at thesurface of the cylinder, so for 0=α .
So for each θ , the following equation has been obtained from Equation 3.2–62:
( ) ( ) ( )
( ) ( ) ( )( ) ( )tCg
gbx
tQ
tP
a
mmAmsB
mmAmcB
*0
10220
10220
2sin
cos
+⋅⋅⋅⋅⋅⋅=
⋅⋅
⋅++
⋅⋅
⋅++
∑
∑∞
=
∞
= θη
ωπ
ωθψθψ
ωθψθψ&
Equation 3.2–63
When putting 2πθ = , so at the intersection of the surface of the cylinder with the free
surface of the fluid where ( ) 0=θg , we obtain the constant ( )tC* :
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
=
∑
∑∞
=
∞
=
tQ
tP
tC
mmAmsB
mmAmcB
ωπψπψ
ωπψπψ
sin22
cos22
10220
10220
*
in which:
( ) ( ) ∑=
−
⋅
+−
⋅−⋅=N
nn
m
a
bmA a
nmn
01202 22
1212
σξπψ
Equation 3.2–64
A substitution of ( )tC* in the equation for each θ-value, results for each θ-value less than2π into the following equation:
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )
( )θη
ωπ
ωπψθψπψθψ
ωπψθψπψθψ
gg
bx
tQ
tP
a
mmAmAmsBsB
mmAmAmcBcB
⋅⋅⋅⋅⋅
⋅
=
⋅⋅
−⋅+−+
⋅⋅
−⋅+−+
∑
∑∞
=
∞
=
2
sin22
cos22
0
10202200
10202200
&
Equation 3.2–65
The right hand side of this equation can be written as:
( ) ( ) ( )
( ) ( ) ( ) tQtPg
tx
ggg
bx b
a
a
a
⋅⋅+⋅⋅⋅=
+⋅⋅⋅⋅⋅−=⋅⋅⋅⋅⋅
⋅
ωωθ
εωξπη
θθη
ωπ
sincos
sin2
00
0&
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
82
εξπη
sin0 ⋅⋅⋅−= ba
axP and εξπ
ηcos0 ⋅⋅⋅−= b
a
axQ
Equation 3.2–66
This provides for each θ-value less than 2π a set of two equations with the unknownparameters mP2 and mQ2 :
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ∑
∑∞
=
∞
=
⋅+⋅=−
⋅+⋅=−
122000
122000
2
2
mmmsBsB
mmmcBcB
QfQg
PfPg
θθπψθψ
θθπψθψ
in which:( ) ( ) ( )202022 πψθψθ mAmAmf +−=
Equation 3.2–67
These equations can also be written as:
( ) ( ) ( )
( ) ( ) ( ) ∑
∑∞
=
∞
=
⋅=−
⋅=−
02200
02200
2
2
mmmsBsB
mmmcBcB
Qf
Pf
θπψθψ
θπψθψ
in which:for :0=m ( ) ( )θθ gf =0
for 0>m : ( ) ( ) ( )202022 πψθψθ mAmAmf +−=
Equation 3.2–68
The series in these two sets of equations converges uniformly with an increasing value of m .For practical reasons the maximum value of m is limited to M , for instance 10=M .
Each θ-value less than 2π will provide an equation for the mP2 and mQ2 series. For a lot of
θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of aleast squares method. Notice that at least 1+M values of θ , less than 2π , are required tosolve these equations.
Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration.Herewith, two sets of 1+M equations have been obtained, one set for mP2 and one set for
mQ2 :
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )∫∑ ∫
∫∑ ∫
⋅⋅−=
⋅⋅⋅
⋅⋅−=
⋅⋅⋅
=
=
2
0
2000
2
0
222
2
0200
0
2
0222
2
2
ππ
ππ
θθπψθψθθθ
θθπψθψθθθ
dfdffQ
dfdffP
nsBsB
M
mnmm
ncBcB
M
mnmm
for: Mn ,...0=
Equation 3.2–69
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
83
Now the mP2 and mQ2 series can be solved by a numerical method and with these values, the
coefficients 0P and 0Q are known now and from the definition of these coefficients inEquation 3.2–66 follows the amplitude ratio of the radiated waves and the forced swayoscillation:
20
20 QPx
b
a
a
+
⋅=
ξπη
Equation 3.2–70
With the solved mP2 and mQ2 values, the velocity potential on the surface of the cylinder( 0=α ) is known too:
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑
=
=
tQ
tPg
M
mmAmsB
M
mmAmcB
a
ωθφθφ
ωθφθφ
ωπηθ
sin
cos
10220
10220
0
in which:( ) ( )( )
( ) ( )( )∑=
−
⋅+⋅⋅
+−
⋅−⋅−
⋅++=N
nn
n
a
b
mA
nmanm
n
m
012
02
22sin2212
1
12sin
θσξ
θθφ
Equation 3.2–71
In this expression, ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at thesurface of the cylinder, so for 0=α .
3.2.2.1 Pressure Distribution During Sway Motions
Now the hydrodynamic pressure on the surface of the cylinder can be obtained from thelinearised equation of Bernoulli:
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅+−
⋅⋅
⋅++
⋅⋅⋅−
=
∂Φ∂
⋅−=
∑
∑∞
=
∞
=
tP
tQg
tp
mmAmcB
mmAmsB
a
ωθφθφ
ωθφθφ
πηρ
θρθ
sin
cos
10220
10220
0
Equation 3.2–72
It is obvious that this pressure is skew-symmetric in θ .
3.2.2.2 Sway Coefficients
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
84
The two-dimensional hydrodynamic horizontal force, acting on the cylinder in the direction ofthe x -axis, can be found by integrating the horizontal component of the hydrodynamicpressure on the surface S of the cylinder:
( ) ( )
( ) θθ
θ
θθ
π
π
dddy
p
dsdsdy
ppFx
⋅⋅⋅=
⋅−
⋅−−+−=
∫
∫+
0
2
0
0
2
0
'
2
Equation 3.2–73
With this the two-dimensional hydrodynamic horizontal force due to sway oscillations can bewritten as follows:
( ) ( )( )tNtMbg
F ax ⋅⋅−⋅⋅⋅
⋅⋅⋅−= ωω
πηρ
sincos 000'
in which:
( ) ( ) ( ) ( )( )
( ) ( )
( ) ( ) ( )( ) ( )∑ ∑∑
∑
∫ ∑
= = =−−
−
=+
=−
⋅⋅−−+
−⋅−⋅⋅−+
⋅+⋅⋅−⋅⋅
+
⋅⋅−⋅⋅−⋅−⋅⋅−=
M
m
N
n
N
iinm
m
a
b
N
mmm
m
a
N
nn
nsB
a
aanim
inQ
amQ
dnanM
1 0 012122222
1
1122
2
0 01200
12221212
1
1214
12sin1211
σξ
σπ
θθθφσ
π
and 0N as obtained from this expression above for 0M , by replacing there ( )θφ sB0 by
( )θφ cB0 and mQ2 by mP2 .
Equation 3.2–74
For the determination of 0M and 0N , it is required that NM ≥ .
With Equation 3.2–74 in some other format:
( ) ( )( )
0
0
000'
cos
sin
sincos
Qx
Px
tNtMbg
F
ba
a
ba
a
ax
⋅⋅⋅
−=
⋅⋅⋅
−=
−+⋅⋅−−+⋅⋅⋅⋅⋅⋅−
=
ξπηε
ξπηε
εεωεεωπ
ηρ
Equation 3.2–75
the two-dimensional hydrodynamic horizontal force can be resolved into components in phaseand out phase with the horizontal displacement of the cylinder:
( ) ( ) ( ) ( ) εωεω
ξπηρ
+⋅⋅⋅−⋅++⋅⋅⋅+⋅
⋅⋅⋅⋅⋅⋅
=
tQNPMtPNQM
xbg
Fab
ax
sincos 00000000
2
20'
Equation 3.2–76
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
85
This hydrodynamic vertical force can also be written as:
( ) ( )εωωεωω +⋅⋅⋅⋅++⋅⋅⋅⋅=
⋅−⋅−=
txNtxM
xNxMF
aa
x
sincos '22
2'22
'22
'22
' &&&
Equation 3.2–77
in which:'
22M 2-D hydrodynamic mass coefficient of sway'
22N 2-D hydrodynamic damping coefficient of swayWhen using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before in Equation 3.2–70, the two-dimensional hydrodynamic mass and dampingcoefficients of sway are given by:
ωρ
ρ
⋅+
⋅−⋅⋅
⋅=
+⋅+⋅
⋅⋅
=
20
20
00002
0'22
20
20
00002
0'22
2
2
QPQNPMb
N
QPPNQMb
M
Equation 3.2–78
The signs of these two coefficients are proper in both, the axes system of Tasai and the shipmotions ( )zyxO ,, co-ordinate system.
The energy delivered by the exciting forces should be equal to the energy radiated by thewaves, so:
( ) ( )2
1 2
0
'22
cgdtxxN
Ta
T
osc
osc ⋅⋅⋅=⋅⋅⋅⋅ ∫
ηρ&&
Equation 3.2–79
in which oscT is the period of oscillation.
With the relation for the wave speed ωgc = at deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced sway oscillation:
2
3
2'
22
⋅
⋅=
a
a
xg
Nη
ωρ
Equation 3.2–80
With this amplitude ratio the two-dimensional hydrodynamic damping coefficient of heave isalso given by:
ωπρ⋅
+⋅
⋅⋅= 2
02
0
20
2'
22
14 QP
bN
Equation 3.2–81
When comparing this expression for '22N with the expression found before, the following
energy balance relation is found:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
86
2
2
0000
π=⋅−⋅ QNPM
Equation 3.2–82
3.2.2.3 Coupling of Sway into Roll
In the case of a sway oscillation generally a roll moment is produced. The hydrodynamicpressure is skew-symmetric in θ .The two-dimensional hydrodynamic moment acting on the cylinder in the clockwise directioncan be found by integrating the roll component of the hydrodynamic pressure on the surfaceS of the cylinder:
( ) ( )
( ) θθθ
θ
θθ
π
π
dddy
yddx
xp
dsdsdy
ydsdx
xppM R
⋅
⋅+⋅⋅⋅−=
⋅
−
⋅++
⋅−⋅−−+=
∫
∫+
00
00
2
0
00
00
2
0
'
2
Equation 3.2–83
With this the two-dimensional hydrodynamic roll moment due to sway oscillations can bewritten as follows:
( ) ( )( )tXtYbg
M RRa
R ⋅⋅−⋅⋅⋅⋅⋅⋅
= ωωπ
ηρsincos
20'
in which:
( ) ( ) ( ) ( )( )
( ) ( ) ( )( ) ( )
( )( ) ( )
( ) ( )∑
∑ ∑
∑ ∑
∑ ∑∑
∫ ∑∑
=−
= +=−+−−−−
=
−
=−−+−−−
= = =−−
= =−−
+
⋅⋅⋅
−−⋅−+−−
+
⋅⋅⋅
−−⋅−−+−
+
⋅⋅−
⋅
⋅+
⋅⋅−−+
−⋅−⋅⋅−⋅
⋅+
⋅⋅−⋅⋅⋅−⋅−⋅⋅⋅
=
N
mmN
n
N
nmiinmin
N
mn
mn
iinmin
mm
a
b
M
m
N
n
N
iinm
m
a
N
n
N
iin
insB
a
R
aaain
iinm
aaain
iinm
Q
aainm
iniQ
dinaaiY
1
012221212
012221212
2
3
1 0 012122222
2
0 0 0121202
22121222
22121222
1
8
2212
22121
2
1
22sin1212
1
σξπ
σ
θθθφσ
π
and RX as obtained from this expression above for RY , by replacing there ( )θφ sB0 by
( )θφ cB0 and mQ2 by mP2 .
Equation 3.2–84
With Equation 3.2–84 in some other format:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
87
( ) ( )( )
0
0
20'
cos
sin
sincos
Qx
Px
tXtYbg
M
ba
a
ba
a
RRa
R
⋅⋅⋅
−=
⋅⋅⋅
−=
−+⋅⋅−−+⋅⋅⋅⋅⋅⋅
=
ξπηε
ξπηε
εεωεεωπ
ηρ
Equation 3.2–85
the two-dimensional hydrodynamic roll moment can be resolved into components in phaseand out phase with the lateral displacement of the cylinder:
( ) ( ) ( ) ( )( )εωεωξπ
ηρ
+⋅⋅⋅−⋅++⋅⋅⋅+⋅
⋅⋅⋅
⋅⋅⋅−=
tQXPYtPXQY
xbg
M
RRRR
ab
aR
sincos 0000
2
220'
Equation 3.2–86
This hydrodynamic roll moment can also be written as:
( ) ( )εωωεωω +⋅⋅⋅⋅++⋅⋅⋅⋅=
⋅−⋅−=
txNtxM
xNxMM
aa
R
sincos '42
2'42
'42
'42
' &&&
Equation 3.2–87
in which:'
42M 2-D hydrodynamic mass coupling coefficient of sway into roll'
42N 2-D hydrodynamic damping coupling coefficient of sway into roll
When using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before, the two-dimensional hydrodynamic mass and damping coupling coefficients ofsway into roll in Tasai's axes system are given by:
ωρ
ρ
⋅+
⋅−⋅⋅
⋅−=
+⋅+⋅
⋅⋅−
=
20
20
003
0'42
20
20
003
0'42
2
2
QPQXPYb
N
QPPXQYb
M
RR
RR
Equation 3.2–88
In the ship motions ( )zyxO ,, co-ordinate system these two coupling coefficients will changesign.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
88
3.2.3 Roll Motions
The determination of the hydrodynamic coefficients of a rolling cross section of a ship in deepand still water at zero forward speed, is based here on work published by Tasai [1961] for theLewis method. Starting points for the derivation these coefficients here are the velocitypotentials and the conjugate stream functions of the fluid as they have been derived by Tasaiand also by de Jong [1973].Suppose an infinite long cylinder in the surface of a fluid, of which a cross section is given inFigure 3.2–1. The cylinder is forced to carry out a simple harmonic roll motion about theorigin O with a frequency of oscillation ω and small amplitude of displacement aβ :
( )γωββ +⋅⋅= ta cos
Equation 3.2–89
in which γ is a phase angle.Respectively, the angular velocity and acceleration of the cylinder are:
( )γωβωβ +⋅⋅⋅−= ta sin& and ( )γωβωβ +⋅⋅⋅−= ta cos2&&
Equation 3.2–90
This forced angular oscillation of the cylinder causes a surface disturbance of the fluid.Because the cylinder is supposed to be infinitely long, the generated waves will be two-dimensional. These waves travel away from the cylinder and a stationary state is rapidlyattained.
Two kinds of waves will be produced:• A standing wave system, denoted here by subscript A .
The amplitudes of these waves decrease strongly with the distance to the cylinder.• A regular progressive wave system, denoted here by subscript B .
These waves dissipate energy. At a distance of a few wavelengths from the cylinder, thewaves on each side can be described by a single regular wave train. The wave amplitude atinfinity aη is proportional to the amplitude of oscillation of the cylinder aβ , provided thatthis amplitude is sufficiently small compared with the radius of the cylinder and the wavelength is not much smaller than the diameter of the cylinder.
The two-dimensional velocity potential of the fluid has to fulfil the following requirements:
1. The velocity potential must satisfy to the equation of Laplace:
02
2
2
22 =
∂Φ∂
+∂
Φ∂=Φ∇
yx
Equation 3.2–91
2. Because the sway motion of the fluid is not symmetrical about the y -axis, this velocitypotential has the following anti-symmetric relation:
( ) ( )yxyx ,, +Φ−=−Φ
Equation 3.2–92
3. The linearised free surface condition in deep water is expressed as follows:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
89
02
=∂Φ∂
+Φ⋅yg
ω for:
2sB
x ≥ and 0=y
Equation 3.2–93
In consequence of the conformal mapping, this free surface condition can be written as:
( ) ( ) 0120
1212 =
∂Φ∂
±⋅⋅−⋅Φ⋅ ∑=
⋅−−− θσ
ξ αN
n
nn
a
b ean for: 0≥α and 2πθ ±=
in which:
sa
b Mg
⋅=2ω
σξ
or gb
b ⋅⋅
=2
02ωξ (non-dimensional frequency squared)
Equation 3.2–94
From the definition of the velocity potential follows the boundary condition on the surface ofthe cylinder S for 0=α :
( )sr
rn ∂
∂⋅⋅=
∂Φ∂ 0
00 βθ &
Equation 3.2–95
in which n is the outward normal of the cylinder surface S and 0r is the radius from theorigin to the surface of the cylinder.Using the stream function Ψ , this boundary condition on the surface of the cylinder ( 0=α )reduces to:
( )
+∂∂⋅=
∂Ψ∂−
2
20
200 yx
ssβθ &
Equation 3.2–96
Integration results into the following requirement for the stream function on the surface of thecylinder:
( ) ( ) ( )tCyx ++⋅−=Ψ 20
200 2
βθ&
Equation 3.2–97
in which ( )tC is a function of time only.
The vertical oscillation at the intersection of the surface of the cylinder and the waterline isdefined by:
( )γωχβχ +⋅⋅=⋅= tb
a sin20
Equation 3.2–98
When defining:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
90
( )( )
( ) ( )( )
( ) ( )( ) 2
012
2
012
20
20
20
12cos11
12sin11
2
⋅−⋅⋅−⋅++
⋅−⋅⋅−⋅−=
+=
∑
∑
=−
=−
N
nn
n
a
N
nn
n
a
na
na
byx
θσ
θσ
θµ
Equation 3.2–99
the stream function on the surface of the cylinder is given by:
( ) ( ) ( )tCb
+⋅⋅−=Ψ θµχθ40
0 &
Equation 3.2–100
For the standing wave system a velocity potential and a stream function satisfying to theequation of Laplace, the non-symmetrical motion of the fluid and the free surface conditionhas to be found.
The following set of velocity potentials, as they are given by Tasai [1961] and de Jong [1973],fulfil these requirements:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Φ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαφωθαφ
ωπη
in which:( ) ( ) ( )( )
( ) ( ) ( )( )∑=
⋅+−−
⋅+−
⋅+⋅⋅⋅
+−
⋅−⋅−
⋅+⋅+=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
2212
122
22sin2212
1
12sin,
θσξ
θθαφ
α
α
Equation 3.2–101
The set of conjugate stream functions is expressed as:
( ) ( ) ( ) ( )
⋅⋅⋅+⋅⋅⋅⋅
⋅⋅
=Ψ ∑∑∞
=
∞
= 122
122 sin,cos,
mmAm
mmAm
aA tQtP
gωθαψωθαψ
ωπη
in which:( ) ( ) ( )( )
( ) ( ) ( )( )∑=
⋅+−−
⋅+−
⋅+⋅⋅⋅
+−
⋅−⋅+
⋅+⋅−=N
n
nmn
n
a
b
mmA
nmeanm
n
me
0
2212
122
22cos2212
1
12cos,
θσξ
θθαψ
α
α
Equation 3.2–102
These sets of functions tend to zero as α tends to infinity.In these expressions the magnitudes of the mP2 and the mQ2 series follow from the boundaryconditions as will be explained further on.
Another requirement is a diverging wave train for α goes to infinity. It is therefore necessaryto add a stream function, satisfying the equation of Laplace, the non-symmetrical motion and
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
91
the free surface condition, representing such a train of waves at infinity. For this, a functiondescribing a two-dimensional horizontal doublet at the origin O is chosen.Tasai [1961] and de Jong [1973] gave the velocity potential of the progressive wave systemas:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωφωφωπη
sin,cos,
in which:( )
( ) ( ) ( )
( )22
022
sincoscos
sin
yxx
dkek
ykkykxej
xej
xkyBs
yBc
+⋅+
⋅⋅+
⋅⋅+⋅⋅−⋅⋅⋅+=⋅
⋅⋅⋅−=⋅
∫∞
⋅−⋅−
⋅−
ν
νννπφ
νπφ
ν
ν
while:1+=j for: 0>x1−=j for: 0<x
g
2ων = (wave number for deep water)
Equation 3.2–103
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Φ ωθαφωθαφωπη
sin,cos,
Equation 3.2–104
The conjugate stream function is given by:
( ) ( ) ( ) ( ) tyxtyxg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωψωψωπη
sin,cos,
in which:( )
( ) ( ) ( )
( )22
022
cossinsin
cos
yxy
dkek
ykkykxe
xe
xkyBs
yBc
+−
⋅⋅+
⋅⋅−⋅⋅+⋅⋅⋅+=
⋅⋅⋅+=
∫∞
⋅−⋅−
⋅−
ν
νννπψ
νπψ
ν
ν
Equation 3.2–105
Changing the parameters provides:
( ) ( ) ( ) ( ) ttg
BsBca
B ⋅⋅+⋅⋅⋅⋅⋅
=Ψ ωθαψωθαψωπη
sin,cos,
Equation 3.2–106
When calculating the integrals in the expressions for Bsψ and Bcψ numerically, theconvergence is very slowly.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
92
Power series expansions, as given by Porter [1960], can be used instead of these last integralsover k . Summations in these expansions converge much faster than the numeric integrationprocedure. This has been showed for the sway case.
The total velocity potential and stream function to describe the waves generated by a swayingcylinder are:
BA
BA
Ψ+Ψ=ΨΦ+Φ=Φ
Equation 3.2–107
So the velocity potential and the conjugate stream function are expressed by:
( )( ) ( ) ( )
( ) ( ) ( )
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑
∑
∑
∞
=
∞
=
∞
=
∞
=
tQ
tPg
tQ
tPg
mmAmBs
mmAmBc
a
mmAmBs
mmAmBc
a
ωθαψθαψ
ωθαψθαψ
ωπηθα
ωθαφθαφ
ωθαφθαφ
ωπηθα
sin,,
cos,,
,
sin,,
cos,,
,
122
122
122
122
Equation 3.2–108
When putting 0=α , the stream function is equal to the expression found before in Equation3.2–100 from the boundary condition on the surface of the cylinder:
( )( ) ( ) ( )
( ) ( ) ( )
( ) ( )tCb
tQ
tPg
mmAmsB
mmAmcB
a
+⋅⋅−=
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Ψ
∑
∑∞
=
∞
=
θµχ
ωθψθψ
ωθψθψ
ωπηθ
4
sin
cos
0
10220
10220
0
&
in which:( ) ( )( )
( ) ( )( )∑=
−
⋅+⋅⋅
+−
⋅−⋅+
⋅+−=N
nn
n
a
b
mA
nmanm
n
m
012
02
22cos2212
1
12cos
θσξ
θθψ
Equation 3.2–109
In this expression, ( )θψ cB 0 and ( )θψ sB 0 are the values of ( )θαψ ,Bc and ( )θαψ ,Bs at thesurface of the cylinder, so for 0=α .
So for each θ , the following equation has been obtained:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
93
( ) ( ) ( )
( ) ( ) ( )( ) ( )tC
gb
tQ
tP
a
mmAmsB
mmAmcB
*0
10220
10220
4sin
cos
+⋅⋅⋅⋅⋅⋅−=
⋅⋅
⋅++
⋅⋅
⋅++
∑
∑∞
=
∞
= θµη
ωπχωθψθψ
ωθψθψ&
Equation 3.2–110
When putting 2πθ = , so at the intersection of the surface of the cylinder with the free
surface of the fluid where ( ) 1=θµ , we obtain the constant ( )tC* :
( ) ( ) ( ) ( )
( ) ( ) ( )
a
mmAmsB
mmAmcB
gb
tQ
tPtC
ηωπχ
ωπψπψ
ωπψπψ
⋅⋅⋅⋅
⋅+
⋅⋅
⋅++
⋅⋅
⋅+=
∑
∑∞
=
∞
=
4
sin22
cos22
0
10220
10220
*
&
in which:
( ) ( ) ∑=
−
⋅
+−
⋅−⋅=N
nn
m
a
bmA a
nmn
01202 22
1212
σξπψ
Equation 3.2–111
A substitution of ( )tC* in the equation for each θ-value, results for any θ-value less than2π into the following equation:
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )
( ) 14
sin22
cos22
0
10202200
10202200
−⋅⋅⋅⋅⋅
⋅−
=
⋅⋅
−⋅+−+
⋅⋅
−⋅+−+
∑
∑∞
=
∞
=
θµη
ωπχ
ωπψθψπψθψ
ωπψθψπψθψ
a
mmAmAmSBsB
mmAmAmcBcB
gb
tQ
tP
&
Equation 3.2–112
The right hand side of this equation can be written as:
( ) ( ) ( )
( ) ( ) ( ) tQtP
tg
bb
a
a
a
⋅⋅+⋅⋅⋅−=
+⋅⋅⋅⋅⋅
−⋅−=−⋅⋅⋅⋅⋅
⋅−
ωωθµ
γωξηχπθµθµ
ηωπχ
sincos1
sin2
114
00
0&
in which:
( )γξηχπ
sin20 ⋅⋅
⋅⋅
= ba
aP and ( )γξηχπ
cos20 ⋅⋅
⋅⋅
= ba
aQ
Equation 3.2–113
This provides for each θ-value less than 2π a set of two equations with the unknownparameters mP2 and mQ2 :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
94
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ∑
∑∞
=
∞
=
⋅+⋅−=−
⋅+⋅−=−
122000
122000
12
12
mmmsBsB
mmmcBcB
QfQ
PfP
θθµπψθψ
θθµπψθψ
in which:( ) ( ) ( )202022 πψθψθ mAmAmf +−=
Equation 3.2–114
These equations can also be written as:
( ) ( ) ( )
( ) ( ) ( ) ∑
∑∞
=
∞
=
⋅=−
⋅=−
02200
02200
2
2
mmmsBsB
mmmcBcB
Qf
Pf
θπψθψ
θπψθψ
in which:for :0=m ( ) ( ) 10 −= θµθf
for 0>m : ( ) ( ) ( )202022 πψθψθ mAmAmf +−=
Equation 3.2–115
The series in these two sets of equations converges uniformly with an increasing value of m .For practical reasons the maximum value of m is limited to M , for instance 10=M .
Each θ-value less than 2π will provide an equation for the mP2 and mQ2 series. For a lot of
θ-values, the best fit values of mP2 and mQ2 are supposed to be those found by means of aleast squares method. Note that at least 1+M values of θ , less than 2π , are required tosolve these equations.
Another favourable method is to multiply both sides of the equations with θ∆ . Then thesummation over θ can be replaced by integration. Herewith, two sets of 1+M equationshave been obtained, one set for mP2 and one set for mQ2 :
( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( )∫∑ ∫
∫∑ ∫
⋅⋅−=
⋅⋅⋅
⋅⋅−=
⋅⋅⋅
=
=
2
0
2000
2
0
222
2
0200
0
2
0222
2
2
ππ
ππ
θθπψθψθθθ
θθπψθψθθθ
dfdffQ
dfdffP
nsBsB
M
mnmm
ncBcB
M
mnmm
for: Mn ,...0=
Equation 3.2–116
Now the mP2 and mQ2 series can be solved by a numerical method and with these values, the
coefficients 0P and 0Q are known now and from the definition of these coefficients followsthe amplitude ratio of the radiated waves and the forced sway oscillation:
20
202 QP
b
a
a
+⋅
⋅=
ξπχη
Equation 3.2–117
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
95
With the solved mP2 and mQ2 values, the velocity potential on the surface of the cylinder( 0=α ) is known too:
( )( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅++
⋅⋅
⋅++
⋅⋅⋅
=Φ
∑
∑
=
=
tQ
tPg
M
mmAmsB
M
mmAmcB
a
ωθφθφ
ωθφθφ
ωπηθ
sin
cos
10220
10220
0
in which:( ) ( )( )
( ) ( )( )∑=
−
⋅+⋅⋅
+−
⋅−⋅−
⋅++=N
nn
n
a
b
mA
nmanm
n
m
012
02
22sin2212
1
12sin
θσξ
θθφ
Equation 3.2–118
In this expression ( )θφ cB0 and ( )θφ sB0 are the values of ( )θαφ ,Bc and ( )θαφ ,Bs at the surfaceof the cylinder, so for 0=α .
3.2.3.1 Pressure Distribution During Roll Motions
Now the hydrodynamic pressure on the surface of the cylinder can be obtained from thelinearised equation of Bernoulli:
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
⋅⋅
⋅+−
⋅⋅
⋅++
⋅⋅⋅−
=
∂Φ∂
⋅−=
∑
∑∞
=
∞
=
tP
tQg
tp
mmAmcB
mmAmsB
a
ωθφθφ
ωθφθφ
πηρ
θρθ
sin
cos
10220
10220
0
Equation 3.2–119
It is obvious that this pressure is skew-symmetric in θ .
3.2.3.2 Roll Coefficients
The two-dimensional hydrodynamic moment acting on the cylinder in the clockwise directioncan be found by integrating the roll component of the hydrodynamic pressure on the surfaceS of the cylinder:
( ) ( )
( ) θθθ
θ
θθ
π
π
dddy
yddx
xp
dsdsdy
ydsdx
xppM R
⋅
⋅+⋅⋅⋅−=
⋅
−
⋅++
⋅−⋅−−+=
∫
∫+
00
00
2
0
00
00
2
0
'
2
Equation 3.2–120
With this the two-dimensional hydrodynamic moment due to roll oscillations can be written asfollows:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
96
( ) ( )( )tXtYbg
M RRa
R ⋅⋅−⋅⋅⋅⋅⋅⋅
= ωωπ
ηρsincos
20'
in which:
( ) ( ) ( ) ( )( )
( ) ( ) ( )( ) ( )
( )( ) ( )
( ) ( )∑
∑ ∑
∑ ∑
∑ ∑∑
∫ ∑∑
=−
= +=−+−−−−
=
−
=−−+−−−
= = =−−
= =−−
+
⋅⋅⋅
−−⋅−+−−
+
⋅⋅⋅
−−⋅−−+−
+
⋅⋅−
⋅
⋅−
⋅⋅−−+
−⋅−⋅⋅−⋅
⋅+
⋅⋅−⋅⋅⋅−⋅−⋅⋅⋅
=
M
mmN
n
N
nmiinmin
N
mn
mn
iinmin
mm
a
b
M
m
N
n
N
iinm
m
a
N
n
N
iin
insB
a
R
aaain
iinm
aaain
iinm
Q
aainm
iniQ
dinaaiY
1
012221212
012221212
2
3
1 0 012122222
2
0 0 0121202
22121222
22121222
1
8
2212
22121
2
1
22sin1212
1
σξπ
σ
θθθφσ
π
and RX as obtained from this expression above for RY , by replacing there ( )θφ sB0 by
( )θφ cB0 and mQ2 by mP2 .
Equation 3.2–121
These expressions are similar to those found before for the hydrodynamic roll moment due tosway oscillations.
With Equation 3.2–121 in some other format:
( ) ( )( )
0
0
20'
2cos
2sin
sincos
Qx
Px
tXtYbg
M
ba
a
ba
a
RRa
R
⋅⋅⋅
⋅=
⋅⋅⋅
⋅=
−+⋅⋅−−+⋅⋅⋅⋅⋅⋅
=
ξπηγ
ξπηγ
γγωγγωπ
ηρ
Equation 3.2–122
the two-dimensional hydrodynamic roll moment can be resolved into components in phaseand out phase with the angular displacement of the cylinder:
( ) ( ) ( ) ( ) γωγω
χξπηρ
+⋅⋅⋅−⋅++⋅⋅⋅+⋅
⋅⋅⋅
⋅⋅⋅⋅=
tQXPYtPXQY
bgM
RRRR
ab
aR
sincos
2
0000
2
220'
Equation 3.2–123
This hydrodynamic roll moment can also be written as:
( ) ( )γωβωγωβω
ββ
+⋅⋅⋅⋅++⋅⋅⋅⋅=
⋅−⋅−=
tNtM
NMM
aa
R
sincos '44
2'44
'44
'44
' &&&
Equation 3.2–124
in which:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
97
'44M 2-D hydrodynamic mass moment of inertia coefficient of roll
'44N 2-D hydrodynamic damping coefficient of roll
When using also the amplitude ratio of the radiated waves and the forced roll oscillation,found before, the two-dimensional hydrodynamic mass and damping coefficients of roll inTasai's axes system are given by:
ωρ
ρ
⋅+
⋅−⋅⋅
⋅+=
+⋅+⋅
⋅⋅+
=
20
20
004
0'44
20
20
004
0'44
8
8
QPQXPYb
N
QPPXQYb
M
RR
RR
Equation 3.2–125
The signs of these two coefficients are proper in both, the axes system of Tasai and the shipmotions ( )zyxO ,, co-ordinate system.
The energy delivered by the exciting moments should be equal to the energy radiated by thewaves, so:
( ) ( )2
1 2
0
'44
cgdtN
Ta
T
osc
osc ⋅⋅⋅=⋅⋅⋅⋅ ∫
ηρββ &&
Equation 3.2–126
in which oscT is the period of oscillation.
With the relation for the wave speed ωgc = in deep water, follows the relation between thetwo-dimensional heave damping coefficient and the amplitude ratio of the radiated waves andthe forced sway oscillation:
2
3
2'
44
⋅
⋅=
a
agN
βη
ωρ
Equation 3.2–127
With this amplitude ratio the two-dimensional hydrodynamic damping coefficient of heave isalso given by:
ωπρ⋅
+⋅
⋅⋅= 2
02
0
40
2'
44
164 QP
bN
Equation 3.2–128
When comparing this expression for '44N with the expression found before, the following
energy balance relation is found:
8
2
00
π=⋅−⋅ QXPY RR
Equation 3.2–129
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
98
3.2.3.3 Coupling of Roll into Sway
In the case of a roll oscillation generally a sway force produced too. The hydrodynamicpressure is skew-symmetric in θ .The two-dimensional hydrodynamic lateral force, acting on the cylinder in the direction of thex -axis, can be found by integrating the horizontal component of the hydrodynamic pressureon the surface S of the cylinder:
( ) ( )
( ) θθ
θ
θθ
π
π
dddy
p
dsdsdy
ppFx
⋅⋅⋅=
⋅−
⋅−−+−=
∫
∫+
0
2
0
0
2
0
'
2
Equation 3.2–130
With this the two-dimensional hydrodynamic horizontal force due to sway oscillations can bewritten as follows:
( ) ( )( )tNtMbg
F ax ⋅⋅−⋅⋅⋅
⋅⋅⋅−= ωω
πηρ
sincos 000'
in which:
( ) ( ) ( ) ( )( )
( ) ( )
( ) ( ) ( )( ) ( )∑ ∑∑
∑
∫ ∑
= = =−−
−
=+
=−
⋅⋅−−+
−⋅−⋅⋅−+
⋅+⋅⋅−⋅⋅
+
⋅⋅−⋅⋅−⋅−⋅⋅−=
M
m
N
n
N
iinm
m
a
b
N
mmm
m
a
N
nn
nsB
a
aanim
inQ
amQ
dnanM
1 0 012122222
1
1122
2
0 01200
12221212
1
1214
12sin1211
σξ
σπ
θθθφσ
π
and 0N as obtained from this expression above for 0M , by replacing there ( )θφ sB0 by
( )θφ cB0 and mQ2 by mP2 .
Equation 3.2–131
For the determination of 0M and 0N , it is required that NM ≥ .
With Equation 3.2–131 in some other format:
( ) ( )( )
0
0
000'
2cos
2sin
sincos
Qx
Px
tNtMbg
F
ba
a
ba
a
ax
⋅⋅⋅
⋅=
⋅⋅⋅
⋅=
−+⋅⋅−−+⋅⋅⋅⋅⋅⋅−
=
ξπηγ
ξπηγ
γγωγγωπ
ηρ
Equation 3.2–132
the two-dimensional hydrodynamic horizontal force can be resolved into components in phaseand out phase with the horizontal displacement of the cylinder:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
99
( ) ( ) ( ) ( )( )γωγωχξπ
ηρ
+⋅⋅⋅−⋅++⋅⋅⋅+⋅
⋅⋅⋅
⋅⋅⋅⋅−=
tQNPMtPNQM
bgF
ab
ax
sincos
2
00000000
2
20'
Equation 3.2–133
This hydrodynamic vertical force can also be written as:
( ) ( )γωβωγωβω
ββ
+⋅⋅⋅⋅++⋅⋅⋅⋅=
⋅−⋅−=
tNtM
NMF
aa
x
sincos '24
2'24
'24
'24
' &&&
Equation 3.2–134
in which:'
24M 2-D hydrodynamic mass coupling coefficient of roll into sway'
24N 2-D hydrodynamic damping coupling coefficient of roll into sway
When using also the amplitude ratio of the radiated waves and the forced sway oscillation,found before, the two-dimensional hydrodynamic mass and damping coupling coefficients ofroll into sway are given by:
ωρ
ρ
⋅+
⋅−⋅⋅
⋅−=
+⋅+⋅
⋅⋅−
=
20
20
00003
0'24
20
20
00003
0'24
8
8
QPQNPMb
N
QPPNQMb
M
Equation 3.2–135
In the ship motions ( )zyxO ,, co-ordinate system these two coupling coefficients will changesign.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
100
3.2.4 Low and High Frequencies
The potential coefficients for very small and very large frequencies in the ship motions( )zyxO ,, co-ordinate system have been given in the following subsections.
3.2.4.1 Near-Zero Frequency Coefficients
The 2-D hydrodynamic mass coefficient for sway of a Lewis cross section is given by Tasai[1961] as:
( ) ( ) 23
21
2
31
'22 31
120 aa
aaD
M s ⋅+−⋅
+−
⋅⋅
=→πρω
Equation 3.2–136
The 2-D hydrodynamic mass coupling coefficient of sway into roll of a Lewis cross section isgiven by Grim [1955] as:
( ) ( )
( ) 23
21
233
2331311
31
'22
'42
31712
54
53
54
1
1316
00
aa
aaaaaaaa
aaD
MM s
⋅+−
⋅−⋅+
⋅+⋅−⋅+−⋅
⋅+−
⋅⋅
⋅→−=→π
ωω
Equation 3.2–137
In Tasai's axes system, '42M will change sign.
The 2-D hydrodynamic mass coefficient of heave of a Lewis cross section goes to infinite, so:( ) ∞=→ 0'
33 ωM
Equation 3.2–138
The 2-D hydrodynamic mass moment of inertia coefficient of roll of a Lewis cross section isgiven by Grim [1955] as:
( ) ( )
( ) ( )
⋅++⋅⋅⋅++⋅
⋅
++⋅
⋅⋅
=→
23331
23
21
4
31
'44
916
198
1
1216
0
aaaaaa
aaB
M s
πρω
Equation 3.2–139
The 2-D hydrodynamic mass coupling coefficient of roll into sway of a Lewis cross section isgiven by Grim [1955] as:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
101
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) 23331
23
21
2313331311
31'44
'24
916
198
1
712
154
153
11
16
00
aaaaaa
aaaaaaaaa
Daa
MMs
⋅++⋅⋅⋅++⋅
⋅−−⋅⋅++⋅⋅⋅++⋅−⋅
⋅+−
⋅⋅→−=→πωω
Equation 3.2–140
In Tasai's axes system, '24M will change sign.
All potential damping values for zero frequency will be zero:( )( ) 00
00'
42
'22
=→
=→
ω
ω
N
N ( ) 00'
33 =→ωN ( )( ) 00
00'
24
'44
=→
=→
ω
ω
N
N
Equation 3.2–141
3.2.4.2 Infinite Frequency Coefficients
The 2-D hydrodynamic mass coefficient of sway of a Lewis cross section is given byLandweber and de Macagno [1957, 1959] as:
( ) ( )
⋅++−⋅
+−
⋅⋅
=∞→ 23
231
2
31
'22 3
161
12
aaaaa
DM s
πρω
Equation 3.2–142
The 2-D hydrodynamic mass coefficient of heave of a Lewis cross section is given by Tasai[1959] as:
( ) ( ) ( )( )23
21
2
31
'33 31
122aa
aaB
M s ⋅++⋅
++⋅
⋅⋅
=∞→πρω
Equation 3.2–143
The 2-D hydrodynamic mass moment of inertia coefficient of roll of a Lewis cross section isgiven by Kumai [1959] as:
( ) ( ) ( )( )23
23
21
4
31
'44 21
12aaa
aaB
M s ⋅++⋅⋅
++⋅
⋅⋅=∞→ πρω
Equation 3.2–144
Information about the 2-D hydrodynamic mass coupling coefficients between sway and rollof a Lewis cross section could not be found in literature, so:
( )( ) ?
?'
24
'42
=∞→
=∞→
ω
ω
M
M
Equation 3.2–145
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
102
All potential damping values for infinite frequency will be zero:
( )( ) 0
0'
42
'22
=∞→
=∞→
ω
ω
N
N ( ) 0'
33 =∞→ωN ( )( ) 0
0'
24
'44
=∞→
=∞→
ω
ω
N
N
Equation 3.2–146
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
103
3.3 Potential Theory of Keil
In this section, the determination of the hydrodynamic coefficients of a heaving, swaying androlling cross section of a ship in shallow at zero forward speed is based on work published byKeil [1974]. His method is based on Lewis conformal mapping of the ships' cross sections tothe unit circle and the shallow water potential theory.Journée [2001] has given a few comparisons of predicted and measured data on verticalmotions at various water depths. Recently, Vantorre and Journée [2003] have given moreextended comparisons of computed results by Keil’s theory with model test data on verticalmotions of a slender and a full ship, sailing at very shallow water.
For a significant part, the detailed description of the shallow water potential theory in thisSection is simply a translation of Keil’s original 1974 German report into the Englishlanguage. However, it has been supplemented with some numerical improvements andoutcomes of discussions with the author in the early eighties. The theory has been presentedhere in a layout, more or less as used in the computer code SEAWAY.
3.3.1 Notations of Keil
Keil’s notations have been maintained here as far as possible:
a Lewis coefficient
indexA source strength
indexA amplitude ratiob Lewis coefficientB breadth of bodyc wave velocity
indexC non-dimensional force or moment
indexE non-dimensional exciting force or moment
indexF hydrodynamic forceg acceleration of gravityG function (real part)h water depthH function (imaginary part)
indicesH fictive moment armHT water depth - draft ratio
"I hydrodynamic moment of inertia
xk wave number in x -direction
yk wave number in y -direction"m hydrodynamic mass
indicesM hydrodynamic moment
indicesN hydrodynamic damping coefficientp pressure
22 yxr += polar coordinate
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
104
t time or integer valueT draughtU velocity amplitude of horizontal oscillationV velocity amplitude of vertical oscillation
xA cross sectional areazyx ,, earth-bounded co-ordinates
indicesY transfer functionsγ Euler constant (= 0.57722)ε phase shiftζ wave amplitudeθ polar co-ordinate or pitch amplitudeλ wavelengthµ wave direction
g2ων = wave number at deep waterλπν 20 = wave number
ρ density of waterϕ roll angle
indicesΦ time-dependent potential
indicesφ part of potential
indicesΨ time-dependent stream function
indicesψ part of stream function ω circular frequency of oscillation
In here, the indices – being used by Keil - are:
E related to excitationH horizontal or related to horizontal motionsj imaginary partn numbering of potential partsQ related to transverse motionsr real partR related to roll motionsV related to vertical or vertical motionsW related to waves
3.3.2 Basic Assumptions
Figure 3.3–1 shows the co-ordinate system as used by Keil and maintained here.The potentials of the incoming waves have been described in Appendix I of this Section.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
105
Figure 3.3–1: Keil’s axes system
The wave number, λπν 20 = , follows from:
[ ]hh
g⋅⋅=
⋅⋅
⋅⋅
== 00
2
tanh2
tanh2 νν
λπ
λπων
Equation 3.3–1
The fluid is supposed to be incompressible and inviscid. The flow caused by the oscillatingbody in the surface of this fluid can be described by a potential flow. The problem will belinearised, i.e., contributions of second and higher order in the definition of the boundaryconditions will be ignored. Physically, this yields an assumption of small amplitude motions.The earth-bounded axes system of the sectional contour is given in Figure 3.3–2-a.
Figure 3.3–2: Definition of sectional contour
Velocities are positive if they are directed in the positive co-ordinate direction:
yvy
=∂Φ∂
zvz
=∂Φ∂
The value of the stream function increases when - going in the positive direction - the flowgoes in the negative y -direction:
21 Ψ<Ψ →
∂Ψ∂
+=∂Φ∂
zy
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
106
43 Ψ>Ψ →
∂Ψ∂
+=∂Φ∂
∂Ψ∂
−=∂Φ∂
∂Ψ∂
−=∂Φ∂
ns
sn
yz
Equation 3.3–2
For the imaginary parts, the symbols i and j have been used: i for geometrical variables(potential and stream function) and j for functions of time.
3.3.3 Vertical Motions
3.3.3.1 Boundary Conditions
The two-dimensional velocity potential of the fluid has to fulfil the following requirements:
1. The fluid is incompressible and the velocity potential must satisfy to the ContinuityCondition and the Equation of Laplace:
02
2
2
22 =
∂Φ∂
+∂
Φ∂=Φ∇
zy
Equation 3.3–3
2. The linearised free surface condition follows from the condition that the pressure at thefree surface is not time-depending but constant:
02
2
=
∂
Φ∂−
∂Φ∂
⋅⋅=∂∂
tzg
tp ρ for:
2B
y ≥ and 0=z
from which follows:
02
=∂Φ∂
+Φ⋅zg
ω or 0=
∂Φ∂
+Φ⋅z
ν for: 2B
y ≥ and 0=z
Equation 3.3–4
3. The seabed is impervious, so the vertical fluid velocity at hz = is zero:
0=∂Φ∂z
for: hz =
Equation 3.3–5
4. The harmonic oscillating cylinder produces a regular progressive wave system, travellingaway from the cylinder, which fulfils the Sommerfeld radiation condition:
0ImRelim 0 =
Φ⋅−Φ
∂∂
⋅∞→
νy
yy
Equation 3.3–6
In here, λπν 20 = is the wave number of the radiated wave.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
107
5. The oscillating cylinder is impervious too; thus at the surface of the body is the fluidvelocity equal to the body velocity, see Figure 3.3–2-b. This yields that the boundaryconditions on the surface of the body are given by:
body
body
bodynbody
dsd
dsdz
ydsdy
z
vn
Ψ−=
⋅∂Φ∂
−⋅∂Φ∂
=
=∂Φ∂
Equation 3.3–7
Two cases have to be distinguished:
a) The hydromechanical loads, which have to be obtained for the vertically oscillatingcylinder in still water with a vertical velocity equal to:
tjeVV ⋅⋅⋅= ω
The boundary condition on the surface of the body becomes:
tj
bodybody
edsdy
Vdsd ⋅⋅⋅⋅=Ψ
− ω
or:( ) CyeVtzy
bodytj
body+⋅⋅−=Ψ ⋅⋅ω,,
Equation 3.3–8
b) The wave loads, which have to be obtained for the restrained cylinder in regular wavesfrom the incoming undisturbed wave potential WΦ and the diffraction potential SΦ :
body
SW
body
SSWW
body
SW
dsd
dsd
dsdz
ydsdy
zdsdz
ydsdy
z
nn
Ψ+
Ψ−=
⋅∂Φ∂
−⋅∂Φ∂
+⋅∂Φ∂
−⋅∂Φ∂
=
=∂Φ∂
+∂Φ∂
0
or:( ) ( )
body
WW
bodyWbodyS
dzy
dyz
tzydtzyd
⋅∂Φ∂
−⋅∂Φ∂
=
Ψ−=Ψ ,,,,
Equation 3.3–9
The stream function of an incoming wave - which travels in the negative y -direction, so090+=µ - is given in Appendix I of this Section by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
108
( ) [ ] [ ] [ ] zhzej ytjW ⋅⋅⋅−⋅⋅⋅
⋅⋅=Ψ ⋅+⋅⋅
000 coshtanhsinh0 ννννωζ νω
Equation 3.3–10
Because only vertical forces have to be determined, only the in y -symmetric part of thepotential and stream functions have to be considered. From this follows the boundarycondition on the surface of the body for beam waves, so wave direction 090+=µ :
( ) ( )
[ ] [ ] [ ] [ ]body
tj
bodyWbodyS
yzhze
tzytzy
⋅⋅⋅⋅⋅−⋅⋅⋅⋅
=
Ψ−=Ψ
⋅⋅0000 sinhcoshtanhsinh
,,,,
νννννωζ ω
Equation 3.3–11
In case of another wave direction, this problem becomes three-dimensional and a streamfunction can not be written. However, boundary condition (Equation 3.3–11) provides us a''quasi stream function'' sΨ
~, i.e. this is the amount of fluid which has to come out of the body
per unit of length, so that - in total - no fluid of the incoming wave enters into the body.This function can be used as an approximation of the problem:
( )
( )
( ) [ ] [ ] [ ]( )
( )( ) [ ] [ ] [ ]( )
⋅⋅⋅⋅−⋅⋅⋅⋅
⋅−⋅+
⋅⋅⋅−⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅=
⋅∂Ψ∂
−⋅∂Ψ∂
=Ψ
∫
∫∫
⋅⋅
1
1
0
1
00000
20
1001010
0
0111
coshtanhsinhsincos
sin1
coshtanhsinhsinsinsin
coscos
,,,~
y
tj
body
z
z
W
y
W
bodys
dyzhzy
zhzy
xe
dzy
dyz
tzyx
νννµν
µν
νννµνµ
µννωζ ω
Equation 3.3–12
3.3.3.2 Potentials
3.3.3.2.1 3-D Radiation Potential
Suppose a three-dimensional oscillating cylindrical body in previously still water. To find thepotential of the resulting fluid motions, an oscillating pressure p at the free surface will
replace this body. The unknown amplitude p of this pressure has to follow from the boundaryconditions.This pressure is not supposed to act over the full breadth of the body; it is supposed to act -over the full length L of the body - only over a small distance 2y∆ on both sides of 0=y ,so:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
109
( ) ( )
( ) 2 and 2 :for 0,,
:or
,,,,,
0
00
Lxyyzzyxp
ezzyxpjtzzyxp tj
≥∆≥==
⋅=⋅−== ⋅⋅ω
Equation 3.3–13
in which 0z is the z -co-ordinate of the fluid surface.
The resulting force P in the z -direction becomes:
( )
( )∫
∫ ∫+
−
+
−
∆+
∆−
∞<⋅=
⋅⋅==
2
2
'
2
2
2
20,,
L
L
L
L
y
y
dxxP
dxdyzzyxpP
Equation 3.3–14
The boundary condition in Equation 3.3–13 can be fulfilled by pressure amplitude( )0,, zzyxp = , which is found by a superposition of an infinite number of harmonic pressures.
From Equation 3.3–14 follows that the pressure amplitude ( )xp can be integrated, so aFourier series expansion follows from:
( ) ( ) ( )( )∫ ∫∞ +∞
∞−
⋅⋅−⋅⋅⋅=0
cos1
xx dkdxkpxp ξξξπ
Because the pressure amplitude p depends on two variables, the Fourier series expansion hasto be two-dimensional too:
( ) ( )
( )( ) ( )( ) xyxy dkdkddxkyk
pzzyxp
⋅⋅⋅⋅−⋅⋅−⋅
⋅⋅== ∫ ∫ ∫ ∫∞ ∞ +∞
∞−
+∞
∞−
ξηξη
ηξπ
coscos
,1
,,0 0
20
in which xk is the wave number in the x -direction and yk is the wave number in the y -direction.According to Equation 3.3–13, the pressure amplitude p disappears for 2yy ∆≥ and
2Lx ≥ , so for this pressure expression remains:
( ) ( )
( )( ) ( )( ) xyxy
L
L
y
y
dkdkddxkyk
pzzyxp
⋅⋅⋅⋅−⋅⋅−⋅
⋅⋅== ∫ ∫ ∫ ∫∞ ∞ +
−
∆+
∆−
ξηξη
ηξπ
coscos
,1,,0 0
2
2
2
220
It is assumed that the value of y∆ is small. This means that η remains small too. Thus, onecan safely suppose that:
( )( ) ( )ykyk yy ⋅≈−⋅ coscos ηwhich results in:
( ) ( ) ( )( ) ( )
( ) ( ) ( )( )∫∫ ∫
∫ ∫ ∫ ∫+
−
∞ ∞
∞ ∞ +
−
∆+
∆−
⋅⋅⋅−⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅−⋅⋅⋅⋅==
2
2
'
0 02
0 0
2
2
2
220
coscos1
coscos,1
,,
L
L
xyxy
xyyx
L
L
y
y
dkdkdxkPyk
dkdkykdxkdpzzyxp
ξξξπ
ξξηηξπ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
110
Equation 3.3–15
This pressure definition leads - as a start - to the following initial definition of the radiationpotential:
( ) ( ) ( )( ) ( )
( )ξ
ξω
ddkdkhkk
hzkk
ykxkkkCetzyx
yx
yx
yx
yxyxtj
r
⋅⋅⋅
⋅+
−⋅+
⋅⋅⋅−⋅⋅⋅=Φ ∫ ∫ ∫∞ ∞ ∞
⋅⋅
22
22
0 0 00
sinh
cosh
coscos,,,,
Equation 3.3–16
in which the function ( )yx kkC , is still unknown.
This expression in Equation 3.3–16 for the radiation potential fulfils the Equation of Laplace:
02
2
2
2
2
2
=∂
Φ∂+
∂Φ∂
+∂
Φ∂zyx
Now, the harmonic pressure at the free surface 1p can be obtained from an integration of the -with the Bernoulli Equation obtained - derivative to the time of the pressure:
( ) ( )( ) ( )
ξω
ξρω
ddkdkhkk
hkkkkg
ykxkkkCetp
yx
yx
yxyx
yxyxtj
⋅⋅⋅
⋅+
⋅+⋅+⋅−
⋅⋅⋅−⋅⋅⋅⋅=∂∂
∫ ∫ ∫∞ ∞ ∞
⋅⋅
22
22222
0 0 0
1
tanh
tanh
coscos,
Equation 3.3–17
The harmonic oscillating pressure is given by:( ) ( ) tjezzyxpjtzzyxp ⋅⋅⋅=⋅−== ω
0101 ,,,,,
Equation 3.3–18
and its amplitude becomes:
( ) ( ) ( )( ) ( )
ξω
ξωρ
ddkdkhkk
hkkkkg
ykxkkkCzzyxp
yx
yx
yxyx
yxyx
⋅⋅⋅
⋅+
⋅+⋅+⋅−
⋅⋅⋅−⋅⋅⋅== ∫ ∫ ∫∞ ∞ ∞
22
22222
0 0 001
tanh
tanh
coscos,,,
Equation 3.3–19
If this pressure amplitude 1p is supposed to be equal to the amplitude p , then combining
Equation 3.3–15 and Equation 3.3–19 provides the unknown function ( )yx kkC , :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
111
( ) ( ) ( ) ( )( )
( )
( ) ( ) ( )( )
ξν
ξω
ρ
ξξξπ
ddkdkhkk
hkkkk
xkkkCykg
zzyxp
dkdkdxkPykzzyxp
yx
yx
yxyx
xyxy
xy
L
L
xy
⋅⋅⋅
⋅+
⋅+⋅+−
⋅−⋅⋅⋅⋅⋅
=
==
⋅⋅⋅−⋅⋅⋅⋅==
∫ ∫∫
∫ ∫∫
∞ ∞∞
∞ +
−
∞
22
2222
0 00
01
0
2
2
'
020
tanh
tanh
cos,cos
,,
coscos1
,,
Equation 3.3–20
Comparing the two integrands provides:
( ) ( )( )
( ) ( )( ) ξξξπ
ξν
ωρξ
dxkP
dhkk
hkkkkgxkkkC
L
Lx
yx
yxyx
xyx
⋅−⋅⋅⋅
=⋅
⋅+
⋅+⋅+−
⋅⋅
⋅−⋅⋅
∫
∫
+
−
∞
2
2
'
2
22
2222
0
cos1
tanh
tanhcos,
or:
( ) ( )( )
( ) ( )( ) ξξξπρ
ω
νξξ
dxkPg
hkkkk
hkkdxkkkC
L
Lx
yxyx
yx
xyx
⋅−⋅⋅⋅⋅⋅
⋅
⋅+⋅+−
⋅+
=⋅−⋅⋅
∫
∫
+
−
∞
2
2
'
2
2222
22
0
cos
tanh
tanhcos,
Equation 3.3–21
When defining:
( ) ( )2
'
0 πρξωξ
⋅⋅⋅
=gP
A
and substituting Equation 3.3–21 in Equation 3.3–16 provides the radiation potential:
( ) ( ) ( )( )
( ) ( )ξ
ν
ξξω
ddkdkhkkkkhkk
ykhzkk
xkAetzyx
yx
yxyxyx
yyx
x
L
L
tjr
⋅⋅⋅
⋅+⋅+−
⋅+⋅
⋅⋅
−⋅+
⋅−⋅⋅⋅=Φ
∫
∫∫
∞
∞+
−
⋅⋅
0222222
22
0
2
2
00
sinhcosh
coscosh
cos,,,
Equation 3.3–22
This potential fulfils both, the radiation condition at infinity and the boundary condition at thefree surface.
3.3.3.2.2 2-D Radiation Potential
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
112
In case of an oscillating two-dimensional body, no waves are travelling in the x -direction, so0=xk and kk y = . The distribution of ( )ξA is constant over the full length of the body from
−∞=ξ until +∞=ξ and the radiation potential - given in Equation 3.3–22 - reduces to:
( ) ( )[ ][ ] [ ] ( ) dkyk
hkkhkhzk
Aetzy tjr ∫
∞⋅⋅ ⋅⋅⋅
⋅⋅−⋅⋅−⋅
⋅⋅=Φ0
00 cossinhcosh
cosh,,
νω
Equation 3.3–23
To fulfil also the Sommerfeld radiation condition in Equation 3.3–6, a term has to be added.For this, use will be made here of the value of the potential given in Equation 3.3–23 at a largedistance from the body:
( ) ( )[ ][ ] [ ] ( )
( ) ( )
⋅+⋅⋅⋅⋅=
⋅⋅⋅⋅⋅−⋅⋅
−⋅⋅⋅=Φ
∫∫
∫∞
⋅⋅−∞
⋅⋅+⋅⋅
∞⋅⋅
00
0
0
00
21
cossinhcosh
cosh,,
dkekFdkekFAe
dkykhkkhk
hzkAetzy
ykiykitj
tjr
ω
ω
ν
with:
( ) ( )[ ][ ] [ ]hkkhk
hzkkF
⋅⋅−⋅⋅−⋅
=sinhcosh
coshν
The treatment of the singularities is visualised in Figure 3.3–3.
Figure 3.3–3: Treatment of singularities
When substituting for k the term liku ⋅+= , the first integral integrates for 0>y over theclosed line I-II-III-IV in the first quadrant of the complex domain and the second integralintegrates for 0>y over the closed line I-V-VI-VII in the fourth quadrant, so:
1. For line I-II-III-IV :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
113
( )
0
............0
=+++=
+++=⋅⋅ ∫∫∫ ∫∫ ⋅⋅+
IVIIIIII
IVIII
R
II
yui
JJJJ
dudududkdueuF
with:
( ) ( )IVIIIIIR
yki JJJdkekF ++−=⋅⋅∫∞
∞→
⋅⋅+
0
lim
The location of the singular point follows from the denominator in the expression for( )kF :
[ ] [ ] 0sinhcosh 000 =⋅⋅−⋅⋅ hh ννννBecause ( ) 0lim =
∞→ IIIRJ and IVJ disappears too for a large y , the singular point itself
delivers a contribution only:( )
( )[ ][ ] [ ] [ ]
[ ] ( )[ ][ ] [ ]
yi
yi
II
ehhh
hzhi
ehhhhh
hzi
iJ
⋅⋅+
⋅⋅+
⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅+=
⋅⋅⋅⋅−⋅−⋅⋅⋅
−⋅⋅⋅−=
⋅⋅−=
0
0
000
00
0000
0
0
coshsinhcoshcosh
coshsinhsinhcosh
Residue
ν
ν
νννννπ
ννννννπ
νπ
and the searched integral becomes for ∞→y :
( ) [ ] ( )[ ][ ] [ ]
yiyki ehhh
hzhidkekF ⋅⋅+
∞⋅⋅+ ⋅
⋅⋅⋅+⋅−⋅⋅⋅
⋅⋅−=⋅⋅∫ 0
000
00
0 coshsinhcoshcosh ν
νννννπ
2. For line I-V-VI-VII:
( )
0
............0
=+++=
+++=⋅⋅ ∫∫∫ ∫∫ ⋅⋅−
VIIVIVI
VIIVI
R
V
yui
JJJJ
dudududkdueuF
with:
( ) ( )VIIVIVR
yki JJJdkekF ++−=⋅⋅∫∞
∞→
⋅⋅−
0
lim
Because ( ) 0lim =∞→ VIR
J and VIIJ disappears too for a large y , the singular point itself
delivers a contribution only:( )
[ ] ( )[ ][ ] [ ]
yi
V
ehhh
hzhi
iJ
⋅⋅−⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅−=
⋅⋅+=
0
000
00
0
coshsinhcoshcosh
Residue
ν
ννννν
π
νπ
and the searched integral becomes for ∞→y :
( ) [ ] ( )[ ][ ] [ ]
yiyki ehhh
hzhidkekF ⋅⋅−
∞⋅⋅− ⋅
⋅⋅⋅+⋅−⋅⋅⋅
⋅⋅+=⋅⋅∫ 0
000
00
0 coshsinhcoshcosh ν
νννννπ
This provides for the potential in Equation 3.3–23 for ∞→y :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
114
( ) [ ] ( )[ ][ ] [ ] ( )y
hhhhzh
Aetzy tjr ⋅⋅
⋅⋅⋅+⋅−⋅⋅⋅
⋅⋅⋅=Φ ⋅⋅0
000
0000 sin
coshsinhcoshcosh
,, νννν
ννπω
Equation 3.3–24
The Sommerfeld radiation condition in Equation 3.3–6 will be fulfilled when:[ ] ( )[ ]
[ ] [ ] ( ) ( ) 0,,Imcoscoshsinh
coshcosh000
000
0000 =Φ⋅−⋅⋅
⋅⋅⋅+⋅−⋅⋅⋅
⋅⋅⋅⋅⋅⋅ tzyyhhh
hzhAe tj νν
ννννννπω
or:( ) ( )
[ ] ( )[ ][ ] [ ] ( )y
hhhhzh
Ae
tzytzy
tj
j
⋅⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅⋅=
Φ=Φ
⋅⋅0
000
000
00
coscoshsinh
coshcosh
,,,,Im
νννν
ννπω
Equation 3.3–25
With Equation 3.3–24 and Equation 3.3–25, the radiation potential becomes:( ) ( )
( )[ ][ ] [ ] ( )
[ ] ( )[ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅
−⋅
⋅⋅=Φ⋅+Φ
∫∞
⋅⋅
yhhh
hzhj
dkykhkkhk
hzk
Aetzyjtzy tjjr
0000
00
0
000
coscoshsinh
coshcosh
cossinhcosh
cosh
,,,,
νννν
ννπ
ν
ω
Equation 3.3–26
From this follows for ∞→y :
( ) [ ] ( )[ ][ ] [ ]
yjtj ehhh
hzhAejtzy ⋅⋅−⋅⋅ ⋅
⋅⋅⋅+⋅−⋅⋅⋅
⋅⋅⋅⋅=∞→Φ 0
000
0000 coshsinh
coshcosh,, νω
νννννπ
This means that Equation 3.3–26 describes a flow, consisting of waves with amplitude:[ ]
[ ] [ ]hhhh
gA
⋅⋅⋅+⋅⋅
⋅⋅
⋅=000
02
0 coshsinhcosh
ννννπωζ
Equation 3.3–27
travelling away from both sides of the cylinder.
From the orthogonality condition:
zy ∂Ψ∂
+=∂Φ∂
follows the stream function:( ) ( )
( )[ ][ ] [ ] ( )
[ ] ( )[ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅
−⋅
⋅⋅−=Ψ⋅+Ψ
∫∞
⋅⋅
yhhh
hzhj
dkykhkkhk
hzk
Aetzyjtzy tjjr
0000
00
0
000
sincoshsinh
sinhcosh
sinsinhcosh
sinh
,,,,
νννν
ννπ
ν
ω
Equation 3.3–28
For an infinite water depth, Equation 3.3–26 and Equation 3.3–28 reduce to:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
115
( ) ( )
( ) ( )
( ) ( )
( ) ( )
⋅⋅⋅⋅+⋅⋅⋅−
⋅⋅=Ψ⋅+Ψ
⋅⋅⋅⋅+⋅⋅⋅−
⋅⋅=Φ⋅+Φ
⋅−∞ ⋅−
⋅⋅∞∞
⋅−∞ ⋅−
⋅⋅∞∞
∫
∫
yejdkykk
e
Aetzyjtzy
yejdkykk
e
Aetzyjtzy
zzk
tjjr
zzk
tjjr
νπν
νπν
ν
ω
ν
ω
sinsin
,,,,
coscos
,,,,
0
000
0
000
Equation 3.3–29
Now, the potential and stream functions can be written as:
( )
( ) [ ] [ ][ ] [ ]
[ ] ( )[ ][ ] [ ] ( )
( )
( ) [ ] [ ][ ] [ ]
[ ] ( )[ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅
−⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅−
−⋅⋅⋅−
⋅⋅=
Ψ⋅+Ψ+Ψ=Ψ
⋅⋅⋅⋅⋅+⋅
−⋅⋅⋅⋅⋅
+⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅−
+⋅⋅⋅−
⋅⋅=
Φ⋅+Φ+Φ=Φ
∫
∫
∫
∫
∞ ⋅−
∞ ⋅−
⋅⋅
∞
∞ ⋅−
∞ ⋅−
⋅⋅
∞
yhhh
hzhj
dkhkkhkzkkzk
ykk
e
dkykk
e
Ae
j
yhhh
hzhj
dkhkkhkzkkzk
ykk
e
dkykk
e
Ae
j
hk
zk
tj
jradr
hk
zk
tj
jradr
0000
00
0
0
0
0000
0000
00
0
0
0
0000
sincoshsinh
sinhcosh
sinhcoshsinhcosh
sin
sin
coscoshsinh
coshcosh
sinhcoshcoshsinh
cos
cos
νννν
ννπ
νν
ν
ν
νννν
ννπ
νν
ν
ν
ω
ω
Equation 3.3–30
In here, ∞Φ r0 is the potential at deep water and rad0Φ is the additional potential due to the
finite water depth. jΦ can be written in the same way.
3.3.3.2.3 Alternative Derivation
Assuming that the real part of the potential at an infinite water depth, ∞Φr , is known, anotherderivation of the 2-D potential is given by Porter [1960]. The additional potential for arestricted water depth, radΦ , will be determined in such a way that it fulfils the free surfacecondition and - together with ∞Φr - also the boundary condition at the seabed.As a start for the real additional potential will be chosen:
( ) ( ) [ ] ( ) ( )[ ] ( ) dkykhzkkCzkkCAetzy tjrad ⋅⋅⋅−⋅⋅+⋅⋅⋅⋅=Φ ∫
∞⋅⋅ coscoshsinh,,
02100
ω
Equation 3.3–31
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
116
From the free surface condition in Equation 3.3–4 follows for 2By ≥ :
( ) [ ] ( ) ( ) [ ] ( ) 0cossinhcosh0
2120 =⋅⋅⋅⋅⋅⋅−⋅+⋅⋅⋅⋅⋅ ∫∞
⋅⋅ dkykhkkkCkkChkkCAe tj νω
The solution of this Fourier integral equation:
( ) ( ) ( )ξξ gdkkkf =⋅⋅⋅∫∞
cos0
is known:
( ) ( ) ( ) ξξξπ
dkgkf ⋅⋅⋅⋅= ∫∞
cos1
0
Also will be obtained:( ) ( ) ( ) [ ] [ ] 0sinhcosh21 =⋅⋅−⋅⋅⋅+⋅= hkkhkkCkkCkf ν
from which follows:
( ) [ ] [ ] ( )kChkkhk
kkC 12 sinhcosh
⋅⋅⋅−⋅⋅
−=
ν
With this will be obtained:( )
( ) [ ] ( )[ ][ ] [ ] ( ) dkyk
hkkhkhzkk
zkkC
Aetzy tjrad
⋅⋅⋅
⋅⋅−⋅⋅−⋅⋅
−⋅⋅
⋅⋅=Φ
∫∞
⋅⋅
cossinhcosh
coshsinh
,,
01
00
ν
ω
The still unknown function ( )kC1 follows from the boundary condition at the seabed:
( )
( ) [ ] ( )
⋅⋅⋅⋅⋅⋅
−⋅⋅⋅−
⋅
⋅⋅−=
=∂
Φ∂+
∂Φ∂
∫
∫∞
∞ ⋅−
⋅⋅
=
∞
0
1
00
00
coscosh
cos
0
dkykhkkkC
dkykk
ek
Ae
zzhk
tj
hz
radr
νω
So:
( ) ( ) [ ]
( ) ( ) [ ] [ ] [ ]hkhkkhkkek
kC
hkke
kC
hk
hk
⋅⋅⋅⋅−⋅⋅⋅−⋅−
=
⋅⋅−=
⋅−
⋅−
coshsinhcosh
cosh
2
1
νν
ν
With this, the real additional potential, as given in Equation 3.3–30, becomes:( )
[ ] [ ][ ] [ ] ( ) dkyk
hkkhkzkkzk
ke
Aetzyhk
tjrad
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅−
⋅⋅=Φ
∫∞ ⋅−
⋅⋅
cossinhcoshcoshsinh
,,
0
00
νν
ν
ω
Equation 3.3–32
The imaginary part can be obtained as described before.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
117
3.3.3.2.4 2-D Multi-Potential
The free surface conditions can not be fulfilled with the potential and the stream function inEquation 3.3–30 only.Additional potentials nΦ are required which fulfil the boundary conditions in Equation 3.3–3
through Equation 3.3–6 and together with 0Φ also fulfil the boundary conditions in Equation3.3–7 through Equation 3.3–9:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ∑
∑
∞
=∞
∞
∞
=
Φ⋅+Φ+Φ⋅
+Φ⋅+Φ+Φ⋅=
Φ⋅+Φ⋅=Φ
1
'''
'0
'0
'00
1
''00
,,,,,,
,,,,,,
,,,,,,
nnjnradnrn
jradr
nnn
tzyjtzytzyA
tzyjtzytzyA
tzyAtzyAtzy
Equation 3.3–33
Use will be made here of multi-potentials given by Grim [1956, 1957] of which - using theSommerfeld radiation condition - the real additional potential nradΦ and the imaginary
potential part njΦ will be determined. This results in:
( )
( ) ( ) ( )
( )
( ) ( ) [ ] [ ][ ] [ ] ( )
( )
[ ]( )[ ]
[ ] [ ] ( )yhhh
hzh
Aetzy
dkykhkkhkzkkzk
ekk
Aetzy
dkykekk
Aetzy
n
ntj
nj
khn
ntj
nrad
kzn
ntj
nr
⋅⋅⋅⋅⋅+⋅
−⋅⋅
⋅⋅
⋅⋅−=Φ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅+
⋅⋅+=Φ
⋅⋅⋅⋅⋅+
⋅⋅+=Φ
⋅⋅
∞−−⋅
⋅⋅
−−⋅∞
⋅⋅∞
∫
∫
0000
0
0
20
0
12
12
0
coscoshsinh
coshcosh
,,
cossinhcoshcoshsinh
,,
cos
,,
νννν
νννπ
ννν
ν
ω
ω
ω
Equation 3.3–34
The orthogonality condition provides the stream functions:( )
( ) ( ) ( )
( )
( ) ( ) [ ] [ ][ ] [ ] ( )
( )
[ ]( )[ ]
[ ] [ ] ( )yhhh
hzh
Aetzy
dkykhkkhkzkkzk
ekk
Aetzy
dkykekk
Aetzy
n
ntj
nj
khn
ntj
nrad
kzn
ntj
nr
⋅⋅⋅⋅⋅+⋅
−⋅⋅
⋅⋅
⋅⋅+=Ψ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅+
⋅⋅−=Ψ
⋅⋅⋅⋅⋅+
⋅⋅+=Ψ
⋅⋅
∞−−⋅
⋅⋅
−−⋅∞
⋅⋅∞
∫
∫
0000
0
0
20
0
12
12
0
sincoshsinh
sinhcosh
,,
sinsinhcoshsinhcosh
,,
sin
,,
νννν
νννπ
ννν
ν
ω
ω
ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
118
Equation 3.3–35
The potentials njΦ and nradΦ disappear in deep water.
3.3.3.2.5 Total Potentials
Only the complex constant nA with ∞≤≤ n0 in the potential has to be determined:
( ) ( ) ( ) ( )
∑
∑
∑
∞
= ∞
∞⋅⋅
∞
= ∞
∞
∞
=∞
⋅++⋅⋅+
⋅−+⋅⋅=
Φ⋅+Φ+Φ⋅⋅+
Φ⋅−Φ+Φ⋅=
Φ⋅+Φ+Φ⋅⋅+=Φ
0
0'''
'''
0
''' ,,,,,,,,
n njnrnradnrnj
njnjnradnrnrtj
n njnrnradnrnj
njnjnradnrnr
nnjnradnrnjnr
AAj
AAe
AAj
AA
tzyjtzytzyAjAtzy
φφφ
φφφω
( ) ( ) ( ) ( )
∑
∑
∑
∞
= ∞
∞⋅⋅
∞
= ∞
∞
∞
=∞
⋅++⋅⋅+
⋅−+⋅⋅=
Ψ⋅+Ψ+Ψ⋅⋅+
Ψ⋅−Ψ+Ψ⋅=
Ψ⋅+Ψ+Ψ⋅⋅+=Ψ
0
0'''
'''
0
''' ,,,,,,,,
n njnrnradnrnj
njnjnradnrnrtj
n njnrnradnrnj
njnjnradnrnr
nnjnradnrnjnr
AAj
AAe
AAj
AA
tzyjtzytzyAjAtzy
ψψψ
ψψψω
Equation 3.3–36
Summarised, the complex total potential can now be written as:( ) ( )
( )( )( )[ ] [ ]
[ ] ( )( )( )[ ] [ ]
( ) ( ) ( )( )( )[ ] [ ]
[ ]( )( )( )
[ ] [ ]
∑ ∫
∫
∞
=
∞−⋅
⋅⋅
∞
⋅⋅
⋅⋅⋅+⋅−⋅+⋅
⋅⋅
⋅⋅−
⋅⋅⋅−⋅⋅
−⋅+⋅⋅⋅−
⋅⋅+
⋅
+
⋅⋅⋅+⋅−⋅+⋅⋅⋅
⋅⋅+
⋅⋅⋅−⋅⋅
−⋅+⋅
⋅⋅+⋅=Ψ⋅+Φ
1
000
0
0
20
0
1222
000
00
0
00
coshsinhcos
cosh
sinhcoshcos
coshsinhcoscosh
sinhcoshcos
,,,,
nn
n
njnr
tj
jrtj
hhhhziy
hj
dkhkkhk
hziykkk
AjA
e
hhhhziyh
j
dkhkkhk
hziyk
AjAetzyitzy
νννν
ννπ
νν
νννννπ
ν
ω
ω
Equation 3.3–37
The coefficients nrA and njA with ∞≤≤ n0 have to be determined in such a way that the
instantaneous boundary conditions on the body surface have been fulfilled. These coefficientsare dimensional and it is very practical to determine them for the amplitude of the flowvelocity V ; also if they then have the dimension [ ]12 +nL :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
119
( )''jr
jrjr
AjAV
V
AjAVAjA
⋅+⋅=
⋅+⋅=⋅+
Then, nnA φ⋅' and nnA ψ⋅' have the dimensions of a length [ ]L .
3.3.3.3 Expansion of Potential Parts
The expansion of the potential parts at an infinite water depth is given by Grim, see Kirsch[1969].
For 022 →+⋅=⋅ zyr νν :
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
⋅⋅
⋅+⋅
⋅+
−⋅⋅
⋅+⋅
⋅+⋅+
⋅=
⋅⋅
⋅+⋅
⋅+
+⋅⋅
⋅+⋅
⋅+⋅+
⋅=
∑
∑
∑
∑
∞
=
∞
=⋅−∞
∞
=
∞
=⋅−∞
yyizmmz
y
yyizmm
r
e
yyizmmz
y
yyizmm
r
e
m
mm
m
m
m
zr
m
mm
m
m
m
zr
νν
νν
νγψ
νν
νννγ
φ
ν
ν
cosIm!
arctan
sinRe!
ln
sinIm!
arctan
cosRe!
ln
1
1
0
1
1
0
Equation 3.3–38
with the Euler constant: 57722.0=γ .
For ∞→+⋅=⋅ 22 zyr νν :
( ) ( ) ( )
( ) ( ) ( )yyy
eyizr
m
yyy
eyizr
m
zM
m
mmmr
zM
m
m
mmr
⋅⋅⋅⋅−
⋅+⋅
⋅−
=
⋅⋅⋅⋅+
⋅+⋅
⋅−=
⋅−
=∞
⋅−
=∞
∑
∑
νπν
ψ
νπν
φ
ν
ν
cosIm!1
sinRe!1
120
120
Equation 3.3–39
Mind you that ( ) ( )m
mm yizr
m⋅+⋅
⋅−
2
!1ν
is semi-convergent.
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
⋅+⋅
−−⋅+⋅−⋅−=
⋅+⋅
−+⋅+⋅−⋅−=
−−−∞
−−−∞
122
122
Re12
Im!121
Im12
Re!121
nnnnr
nnnnr
ziyn
ziyn
ziyn
ziyn
νψ
νφ
Equation 3.3–40
For the expansion of the remaining potential parts use has been made of the followingrelations as derived in Appendix II:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
120
[ ] ( ) ( ) ( )
[ ] ( ) ( ) ( )
[ ] ( ) ( ) ( )
[ ] ( ) ( ) ( ) ∑
∑
∑
∑
∞
=
++
∞
=
++
∞
=
∞
=
⋅+⋅+
=⋅⋅⋅
⋅+⋅+
=⋅⋅⋅
⋅+⋅=⋅⋅⋅
⋅+⋅=⋅⋅⋅
0
1212
0
1212
0
22
0
22
Im!12
sincosh
Re!12
cossinh
Im!2
sinsinh
Re!2
coscosh
t
tt
t
tt
t
tt
t
tt
yiztk
ykzk
yiztk
ykzk
yizt
kykzk
yizt
kykzk
With these relations follows from Equation 3.3–32:
( ) [ ] [ ]( )
( ) ( ) ( ) ( )
( ) ( ) [ ] [ ]( )( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )∑
∑ ∫
∑∑
∫
∞
=
+
∞
= +
∞ ⋅−+
∞
=
∞
=
++
∞ ⋅−
⋅+⋅+−⋅+⋅⋅++−=
⋅+⋅+−⋅+⋅
⋅
⋅
⋅⋅−⋅⋅⋅−⋅
⋅+=
⋅+⋅⋅−
⋅+⋅+
⋅
⋅
⋅
⋅⋅−⋅⋅⋅−=
0
212
0 212
0
12
0
22
0
1212
0
0
Re12Re!1212
Re12Re
sinhcosh!121
Re!2
Re!12
sinhcosh
t
tt
t tt
hkt
t
tt
t
tt
hk
rad
yiztyizttG
yiztyiz
dkhkkhkk
ekt
yizt
kkyiz
tk
dkhkkhkk
e
ν
ν
νν
ν
ννφ
Equation 3.3–41
It is obvious that:( )
( ) ( ) ( ) ( ) ( )∑∞
=
+
⋅+⋅−⋅+⋅+⋅++
−=0
1220 ImIm12
!1212
t
ttrad yizyizt
ttG νψ
Equation 3.3–42
The function:
( ) ( ) [ ] [ ]( ) dkhkkhkk
ektG
hkt
⋅⋅⋅−⋅⋅⋅−
⋅= ∫
∞ ⋅−
0 sinhcoshννwill be treated in the next Section.
Further follows from Equation 3.3–34:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
121
( ) ( )
( ) [ ] [ ]( )
( ) ( ) ( ) ( )
( )( ) [ ] [ ]( )
( ) [ ] [ ]( )( ) ( ) ( ) ( )
( ) ( )( )
( ) ( ) ( ) ( )∑
∑ ∫
∫
∑∑
∫
∞
= +
∞
=
+
∞ ⋅−−+
∞ ⋅−++
∞
=
∞
=
++
∞ ⋅−−
⋅+⋅+−⋅+⋅
⋅+
−+⋅−++=
⋅+⋅+−⋅+⋅
⋅
⋅⋅⋅−⋅⋅⋅−
⋅⋅
−⋅⋅⋅−⋅⋅⋅−
⋅
⋅+=
⋅+⋅⋅−
⋅+⋅+
⋅
⋅
⋅
⋅⋅−⋅⋅⋅−⋅⋅−
=
0 212
2
0
212
0
1222
0
122
0
22
0
1212
0
1222
Re12Re
!12122122
Re12Re
sinhcosh
sinhcosh
!121
Re!2
Re!12
sinhcosh
t tt
t
tt
hknt
hknt
t
tt
t
tt
hkn
nrad
yiztyiz
tntGntG
yiztyiz
dkhkkhkk
ek
dkhkkhkk
ek
t
yizt
kkyiz
tk
dkhkkhkk
ekk
ν
ν
ν
ννν
νν
ν
νννφ
Equation 3.3–43
It is clear that:
( ) ( )( )
( ) ( ) ( ) ( )∑
∞
= +
⋅+⋅+−⋅+⋅
⋅+
−+⋅−++−=
0 212
2
Im12Im
!12122122
t tt
nrad
yiztyiz
tntGntG
ν
νψ
Equation 3.3–44
For the imaginary parts can be written:[ ]
[ ] [ ]
( ) ( ) [ ] ( ) ( ) [ ]
[ ] [ ]
( ) ( ) [ ] ( ) ( )
⋅+⋅+
⋅⋅−
⋅+⋅
⋅⋅⋅⋅+⋅
⋅⋅−=
⋅+⋅+
⋅⋅−
⋅+⋅
⋅⋅⋅⋅+⋅
⋅⋅+=
∑∑
∑∑
∞
=
++∞
=
∞
=
++∞
=
0
1212
00
0
22
0
000
02
0
0
1212
00
0
22
0
000
02
0
Im!12
tanhIm!2
coshsinhcosh
Re!12
tanhRe!2
coshsinhcosh
t
tt
t
tt
j
t
tt
t
tt
j
yizt
hyizt
hhhh
yizt
hyizt
hhhh
νν
ν
ννννπψ
νν
ν
ννννπ
φ
Equation 3.3–45
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
122
[ ] [ ]
( ) ( ) [ ] ( ) ( ) ( ) ( )
[ ] [ ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) jn
t
tt
t
tt
n
nj
jn
t
tt
t
tt
n
nj
yizt
hyizt
hhh
yizt
hyizt
hhh
022
012
0
0
1212
00
0
22
0
000
20
022
012
0
0
1212
00
0
22
0
000
20
Im!12
tanhIm!2
coshsinh
Re!12
tanhRe!2
coshsinh
ψννν
νν
ν
ννννπ
ψ
φννν
ννν
ννννπ
φ
⋅−⋅−=
⋅+⋅+
⋅⋅−
⋅+⋅
⋅⋅⋅⋅+⋅
⋅+=
⋅−⋅−=
⋅+⋅+
⋅⋅−
⋅+⋅
⋅⋅⋅⋅+⋅
⋅−=
−⋅
∞
=
++∞
=
−⋅
∞
=
++∞
=
∑∑
∑∑
Equation 3.3–46
3.3.3.4 Function G(t)
The function:
( ) ( ) [ ] [ ]( ) dkhkkhkk
ektG
hkt
⋅⋅⋅−⋅⋅⋅−
⋅= ∫
∞ ⋅−
0 sinhcoshνν
with unit [ ]tL −1 has two singular points: ν=k and 0ν=k , see Figure 3.3–4.
Figure 3.3–4: Singularities in the G-function
Thus, it is not possible to solve this integral directly.
First, this integral will be normalised:( ) ( )
( ) [ ] [ ]( ) duuuuhhu
eu
htGtGut
t
⋅⋅−⋅⋅⋅⋅−
⋅=
⋅=
∫∞ −
−
0
1'
sinhcoshνν
Equation 3.3–47
A substitution of:σς ⋅⋅⋅=⋅+= ieviuw 2
2
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
123
provides:
( ) [ ] [ ]
0
...
...............
sinhcosh
0
0
=
++++=
++++=
⋅⋅−⋅⋅⋅⋅−
⋅=
∫
∫∫∫∫∫
∫
∞
∞
−
IVIIIIII
IVIIIIII
wt
JJJJdu
dwdwdwdwdu
dwwwwhhw
ewJ
νν
From this follows:
( ) [ ] [ ] IVIIIIII
ut
JJJJduuuuhhu
eu+++−=⋅
⋅−⋅⋅⋅⋅−⋅
∫∞ −
Resinhcosh0 νν
IJ and IIJ are imaginary because they are residues and 0=IIIJ for ∞→R .
So, it remains:
( ) [ ] [ ]
( ) [ ] [ ]
⋅⋅−⋅⋅⋅⋅−
⋅−=
−=
⋅⋅−⋅⋅⋅⋅−
⋅
∫
∫
−
∞ −
IV
wt
IV
ut
dwwwwhhw
ew
J
duuuuhhu
eu
sinhcoshRe
Re
sinhcosh0
νν
νν
With the complex function:
( ) [ ] [ ] wwwhhw ~sinh~~cosh~ ⋅−⋅⋅⋅⋅− νν with real: σς ⋅⋅⋅= iew 22
~
the nominator of this integral will be made real by removing.
So:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
124
( ) ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ]
( ) ( )
( )( )
[ ] [ ]( )( )
∫
∫
∞−
−
−
⋅
+−+
−+⋅
⋅−⋅−
−−−
+⋅+
+⋅−
+++
−⋅
⋅
⋅
⋅
−=
⋅
⋅−⋅⋅⋅⋅−
⋅⋅−⋅⋅⋅⋅−
⋅−⋅⋅⋅⋅−⋅⋅
−=
0
222
222
2
2
2
'
sin2cos2
tanh22cosh
2
sin4
tan1cos4
tan1
sin4
tancos4
sin4
tan1cos4
tan12
2
4cos
~sinh~~cosh~sinhcosh
~sinh~~cosh~Re
ς
ςςνςςν
ςςνςνς
νςνς
ςπςπς
ςπςςν
ςπ
ςπ
ν
ς
π
νννννν
ς
ς
d
hh
hh
hh
te
t
th
te
th
t
dw
wwwhhw
wwwhhw
wwwhhw
ew
tG
t
IV
wt
Equation 3.3–48
Because t is always odd:
( ) ( ) ( ) ( )( )
[ ] [ ]( )( )
ςς
ςςνςςν
ςςνςνς
νςνςςςςνςςςνπ ς
d
hh
hh
hh
hehttG
t
⋅
⋅
+−+
−+⋅
⋅−−+−−+
⋅−= ∫∞ −
2
sin2cos2
tanh22cosh
2
sincos4cos2sin44
cos0
222
222
2
22'
for ,......9,5,1=t
( ) ( ) ( ) ( )( )
[ ] [ ]( )( )
ςς
ςςνςςν
ςςνςνς
νςνςςςςνςςςνπ ς
d
hh
hh
hh
hehttG
t
⋅
⋅
+−+
−+⋅
⋅−−−−−+
⋅−= ∫∞ −
2
sin2cos2
tanh22cosh
2
sincos4sin2cos44
cos0
222
222
2
22'
for ,......11,7,3=t
For 1>t the function ( )tG ' becomes finite.
However, ( )1'G does not converge for 0→hν ; the integral increases monotone withdecreasing h⋅ν . This will be investigated first.
( ) ( ) ( ) [ ] [ ]( ) duuuuhhu
euGG
u
⋅⋅−⋅⋅⋅⋅−
⋅== ∫
∞ −
0
'
sinhcosh11
νν
Equation 3.3–49
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
125
This integral converges fast for small h⋅ν -values. This will be approximated by:
( ) ( ) ( ) duuhhu
euG
u
h⋅
−⋅⋅⋅−⋅
= ∫∞ −
→0
2101lim
ννν
Equation 3.3–50
This can be written as:
( ) ( )
( )
⋅⋅+
⋅+⋅⋅⋅
+⋅⋅−
⋅−⋅⋅⋅
+⋅⋅−
⋅⋅−
=
∫
∫
∫
∞ −
∞ −
∞ −
→→
duhu
ehh
duhu
ehh
duhu
eh
G
u
u
u
hh
0
0
0
010
121
121
11
lim1lim
ννν
ννν
νν
νν
From:
⋅++⋅−=⋅
− ∑∫∞
=
−∞ −
10 !ln
m
ma
u
mma
aeduau
e γ
follows:
( )
( ) ( )
( )( ) ( )
( )( ) ( ) ( )
⋅⋅
⋅−+⋅
+⋅⋅+⋅⋅
−
+
⋅⋅
+⋅
+⋅⋅−⋅⋅
+
+
⋅⋅
+⋅+⋅⋅−
−
=
∑
∑
∑
∞
=
⋅+
∞
=
⋅−
∞
=
⋅−
→→
1
2
1
2
1
010
!1
2ln
12
!2ln
12
!ln
1
lim1lim
m
m
mh
m
mh
m
mh
hh
mmhh
hh
e
mmhh
hh
e
mmh
hh
e
G
ννγνν
ννγνν
ννγν
ν
ν
ν
νν
or:
( )
( ) ( )
( )( )
( ) ( )
( )( )
( ) ( ) ( )
⋅⋅
⋅−+⋅−⋅
+⋅⋅−⋅⋅
⋅⋅−
−
⋅⋅
+⋅+⋅
+⋅⋅−⋅⋅
⋅⋅+
+
⋅⋅
+⋅+⋅⋅−
−
=
∑
∑
∑
∞
=
⋅+
∞
=
⋅−
∞
=
⋅−
→→
2
2
2
2
1
010
!1
2ln
12
1
!2ln
12
1
!ln
1
lim1lim
m
m
mh
m
mh
m
mh
hh
mmh
hh
hh
eh
mmh
hh
hh
eh
mmh
hh
e
G
νννγνν
ν
νννγνν
ν
ννγν
ν
ν
ν
νν
or:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
126
( )
( ) ( )
[ ] [ ] ( )
[ ] [ ]( )
( ) ( )
( ) ( ) ( ) ( )
⋅⋅
⋅−⋅+⋅
⋅⋅−⋅−
⋅⋅
⋅+⋅
⋅⋅+
⋅⋅−⋅
+⋅
⋅−
⋅−⋅
+
+
⋅
+⋅
⋅⋅
−⋅⋅⋅−
+
⋅⋅
+⋅+⋅⋅−
−
=
∑∑
∑∑
∑
∞
=
⋅+∞
=
−
⋅+
∞
=
⋅−∞
=
−
⋅−
∞
=
⋅−
→→
2
2
2
21
2
2
2
21
1
01
0
!1
!1
!!
121sinh
1cosh
2lnsinh
cosh1
1
!ln
1
lim1lim
m
m
mh
m
m
mh
m
m
h
m
m
h
m
mh
hh
mmh
emm
he
mmh
emm
he
hhh
hh
hh
hh
h
mmh
hh
e
G
νν
νν
ννν
νν
νγν
ννν
ννγν
νν
νν
ν
νν
or:( ) ( )hG
h⋅−−=
→νγ
νln11lim 10
Equation 3.3–51
The imaginary part of integral in Equation 3.3–48 has been treated in Appendix III.
3.3.3.5 Hydrodynamic Loads
The hydrodynamic loads can be found from an integration of the pressures on the hull of theoscillating body in (previously) still water. With a known potential, these pressures can befound from the linear part of the instationary pressures as follows from the Bernoulli equation:
Φ⋅⋅⋅−=∂Φ∂
⋅−=
−=
ρω
ρ
jt
ppp statdyn
The potential is in-phase with the oscillation velocity. To obtain the phase of the pressureswith respect to the oscillatory motion a phase shift of 090− is required, which means amultiplication with j− . Then the pressure is:
Φ⋅⋅−= ωρdynpThe hydrodynamic force on the body is equal to the integrated pressure on the body. In thetwo-dimensional case, this is a force per unit length.The vertical force becomes:
( )( ) ∑∫
∫
∞
=
⋅⋅ ⋅⋅−⋅⋅+⋅−⋅⋅⋅⋅⋅−=
⋅+⋅⋅⋅=
⋅=
⋅+=
0
''''
''
n Snrnjnjnrnjnjnrnr
tj
VjVr
S
VjVrV
dyAAjAAeV
FjFV
dyp
FjFF
φφφφωρ
ωρ
ω
Equation 3.3–52
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
127
The real part of this force is equal to the hydrodynamic mass coefficient times the oscillatoryacceleration, from which the hydrodynamic mass coefficient follows:
VF
bF
m VrVr
⋅==
ω"
or non-dimensional:
VB
F
B
mC Vr
V
⋅⋅⋅⋅=
⋅⋅=
ωπρπρ 22
"
88
Equation 3.3–53
The imaginary part of the force must be equal to the hydrodynamic damping coefficient timesthe oscillatory velocity, from which the damping coefficient follows:
V
FN
Vj
V =
Instead of this coefficient, generally the ratio between amplitude of the radiated wave ζ and
the oscillatory motion z will be used. The energy balance provides:
[ ][ ]
[ ][ ] [ ] Vj
V
group
V
V
Fhhh
hV
Nhh
hg
cN
g
zA
⋅⋅⋅⋅+⋅
⋅⋅
⋅⋅=
⋅⋅⋅+⋅⋅
⋅⋅⋅
⋅⋅
=
⋅⋅
⋅=
=
000
022
00
00
2
2
2
coshsinhcosh
2sinh22sinh
2
νννν
ωρν
ννν
ρνω
ρω
ζ
Equation 3.3–54
In deep water, the hydrodynamic mass for 0→ν becomes infinite, because the potential inEquation 3.3–38 becomes:
( )rr ⋅+=∞→νγφ
νlnlim 00
Equation 3.3–55
and the non-dimensional mass of a circle becomes:
( ) ( )
−⋅
+⋅−−⋅=
⋅⋅=
∑∞
=
→→
122
2
"
00
141
ln8
8
limlim
n
V
nnr
B
mC
νγπ
πρνν
and the amplitude ratio in this deep water case becomes:
BdAd V
=
→ νν 0
lim
The hydrodynamic mass for 0→ν in shallow water remains finite. Because the multi-potentials - just as in deep water – provide finite contributions, the radiation potential has to
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
128
be discussed only, which is decisive (infinite mass) in deep water. The change-over borderline''deep to shallow'' water provides for this radiation potential:
( ) ( ) ( )
+=
⋅−⋅+=+∞→
hr
hrradr
ln
lnlnlim 000
γ
ννγφφν
Equation 3.3–56
It is obvious that Equation 3.3–56 - just as Equation 3.3–55 - provides an infinite value.
When the contributions of the multi-potentials (which disappear here for the borderline case0→ν ) are ignored, it follows from Equation 3.3–27 for the amplitude ratio in shallow water:
[ ][ ] [ ]
[ ][ ] [ ]
[ ][ ] [ ]
[ ] [ ][ ] [ ]hhh
hhA
hhhh
VA
hhhh
zA
hhhh
zgA
zAV
⋅⋅⋅+⋅⋅⋅⋅⋅
⋅⋅=
⋅⋅⋅+⋅⋅
⋅⋅⋅=
⋅⋅⋅+⋅⋅
⋅⋅⋅
⋅=
⋅⋅⋅+⋅⋅
⋅⋅⋅
⋅=
=
000
000'0
000
02
0
000
02
0
000
02
0
coshsinhcoshsinh
coshsinhcosh
coshsinhcosh
coshsinhcosh
ννννννπ
νννννπ
νννν
ωπν
ννννπω
ζ
Because:
1lim'
0
0=
⋅→ B
A πν
follows:[ ] [ ]
[ ] [ ]hhhhh
BAV⋅⋅⋅+⋅
⋅⋅⋅⋅⋅=
→000
000
0 coshsinhcoshsinh
limννν
νννν
and:
[ ] [ ]∞=
⋅⋅⋅+⋅
⋅=
⋅=
→
→→
hhhB
dd
dAd
dAd VV
0000
0
000
coshsinh1
lim2
limlim
0
0
ννν
νν
νν
ν
νν
Thus, ( )2BAV ⋅ν has at 02 =⋅ Bν a vertical tangent.
The fact that the hydrodynamic mass goes to infinity for zero frequency can be explainedphysically as follows. The smaller the frequency becomes, the longer becomes the radiatedwave and the faster travels it away from the cylinder. In the borderline case 0→ν has thewave an infinite length and it travels away - just as the pressure (incompressible fluid) - withan infinite velocity. This means that all fluid particles are in phase with the motions of thebody. This means that the hydrodynamic force is in phase with the motion of the body, whichholds too that:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
129
0arctanarctan '
'
=
=
=∞=
Vr
Vj
Vr
ViHT F
FFFε
This condition is fulfilled only when 0' =VjF or ∞== ρ"' mFVj .
However, 'VjF is finite:
2
22
4
2'
νωωρωρV
VVVj
VjA
AgN
VF
F =⋅=⋅
=⋅⋅
=
Because νν
⋅=→
BAV00
lim follows 2'
00
lim BFVj =→ν
. The term ρ"' mFVj = has to be infinite.
The finite value of the hydrodynamic mass at shallow water is physically hard to interpret. Afull explanation is not given here. However, it has been shown here that the result makes somesense. At shallow water can the wave (even in an incompressible fluid) not travel with aninfinite velocity; its maximum velocity is hg ⋅ . In case of long waves at shallow water, theenergy has the same velocity. From that can be concluded that at low decreasing frequenciesthe damping part in the hydrodynamic force will increase. This means that:
0arctan '
'
≠
=∞≠
Vr
ViHT
FFε
So, ρ"' mFVr = has to be finite.
3.3.3.6 Wave Loads
The wave forces EF on the restrained body in waves consist of:• forces 1F in the undisturbed incoming waves (Froude-Krylov hypothesis) and• forces caused by the disturbance of the waves by the body:
• one part 2F in phase with the accelerations of the water particles and• another part 3F in phase with the velocity of the water particles.
Thus:
321 FjFFFE ⋅++=
These forces will be determined from the undisturbed wave potential WΦ and the disturbance
potential SΦ . As mentioned before, for 090≠µ only an approximation will be found.
( )
[ ] [ ] [ ] ( )
( ) ∫
∑∫
∫
⋅
⋅+⋅⋅+⋅−⋅⋅
+⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅
⋅⋅⋅−=
⋅Φ+Φ⋅⋅−=
⋅++=
∞
=
⋅⋅⋅−
⋅⋅
S
n Snrnjnjnrnjnjnrnr
xj
tj
SSW
E
dy
AAjAAV
yzhz
e
e
dy
FjFFF
0
''''
0000
cos
321
sincossinhtanhcosh
0
φφφφ
µνννννωζ
ωρ
ωρ
µν
ω
Equation 3.3–57
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
130
In here:
( )
[ ] [ ] [ ] ( )
dyAAeVF
dyAAeVF
dyyzhz
eF
n S
nrnjnjnrtj
n S
njnjnrnrtj
S
xtj
⋅⋅+⋅⋅⋅⋅⋅−=
⋅⋅−⋅⋅⋅⋅⋅−=
⋅⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅⋅
−=
∑∫
∑∫
∫
∞
=
⋅⋅
∞
=
⋅⋅
⋅⋅−⋅⋅
0
''3
0
''2
0000
cos2
1
sincossinhtanhcosh
0
φφωρ
φφωρ
µννννν
ζωρ
ω
ω
µνω
Using ωζ ⋅=V , the non-dimensional amplitudes are:
[ ] [ ] [ ] ( )
dyAAB
BgF
E
dyAAB
BgF
E
dyyzhzB
BgF
E
n S
nrnjnjnr
n S
njnjnrnr
S
⋅⋅+⋅⋅−=
⋅⋅⋅=
⋅⋅−⋅⋅−=
⋅⋅⋅=
⋅⋅⋅⋅⋅⋅⋅−⋅⋅−=
⋅⋅⋅=
∑∫
∑∫
∫
∞
=
∞
=
0
''
33
0
''
22
0000
11
sincossinhtanhcosh1
φφνζρ
φφνζρ
µνννν
ζρ
Equation 3.3–58
In case of 090=µ , so beam waves, the theory of Haskind-Newman – see Haskind [1957] or
Newman [1962] - can be used too to determine the amplitudes 1E , 2E and 3E .When tj
WW e ⋅⋅⋅=Φ ωφ is the potential of the incoming wave and tjSS e ⋅⋅⋅=Φ ωφ is the potential
of the disturbance by the body at a large distance from the body with velocity amplitude1=V , then:
dzyy
eFh
WW
tjE ⋅
∂
∂⋅−
∂∂
⋅⋅⋅⋅−= ∫⋅⋅
0
φφφφωρ ω
Equation 3.3–59
According to Equation 3.3–105 in Appendix I is:
[ ] [ ] [ ] ( )yzhzW ⋅⋅⋅⋅⋅−⋅⋅⋅
= 0000 cossinhtanhcosh νννννωζφ
From the previous subsections follows the asymptotic expression for the disturbance potentialin still water with 1=V :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
131
( )[ ]
[ ] [ ][ ] [ ] [ ] ( )
( ) ( ) ( )
⋅⋅+⋅−−⋅+
⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅⋅+⋅
⋅⋅=
∑∞
=
−
∞→
1
120
''220
'0
'0
0000
000
02
sinsinhtanhcosh
coshsinhcosh
n
nnjnrjr
y
AjAAjA
yzhz
hhhh
ννν
ννννννν
νπφ
Substituting this in Equation 3.3–59, provides:[ ]
[ ] [ ]
[ ] [ ] [ ]
( ) ( ) ( )
( ) ( ) ( )
⋅⋅+⋅−−⋅+
⋅⋅⋅⋅⋅−=
⋅⋅+⋅−−⋅+
⋅⋅⋅⋅⋅−⋅
⋅⋅⋅⋅+⋅
⋅⋅⋅⋅⋅⋅⋅⋅−=
∑
∑
∫
∞
=
−
⋅⋅
∞
=
−
⋅⋅
1
120
''220
'0
'0
1
120
''220
'0
'0
0
2000
000
02
0
sinhtanhcosh
coshsinhcosh2
n
nnjnrjr
tj
n
nnjnrjr
h
tjE
AjAAjA
eg
AjAAjA
dzzhz
hhhh
egF
ννν
πζρ
ννν
ννν
ννννπνζρ
ω
ω
Non-dimensional:
( ) ( )
( ) ( )
⋅⋅−−⋅−=
⋅⋅⋅=
⋅⋅−−⋅−=
⋅⋅⋅=+
∑
∑
∞
=
−
∞
=
−
1
120
'220
'0
3
1
120
'220
'0
21
Im
Re
n
nnjj
E
n
nnrr
E
AAB
BgF
E
AAB
BgF
EE
νννπ
ζρ
νννπ
ζρ
Equation 3.3–60
3.3.3.7 Solution
The Lewis transformation of a cross section is given by:θθθ 3⋅−⋅−⋅+ ⋅+⋅+=⋅+ iii ebeaeziy
Equation 3.3–61
Then, the co-ordinates of the cross section are:( ) ( ) ( )( ) ( ) ( )θθ
θθ3sinsin1
3coscos1
⋅−⋅−=⋅+⋅+=
baz
bay
Equation 3.3–62
Then:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
132
( ) ( )θθθ 3⋅+⋅+⋅− ⋅+⋅+⋅=⋅−⋅=⋅+ iii ebeaeiziyiyiz
Equation 3.3–63
All calculations will be carried out in the Lewis domain. Scale factors are given in the tablebelow.
Ship Lewis formform Lewis
Ship
Breadth BR ( )ba ++⋅ 12 ( ) baBR ++⋅ 12Draught TI ba +−1 ( )baTI +−1Water depth TIHT ⋅ ( )baHTWT +−⋅= 1 ( )baTI +−1Wave number
0ν ( )baTIWF +−⋅= 1/0ν ( ) TIba +−1Acceleration g g 1Forces
GF F ( ) 21 baTI +−
Table 3.3–1: Lewis form parameters
3.3.3.8 Determination of Source Strengths An
The yet unknown complex coefficients nA ( ∞≤≤ n0 ), the source strengths of the by the flowgenerated singularity, can be determined by substituting the stream function in Equation 3.3–36 and the co-ordinates of the cross section in the relevant boundary conditions in Equation3.3–7 through Equation 3.3–9.
tzytzybodybodybody
,,,, Ψ=Ψ
Equation 3.3–64
To determine the unknowns nA , an equal number of equations has to be formulated. Becauseonly Lewis forms are used here, a simple approach is possible.All stream function parts and boundary conditions can be given as a Fourier series:
( ) ( )[ ] ∑∞
=
⋅+⋅+⋅⋅=0
12cos2sinm
mnmnn mdmc θθψ
or with:
( )( ) ( )( )[ ] ( )∑
∞
=
⋅⋅+−⋅
+⋅+⋅−=⋅+1
22
2
2sin124
1222112cosk
kmkk
mm θππ
θθ
in:
( ) ∑∞
=
⋅⋅+=1
0 2sinm
nmnn maa θψ
The solution of the by equating coefficients generated equations provide the unknowns 'nA .
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
133
3.3.4 Horizontal Motions
3.3.4.1 Boundary Conditions
The first four assumptions for the vertical motions are valid for horizontal motions too. Thepotential must fulfil the motion-dependent boundary conditions, which have been substitutedin Equation 3.3–3 through Equation 3.3–6. However, the fifth boundary condition needs herea new formulation.
Because two motions (a translation and a rotation) are considered here, follows from:
bodynbody
vn
=∂Φ∂
also in still water two boundary conditions:
1. For sway:
body
tj
body
body
bodynbody
dsdz
eU
dsd
dsdz
ydsdy
z
vn
⋅⋅−
Ψ−=
⋅∂Φ∂
−⋅∂Φ∂
=
=∂Φ∂
⋅⋅ω
or:
bodytj
bodydzeUd ⋅⋅=Ψ ⋅⋅ω
from which follows:( ) CzeUtzy
bodytj
body+⋅⋅=Ψ ⋅⋅ω,,
Equation 3.3–65
2. For roll:
body
tj
body
body
bodynbody
rdsdr
e
dsd
dsdz
ydsdy
z
vn
⋅⋅⋅⋅=
Ψ−=
⋅∂Φ∂
−⋅∂Φ∂
=
=∂Φ∂
⋅⋅ωωφ
or:
bodytj
bodydrred ⋅⋅⋅⋅−=Ψ ⋅⋅ωωφ
from which follows:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
134
( ) Czyetzybody
tjbody
++⋅⋅⋅
−=Ψ ⋅⋅ 22
2,, ωωφ
Equation 3.3–66
For the restrained body in waves, only the force in the horizontal direction and the momentabout the longitudinal axis of the body will be calculated. One gets in beam waves only the iny -point-symmetric part of the potential and the in y -symmetric part of the stream function ofthe wave (see Appendix I), respectively:
( )
[ ] [ ] [ ] ( )body
tjbodyS
yzhz
etzy
⋅⋅⋅⋅⋅−⋅
⋅⋅⋅
=Ψ ⋅⋅
0000 coscoshtanhsinh
,,
νννννωζ ω
Equation 3.3–67
and in oblique waves:
( ) ( )
( )[ ] [ ] [ ] ( )
( ) [ ] [ ] [ ] body
y
tjbodyS
dyzhzy
zhz
y
xetzyx
∫ ⋅⋅⋅⋅−⋅⋅⋅⋅
⋅−⋅−
⋅⋅⋅−⋅
⋅⋅⋅⋅+
⋅⋅⋅⋅⋅⋅
=Ψ ⋅⋅
1
00000
20
10010
10
011
coshtanhsinhsinsin
sin1
coshtanhsinh
sincossin
cossin,,,
νννµν
µν
νννµνµ
µννωζ ω
Equation 3.3–68
3.3.4.2 Potentials
3.3.4.2.1 2-D Radiation Potential
In a similar way as Equation 3.3–22 for heave, the three-dimensional radiation potential forsway and roll can be derived as:
( ) ( ) ( )( )
( )
( ) ξ
ν
ξξω
ddkdkyk
hkkkkhkk
hzkkkk
xkAetzyx
yxy
yxyxyx
yxyx
x
L
L
tjr
⋅⋅⋅⋅
⋅
⋅+⋅+−
⋅+⋅
−⋅+⋅+
⋅−⋅⋅⋅=Φ
∫
∫∫
∞
∞+
−
⋅⋅
sin
sinhcosh
cosh
cos,,,
0222222
2222
0
2
2
00
Equation 3.3–69
This expression reduces for the two-dimensional case ( 0=xk and kk y = ) into:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
135
( )( )[ ]
[ ] [ ] ( ) dkykhkkhk
hzkk
Aetzy tjr
⋅⋅⋅⋅⋅−⋅⋅
−⋅⋅
⋅⋅=Φ
∫∞
⋅⋅
sinsinhcosh
cosh
,,
0
00
ν
ω
Equation 3.3–70
With the Sommerfeld radiation condition in Equation 3.3–6 and Appendix III, the totalradiation potential becomes:
( ) ( )( )[ ]
[ ] [ ] ( )
( )[ ] [ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
⋅⋅−⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅
−⋅⋅
⋅⋅=Φ⋅+Φ
∫∞
⋅⋅
yhhh
hhzj
dkykhkkhk
hzkk
Aetzyjtzy tjjr
0000
000
0
000
sincoshsinh
coshcosh
sinsinhcosh
cosh
,,,,
νννν
νννπ
ν
ω
Equation 3.3–71
and the stream function becomes:( ) ( )
( )[ ][ ] [ ] ( )
( )[ ] [ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
⋅⋅−⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅
−⋅⋅
⋅⋅=Ψ⋅+Ψ
∫∞
⋅⋅
yhhh
hhzj
dkykhkkhk
hzkk
Aetzyjtzy tjjr
0000
000
0
000
coscoshsinh
coshsinh
cossinhcosh
sinh
,,,,
νννν
νννπ
ν
ω
Equation 3.3–72
Potential and stream function are divided in:
( )
[ ] [ ][ ] [ ] ( )
( )[ ] [ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
⋅⋅−⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅−
⋅
+⋅⋅⋅−
⋅
⋅⋅=Φ⋅+Φ+Φ=Φ
∫
∫∞ ⋅−
∞ ⋅−
⋅⋅∞
yhhh
hhzj
dkykhkkhkzkkzk
kek
dkykk
ek
Aej
hk
zk
tjjradr
0000
000
0
0
00000
sincoshsinh
coshcosh
sinsinhcoshcoshsinh
sin
νννν
νννπ
νν
ν
ν
ω
Equation 3.3–73
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
136
( )
[ ] [ ][ ] [ ] ( )
( )[ ] [ ][ ] [ ] ( )
⋅⋅⋅⋅⋅+⋅
⋅⋅−⋅⋅⋅⋅
+⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅−
⋅
+⋅⋅⋅−
⋅
⋅⋅=Ψ⋅+Ψ+Ψ=Ψ
∫
∫∞ ⋅−
∞ ⋅−
⋅⋅∞
yhhh
hhzj
dkykhkkhkzkkzk
kek
dkykk
ek
Aej
hk
zk
tjjradr
0000
000
0
0
00000
coscoshsinh
coshsinh
cossinhcoshsinhcosh
cos
νννν
νννπ
νν
ν
ν
ω
Equation 3.3–74
3.3.4.2.2 2-D Multi-Potential
The two-dimensional multi-pole potential becomes:
( ) ( ) ( )
( ) ( )
[ ] [ ][ ] [ ] ( )
( )
[ ]( )[ ]
[ ] [ ] ( )yhhh
hzh
Aetzy
dkykhkkhkzkkzk
ekkAetzy
dkykekkAetzy
n
ntj
nj
hknn
tjnrad
zknn
tjnr
⋅⋅⋅⋅⋅+⋅
−⋅⋅
⋅⋅
⋅⋅+=Φ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅+⋅⋅−=Φ
⋅⋅⋅⋅⋅+⋅⋅−=Φ
+
⋅⋅
⋅−−∞
⋅⋅
⋅−−∞
⋅⋅∞
∫
∫
0000
0
0
120
12
0
12
0
sincoshsinh
coshcosh
,,
sinsinhcoshcoshsinh
,,
sin,,
νννν
νν
νπ
νν
ν
ν
ω
ω
ω
Equation 3.3–75
The related stream function is:
( ) ( ) ( )
( ) ( )
[ ] [ ][ ] [ ] ( )
( )
[ ]( )[ ]
[ ] [ ] ( )yhhh
hzh
Aetzy
dkykhkkhkzkkzk
ekkAetzy
dkykekkAetzy
n
ntj
nj
hknn
tjnrad
zknn
tjnr
⋅⋅⋅⋅⋅+⋅
−⋅⋅
⋅⋅
⋅⋅+=Ψ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−⋅⋅
⋅⋅⋅+⋅⋅−=Ψ
⋅⋅⋅⋅⋅+⋅⋅+=Ψ
+
⋅⋅
⋅−−∞
⋅⋅
⋅−−∞
⋅⋅∞
∫
∫
0000
0
0
120
12
0
12
0
coscoshsinh
sinhcosh
,,
cossinhcoshsinhcosh
,,
cos,,
νννν
νν
νπ
νν
ν
ν
ω
ω
ω
Equation 3.3–76
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
137
3.3.4.2.3 Total Potentials
With exception of the complex constant nA with ∞≤≤ n0 , the potential is known:
( ) ( ) ( )( )( )
[ ] [ ][ ] ( )( )
[ ] [ ]
( )
( ) ( )( ) [ ] [ ]
[ ]( )( )
[ ] [ ]
∑ ∫
∫
∞
=+
∞+
⋅⋅
∞
⋅⋅
⋅⋅⋅+⋅−⋅+⋅
⋅⋅
⋅⋅
−⋅⋅⋅−⋅⋅
−⋅+⋅⋅⋅−
⋅⋅+
⋅−
⋅⋅⋅+⋅−⋅+⋅⋅⋅
⋅⋅⋅
+⋅⋅⋅−⋅⋅
−⋅+⋅⋅
⋅⋅+⋅+=Ψ⋅+Φ
1
000
0
0
120
0
1222
000
000
0
0
coshsinhsin
cosh
sinhcoshsin
coshsinhsincosh
sinhcoshsin
,,,,
nn
n
njnr
tj
ojrtj
hhhhziy
hj
dkhkkhk
hziykkk
AjA
e
hhhhziyh
j
dkhkkhk
hziykk
AjAetzyitzy
νννν
ννπ
νν
ννννννπ
ν
ω
ω
Equation 3.3–77
Writing this in a similar way as for heave provides:
( )( )
( ) ∑∞
= ∞
∞⋅⋅
⋅++⋅⋅−
⋅−+⋅⋅+=Φ
0
,,n njnrnradnrnj
njnjnradnrnrtj
AAj
AAetzy
φφφ
φφφω
Equation 3.3–78
and
( )( )
( ) ∑∞
= ∞
∞⋅⋅
⋅++⋅⋅+
⋅−+⋅⋅+=Ψ
0
,,n njnrnradnrnj
njnjnradnrnrtj
AAj
AAetzy
ψψψ
ψψψω
Equation 3.3–79
with:( )
( ) rollfor
swayfor ''
''
jrjr
jrjr
AjAAjA
AjAUAjA
⋅+⋅⋅=⋅+
⋅+⋅=⋅+
ωφ
'nA has dimension [ ]22 +nL .
nnA φ⋅' and nnA ψ⋅' have for sway dimension [ ]L and for roll dimension [ ]2L .
The determination of the coefficients 'nA follow from the boundary conditions at the body
contour.
3.3.4.3 Expansion of Potential Parts
For 022 →+⋅=⋅ zyr νν :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
138
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
⋅⋅
⋅+⋅
⋅+
+⋅⋅
⋅+⋅
⋅+⋅+
⋅⋅−+
+=
⋅⋅
⋅+⋅
⋅+
−⋅⋅
⋅+⋅
⋅+⋅+
⋅⋅++
−=
∑
∑
∑
∑
∞
=
∞
=⋅−∞
∞
=
∞
=⋅−∞
yyizmmz
y
yyizmm
r
ezy
z
yyizmmz
y
yyizmm
r
ezy
y
m
mm
m
m
m
zr
m
mm
m
m
m
zr
νν
νν
νγνψ
νν
νννγ
νφ
ν
ν
sinIm!
arctan
cosRe!
ln
cosIm!
arctan
sinRe!
ln
1
1
220
1
1
220
Equation 3.3–80
with the Euler constant: 57722.0=γ .
For ∞→+⋅=⋅ 22 zyr νν :
( ) ( ) ( )
( ) ( ) ( )yyy
eyizr
mzy
z
yyy
eyizr
mzy
y
zM
m
mmmr
zM
m
m
mmr
⋅⋅⋅⋅⋅−
⋅+⋅
⋅−
⋅−+
+=
⋅⋅⋅⋅⋅−
⋅+⋅
⋅−
⋅++
−=
⋅−
=∞
⋅−
=∞
∑
∑
ννπν
νψ
ννπν
νφ
ν
ν
sinRe!1
cosIm!1
12220
12220
Equation 3.3–81
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
⋅+⋅−⋅+⋅⋅−=
⋅+⋅+⋅+⋅⋅−=
−+−+∞
−+−+∞
nnnnr
nnnnr
ziyn
ziyn
ziyn
ziyn
2121
2121
Re2
Im!21
Im2
Re!21
νψ
νφ
Equation 3.3–82
( )( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ∑
∑∞
=
+
∞
=
+
⋅+⋅+
⋅−⋅+⋅++
=
⋅+⋅+⋅−⋅+⋅++=
0
2120
0
2120
Re!212
Re!1232
Im!212Im
!1232
t
ttrad
t
ttrad
yizttG
yizttG
yizttGyiz
ttG
νψ
νφ
Equation 3.3–83
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )
∑
∑
∞
=
+
∞
=
+
⋅+⋅−+⋅−++
⋅−
⋅+⋅+
++⋅−+++
=
⋅+⋅−+⋅−++
⋅−
⋅+⋅+
++⋅−+++
=
0 22
122
0 22
122
Re!2
122122
Re!12
122322
Im!2
122122
Im!12
122322
t t
t
nrad
t t
t
nrad
yizt
ntGntG
yizt
ntGntG
yizt
ntGntG
yizt
ntGntG
νν
ν
ψ
νν
ν
φ
Equation 3.3–84
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
139
[ ][ ] [ ]
( ) ( )
[ ] ( ) ( ) [ ]
[ ] [ ]
( ) ( )
[ ] ( ) ( )
⋅+⋅⋅⋅−
⋅+⋅+
⋅⋅⋅⋅+⋅
⋅⋅⋅+=
⋅+⋅⋅⋅−
⋅+⋅+
⋅⋅⋅⋅+⋅
⋅⋅⋅+=
∑
∑
∑
∑
∞
=
∞
=
++
∞
=
∞
=
++
0
22
00
0
1212
0
000
02
0
0
22
00
0
1212
0
000
02
0
Re!2
tanh
Re!12
coshsinhcosh
Im!2
tanh
Im!12
coshsinhcosh
t
tt
t
tt
oj
t
tt
t
tt
oj
yizt
h
yizt
hhhh
yizt
h
yizt
hhhh
νν
ν
νννννπψ
νν
ν
νννννπφ
Equation 3.3–85( ) ( )( ) ( ) jn
nj
jn
nj
022
012
0
022
012
0
ψνννψ
φνννφ
⋅−⋅=
⋅−⋅=−
−
Equation 3.3–86
3.3.4.4 Zero-Frequency Potential
Grim [1956, 1957] gives for the horizontal motions at zero frequency the complex potential:
( ) ( )
( ) ( ) ( ) ( )
⋅−⋅++⋅+⋅+
+⋅+⋅=⋅+∑∑ ∞
=
+−+−
+−∞
=
1
1212
12
0 22m
nn
n
nn mhiziymhiziy
ziyAi ψφ
Equation 3.3–87
For Lewis forms this becomes:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
140
( ) ( ) ( )
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( )
⋅⋅⋅
⋅
+⋅
+
++
⋅⋅⋅−⋅
+
⋅⋅⋅
⋅
+⋅−
⋅=
⋅
⋅+⋅+
⋅
+
++⋅−
⋅⋅⋅⋅⋅
+
⋅+⋅⋅
+⋅−⋅
⋅=⋅+
∑∑ ∑∑
∑ ∑
∑
∑ ∑
∑
∑
∞
=
∞
=
+
= =
⋅+−−−
++−+
∞
= =
⋅+++⋅−−
∞
=
∞
=
∞
=+⋅−⋅−⋅+
+−
∞
=
⋅−⋅−⋅+⋅−
∞
=
1 0
12
0 0
1222
12
0 0
1222
0
1 0123
12
0
4212
0
12
12
122
21
2
21
2
12
1221
22
21
m p
p
l
l
k
klpikkl
npnp
p
p
l
lpnillpp
nn
m ppiii
p
n
p
piipni
nn
ebak
l
l
p
p
pn
Hm
ebal
p
p
pn
A
Hmebeae
p
np
Hmii
ebeap
pne
Ai
θ
θ
θθθ
θθθ
ψφ
Equation 3.3–88
These sums converge as long as:
( ) 1122
33
<+−⋅⋅⋅+⋅+
=⋅
⋅+⋅+ ⋅−⋅−⋅+⋅−⋅−⋅+
baHTmebeae
Hmebeae iiiiii θθθθθθ
Equation 3.3–89
Because 1≥m , it follows the condition:
baebea
HTii
+−⋅+⋅+
≥⋅⋅−⋅−
11
242! θθ
Equation 3.3–90
The potential converges too when:
TB
baba
HT⋅
=+−++
≥⋅21
12 or hB ⋅≤ 4
Equation 3.3–91
3.3.4.5 Hydrodynamic Loads
The hydrodynamic force at sway oscillations in still water becomes:
( ) dzAAjAA
eU
FjFF
n Snrnjnjnrnjnjnrnr
tj
QjQrQ
⋅⋅+⋅⋅+⋅−⋅
⋅⋅⋅⋅−=
⋅+=
∑∫∞
=
⋅⋅
0
'''' φφφφ
ωρ ω
Equation 3.3–92
and at roll oscillations:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
141
( ) dzAAjAA
e
FjFF
n Snrnjnjnrnjnjnrnr
tj
RjRrR
⋅⋅+⋅⋅+⋅−⋅
⋅⋅⋅⋅−=
⋅+=
∑∫∞
=
⋅⋅
0
''''
2
φφφφ
φωρ ω
Equation 3.3–93
The hydrodynamic moment at sway oscillations in still water becomes:
( ) ( )dzzdyyAAjAA
eU
MjMM
n Snrnjnjnrnjnjnrnr
tj
QjQrQ
⋅+⋅⋅⋅+⋅⋅+⋅−⋅
⋅⋅⋅⋅−=
⋅+=
∑∫∞
=
⋅⋅
0
'''' φφφφ
ωρ ω
Equation 3.3–94
and at roll oscillations:
( ) ( )dzzdyyAAjAA
e
MjMM
n Snrnjnjnrnjnjnrnr
tj
RjRrR
⋅+⋅⋅⋅+⋅⋅+⋅−⋅
⋅⋅⋅⋅−=
⋅+=
∑∫∞
=
⋅⋅
0
''''
2
φφφφ
φωρ ω
Equation 3.3–95
Of course, the coefficients 'nrA and '
njA of sway and roll will differ.
Fictive moment levers are defined by:
RrRr
QrQr
FI
H
mUM
H
φωω
⋅⋅=
⋅⋅=
2"
"
Rj
RRj
Q
QjQj
F
BNH
NUM
H
⋅⋅⋅⋅
=
⋅=
2
φω
Non-dimensional values for the sway motions are:
[ ][ ] [ ]
Qj
QjQj
Qr
QrQr
QjQ
QrH
FTM
TH
FTM
TH
Fhhh
hUy
A
TU
F
T
mC
⋅=
⋅=
⋅⋅⋅⋅+⋅
⋅⋅
⋅⋅==
⋅⋅⋅⋅=
⋅⋅=
000
022
2
22
22
"
coshsinhcosh
22
νννν
ωρνζ
πωρπρ
Equation 3.3–96
and for roll motions:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
142
[ ][ ] [ ]
Rj
RjRj
Rr
RrRr
RjR
RrR
FT
MT
H
FT
MT
H
Mhhh
h
BBA
T
M
T
IC
⋅=
⋅=
⋅⋅⋅⋅+⋅
⋅⋅
⋅⋅⋅⋅
=⋅
=
⋅⋅⋅⋅=
⋅⋅=
000
02
22
2
22
22
424
"
coshsinhcosh4
4
88
νννν
φωρν
φ
ζ
φπωρπρ
Equation 3.3–97
3.3.4.6 Wave Loads
The wave loads are separated in contributions of the undisturbed wave and diffraction:
[ ] [ ] [ ]( ) ( )
( ) ∫
∑
∫
⋅
⋅+⋅⋅+⋅−⋅⋅+
⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅
−
⋅⋅⋅−=
⋅Φ+Φ⋅⋅−=
⋅++=
∞
=
⋅⋅⋅−
⋅⋅
S
nnrnjnjnrnjnjnrnr
xj
tj
SSW
E
dz
AAjAAU
yzhz
e
e
dz
FjFFF
0
''''
0000
cos
321
sinsinsinhtanhcosh
0
φφφφ
µνννννωζ
ωρ
ωρ
µν
ω
Equation 3.3–98
[ ] [ ] [ ]( ) ( )
( ) ( )∫
∑
⋅+⋅⋅
⋅+⋅⋅+⋅−⋅⋅+
⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅
−
⋅⋅⋅−=
⋅++=
∞
=
⋅⋅⋅−
⋅⋅
S
nnrnjnjnrnjnjnrnr
xj
tj
E
dzzdyy
AAjAAU
yzhz
e
e
MjMMM
0
''''
0000
cos
321
sinsinsinhtanhcosh
0
φφφφ
µνννννωζ
ωρ
µν
ω
Equation 3.3–99
The separate parts are:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
143
( )
[ ] [ ] [ ]( ) ( )
dzAAeUF
dzAAeUF
dzyzhz
eF
n Snrnjnjnr
tj
n S
njnjnrnrtj
S
xtj
⋅⋅+⋅⋅⋅⋅⋅−=
⋅⋅−⋅⋅⋅⋅⋅−=
⋅⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅⋅
+=
∑∫
∑∫
∫
∞
=
⋅⋅
∞
=
⋅⋅
⋅⋅−⋅⋅
0
''3
0
''2
0000
cos2
1
sinsinsinhtanhcosh
0
φφωρ
φφωρ
µννννν
ζωρ
ω
ω
µνω
( )
[ ] [ ] [ ]( ) ( ) ( )
( )
( )dzzdyyAAeUM
dzzdyyAAeUM
dzzdyyyzhz
eM
n Snrnjnjnr
tj
n S
njnjnrnrtj
S
xtj
⋅+⋅⋅⋅+⋅⋅⋅⋅⋅−=
⋅+⋅⋅⋅−⋅⋅⋅⋅⋅−=
⋅+⋅⋅⋅⋅⋅⋅⋅⋅−⋅
⋅⋅⋅⋅
+=
∑∫
∑∫
∫
∞
=
⋅⋅
∞
=
⋅⋅
⋅⋅−⋅⋅
0
''3
0
''2
0000
cos2
1
sinsinsinhtanhcosh
0
φφωρ
φφωρ
µννννν
ζωρ
ω
ω
µνω
Dimensionless:
[ ]
[ ] [ ] [ ]( ) ( )
[ ]
[ ] dzAAA
h
AgF
E
dzAAA
h
AgF
E
dzyzhz
Ah
AgF
E
n Snrnjnjnr
x
x
n Snjnjnrnr
x
x
S
x
x
⋅⋅+⋅⋅⋅
−=
⋅⋅⋅⋅=
⋅⋅−⋅⋅⋅
−=
⋅⋅⋅⋅=
⋅⋅⋅⋅⋅⋅⋅−⋅
⋅⋅
⋅+=
⋅⋅⋅⋅=
∑∫
∑∫
∫
∞
=
∞
=
0
''0
0
33
0
''0
0
22
0000
0
0
11
tanh
tanh
sinsinsinhtanhcosh
tanh
φφν
νζρ
φφν
νζρ
µνννν
νν
νζρ
( )21
21
FFTMM
TH W r
+⋅+
= 3
3
FTM
TH W j
⋅=
Equation 3.3–100
The Haskind-Newman relations – see Newman [1962] - are valid too here:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
144
( ) ( )
( ) ( )
⋅⋅−+⋅=
⋅⋅⋅⋅=
⋅⋅−+⋅=
⋅⋅⋅⋅=+
∑
∑
∞
=
−
∞
=
−
1
120
'220
'0
0
3
1
120
'220
'0
0
21
Im
Re
n
nnjj
x
x
E
n
nnrr
x
x
E
AAA
AgF
E
AAA
AgF
EE
νννπ
νζρ
νννπ
νζρ
Equation 3.3–101
3.3.4.7 Solution
To determine the unknowns nA , an equal number of equations have to be formulated. BecauseLewis forms are used only here, a simple approach is possible.All stream function parts and boundary conditions can be given as a Fourier series:
( )( ) ( ) ∑∞
=
⋅⋅+⋅+⋅=0
2cos12sinm
nmnmn mdmc θθψ
or with
( ) ( ) ( ) ( ) ( )( )∑∞
=
⋅+⋅+⋅+−⋅++
⋅+=⋅0
2
12sin12122122
1612cos
k
kkmkmk
mm θ
πθ
in:
( )( ) ∑∞
=
⋅+⋅=0
12sinm
nmn ma θψ
The solution of the by equating coefficients generated equations provide the unknowns 'nA .
3.3.5 Appendices
3.3.5.1 Appendix I: Undisturbed Wave Potential
The general expression of the complex potential of a shallow water wave, travelling in thenegative y -direction, is:
( )( ) [ ]
[ ]( )[ ] ( )
( )[ ] ( )
[ ]( )[ ] ( )
( )[ ] ( )
⋅+⋅⋅−⋅⋅−
⋅+⋅⋅−⋅⋅
⋅⋅⋅
=
⋅+⋅⋅−⋅⋅−⋅+⋅⋅−⋅
⋅⋅⋅
⋅=
=⋅
⋅+−⋅+⋅⋅⋅=Ψ⋅+Φ
tyhzi
tyhz
h
tyhzi
tyhz
h
ch
thziyci WW
ωννωνν
ννωζ
ωννωνν
ννωζ
νω
νων
ζ
00
00
0
00
00
00
00
0
sinsinh
coscosh
cosh
sinsinh
coscosh
sinh
:ith wsinh
cos
Equation 3.3–102
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
145
[ ] ( )[ ] ( )
[ ] ( )[ ]
[ ] [ ] [ ] yjtj
yjtj
W
ezhze
ehzeh
tyhzh
⋅⋅⋅⋅
⋅⋅⋅⋅
⋅⋅⋅⋅−⋅⋅⋅⋅
=
⋅−⋅⋅⋅⋅⋅
⋅=
⋅+⋅⋅−⋅⋅⋅⋅
⋅+=Φ
0
0
000
00
000
sinhtanhcosh
coshcosh
coscoshcosh
νω
νω
ννννωζ
νννωζ
ωννννωζ
[ ] ( )[ ] ( )
[ ] ( )[ ]
[ ] [ ] [ ] yjtj
yjtj
W
ezhzej
ehzeh
j
tyhzh
⋅⋅⋅⋅
⋅⋅⋅⋅
⋅⋅⋅⋅−⋅⋅⋅⋅
⋅=
⋅−⋅⋅⋅⋅⋅
⋅⋅=
⋅+⋅⋅−⋅⋅⋅⋅
⋅−=Ψ
0
0
000
00
000
coshtanhsinh
sinhcosh
sinsinhcosh
νω
νω
ννννωζ
νννωζ
ωννννωζ
For the vertical motions is the in y -symmetrical part of the potential significant. For thehorizontal motions is the in y -point-symmetrical part (multiplied with j , so a phase shift of
090 ) important.
[ ] [ ] [ ] ( )
[ ] [ ] [ ] ( )yzhze
yzhze
tjW V
tjW V
⋅⋅⋅⋅⋅−⋅⋅⋅⋅−=Ψ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅
+=Φ
⋅⋅
⋅⋅
0000
0000
sincoshtanhsinh
cossinhtanhcosh
νννννωζ
νννννωζ
ω
ω
[ ] [ ] [ ] ( )
[ ] [ ] [ ] ( )yzhze
yzhze
tjW H
tjW H
⋅⋅⋅⋅⋅−⋅⋅⋅⋅
−=Ψ
⋅⋅⋅⋅⋅−⋅⋅⋅⋅
−=Φ
⋅⋅
⋅⋅
0000
0000
coscoshtanhsinh
sinsinhtanhcosh
νννννωζ
νννννωζ
ω
ω
Equation 3.3–103
When the wave travels in the wx -direction, the potential becomes:
( ) [ ] [ ] [ ] zhze wxtjW ⋅⋅⋅−⋅⋅⋅
⋅=Φ ⋅−⋅⋅
000 sinhtanhcosh0 ννννωζ νω
With:
µµµµ
cossin
sincos
⋅+⋅=⋅−⋅=
yxy
yxx
w
w
Equation 3.3–104
the potential becomes:
( ) [ ] [ ] [ ] zhzee xyjtjW V ⋅⋅⋅−⋅⋅⋅⋅
⋅=Φ ⋅−⋅⋅⋅⋅⋅
000cossin sinhtanhcosh0 ννν
νωζ µµνω
Equation 3.3–105
This results for the vertical motions in:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
146
( ) [ ] [ ] [ ] ( )µνννννωζ µνω sincossinhtanhcosh 0000
cos0 ⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅
+=Φ ⋅⋅−⋅⋅ yzhze xtjW V
Equation 3.3–106
and for the horizontal motions in:
( ) [ ] [ ] [ ] ( )µνννννωζ µνω sinsinsinhtanhcosh 0000
cos0 ⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅
−=Φ ⋅⋅−⋅⋅ yzhze xtjW H
Equation 3.3–107
3.3.5.2 Appendix II: Series Expansions of Hyperbolic Functions
With:
β⋅±
⋅±
⋅=
⋅+=⋅±i
zy
i
er
ezyyizarctan
22
the following series expansions can be found.
[ ] ( ) [ ] [ ]( )[ ] ( )[ ]
[ ] [ ] ( )
( )( )
( )( )
( ) ( )
( ) ( ) ∑
∑
∑∑
∞
=
∞
=
∞
=
⋅⋅−∞
=
⋅⋅+
⋅−⋅+
⋅+⋅=
⋅⋅⋅
=
⋅⋅
+⋅⋅
⋅=
⋅⋅+⋅⋅⋅=
⋅−⋅+⋅+⋅⋅=
⋅⋅⋅⋅=⋅⋅⋅
0
22
0
2
0
22
0
22
Re!2
2cos!2
!2!221
coshcosh21
coshcosh21
coshcoshcoscosh
t
tt
t
t
t
tit
t
tit
ii
yizt
k
ttrk
etrk
etrk
erkerk
yizkyizk
ykizkykzk
β
ββ
ββ
[ ] ( ) [ ] [ ]( )[ ] ( )[ ]
[ ] [ ] ( )( )
( ) ( )( )
( )
( )( ) ( )( )
( ) ( ) ∑
∑
∑∑
∞
=
++
∞
=
+
∞
=
⋅+⋅−+∞
=
⋅+⋅++
⋅−⋅+
⋅+⋅+
=
⋅+⋅+
⋅=
⋅+
⋅+⋅
+⋅
⋅=
⋅⋅+⋅⋅⋅=
⋅−⋅+⋅+⋅⋅=
⋅⋅⋅⋅=⋅⋅⋅
0
1212
0
12
0
1212
0
1212
Re!12
12cos!12
!12!1221
sinhsinh21
sinhsinh21
coshsinhcossinh
t
tt
t
t
t
tit
t
tit
ii
yiztk
ttrk
etrk
etrk
erkerk
yizkyizk
ykizkykzk
β
ββ
ββ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
147
[ ] ( ) [ ] [ ]( )[ ] ( )[ ]
[ ] [ ] ( )
( )( )
( )( )
( ) ( )
( ) ( ) ∑
∑
∑∑
∞
=
∞
=
∞
=
⋅⋅−∞
=
⋅⋅+
⋅−⋅+
⋅+⋅=
⋅⋅⋅
=
⋅⋅
−⋅⋅
⋅−=
⋅⋅−⋅⋅⋅−=
⋅−⋅−⋅+⋅⋅−=
⋅⋅⋅⋅⋅−=⋅⋅⋅
0
22
0
2
0
22
0
22
Im!2
2sin!2
!2!22
coshcosh2
coshcosh2
sinhsinhsinsinh
t
tt
t
t
t
tit
t
tit
ii
yizt
k
ttrk
etrk
etrki
erkerki
yizkyizki
ykizkiykzk
β
ββ
ββ
[ ] ( ) [ ] [ ]( )[ ] ( )[ ]
[ ] [ ] ( )( )
( ) ( )( )
( )
( )( ) ( )( )
( ) ( ) ∑
∑
∑∑
∞
=
++
∞
=
+
∞
=
⋅+⋅−+∞
=
⋅+⋅++
⋅−⋅+
⋅+⋅+
=
⋅+⋅+
⋅=
⋅+
⋅−⋅
+⋅
⋅−=
⋅⋅−⋅⋅⋅−=
⋅−⋅−⋅+⋅⋅−=
⋅⋅⋅⋅⋅−=⋅⋅⋅
0
1212
0
12
0
1212
0
1212
Im!12
12sin!12
!12!122
sinhsinh2
sinhsinh2
sinhcoshsincosh
t
tt
t
t
t
tit
t
tit
ii
yiztk
ttrk
etrk
etrki
erkerki
yizkyizki
ykizkiykzk
β
ββ
ββ
3.3.5.3 Appendix III: Treatment of Singular Points
The determination of j0Φ and its terms - which can be added to r0Φ in Equation 3.3–23 with
which the by jr j 00 Φ⋅+Φ described flow of the waves (travelling from both sides of the body
away) is given - is also possible in another way. This approach is based on work carried out byRayleigh and is given in the literature by Lamb [1932] for an infinite water depth.In this approach, a viscous force w⋅⋅ µρ will be included in the Euler equations, µ is thedynamic viscosity and w is the velocity. Because the fluid is assumed to be non-viscous, in alater stage this dynamic viscosity µ will be set to zero.From the Euler equation follows with this viscosity force the with time changing pressurechange:
∂Φ∂
−∂Φ∂
⋅−∂Φ∂
⋅⋅=∂∂
→ 2
2
0lim
ttzg
tp µρ
µ
From this follows the approach as given in a subsection before for the two-dimensionalradiation potential:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
148
( ) ( )
( )[ ][ ] [ ]
( )
( )
[ ] [ ]
[ ] [ ]( )
( ) ( ) jadradjrtj
hk
zk
tj
tjjr
jjAe
dkykhkkhk
gj
zkkzkg
j
kg
j
e
dkykk
gj
e
Ae
dkykhkkhk
gj
hzk
Aetzyjtzy
00000
0
0
0
0
00
000
cossinhcosh
coshsinh
cos
lim
cossinhcosh
coshlim
,,,,
φφφφ
µων
µων
µων
µων
µων
ω
µ
ω
µ
ω
⋅++⋅+⋅⋅=
⋅⋅⋅⋅⋅−⋅⋅
⋅⋅−
⋅⋅−⋅⋅
⋅⋅−
⋅−⋅⋅−
+⋅⋅⋅−
⋅⋅−
⋅⋅=
⋅⋅⋅⋅⋅−⋅⋅
⋅⋅−
−⋅
⋅⋅=Φ⋅+Φ
∞∞⋅⋅
∞ ⋅−
∞ ⋅−
→
⋅⋅
∞
→
⋅⋅
∫
∫
∫
The first integral leads to the potential in Equation 3.3–29 and the second integral can beexpanded as follows:
[ ] [ ]
[ ] [ ]( )
( ) ( )
( ) ( )
[ ] [ ]
( ) ( )
( ) ( )
( ) ( )
+⋅++
⋅
⋅+
−+⋅+
⋅
⋅⋅−
−=
⋅
⋅⋅−⋅⋅
⋅⋅−⋅
−
⋅⋅−
⋅
⋅
⋅+
−+⋅+
⋅
⋅⋅−
=
⋅⋅⋅⋅⋅−⋅⋅
⋅⋅−
⋅⋅−⋅⋅
⋅⋅−
⋅−
⋅⋅−
=
⋅+
∑
∑∫
∫
∞
=
+
→
∞
=
∞ +⋅−
+
→
∞ ⋅−
→
0
212
0
0
0
12
212
0
00
00
1212
!2Re
!12Re
lim
sinhcosh
!2Re
!12Re
lim
cossinhcosh
coshsinh
lim
t
tt
t
thk
tt
hk
jadrad
tHjtG
tyiz
tyiz
gj
dk
hkkhkg
jkg
j
ke
tyiz
tyiz
gj
dkykhkkhk
gj
zkkzkg
j
kg
j
e
j
µων
µωνµων
µων
µων
µων
µων
φφ
µ
µ
µ
The function:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
149
( ) ( )
[ ] [ ]
⋅⋅⋅−⋅⋅
⋅⋅−
⋅
⋅⋅−−
=⋅+ ∫∞
⋅−
→0
0
sinhcosh
lim
dkhkkhk
gi
k
gik
e
tHitGt
hk
µων
µων
µ
will be normalised as done before:
( ) ( ) ( ) ( )( )
[ ] [ ]
⋅
⋅−⋅
⋅⋅⋅−⋅
⋅
⋅⋅⋅−⋅−
⋅=
=⋅+=⋅+
∫∞ −
→
−
00
1''
sinhcosh
lim du
uuug
hih
gh
ihu
eu
tHitGhtHitG
ut
t
µων
µωνµ
This is a complex integral and must be solved in the complex domain with viuw ⋅+= .The integrand has a singularity for:
gh
ihw⋅⋅
⋅−⋅=µων1
and 2w is the solution of the equation:
[ ] [ ] 0sinhcosh =⋅−⋅
⋅⋅⋅−⋅ www
gh
ihµων
see Figure 3.3–5-a.
Figure 3.3–5: Treatment of singularities
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
150
( ) [ ] [ ]( )
0
...lim
.........limsinhcosh
lim
00
00
110
=
++=
++=⋅⋅−⋅⋅−
⋅
∫
∫∫ ∫∫
→
→
−
→
III
R
II
R
I
tw
JJdu
dwdwdudwwwwwww
we
µ
µµ
Also:
( ) ( )
[ ] [ ]
IIIR
ut
JJ
du
uuug
hih
gh
ihu
eutHitG
+=
⋅
⋅−⋅
⋅⋅⋅−⋅
⋅
⋅⋅⋅−⋅−
⋅=⋅+
∞→→
∞ −
→ ∫
and 0
00
''
lim
sinhcosh
lim
µ
µ
µων
µων
Because:0lim =
∞→ IRJ
follows:
( ) ( ) ( ) [ ] [ ]( )
⋅⋅−⋅⋅−
⋅−=⋅+ ∫−
→II
wt
dwwwwwww
ewtHitGsinhcosh
lim11
0
''
µ
The real part of this integral:
( ) ( ) [ ] [ ]( )
⋅⋅−⋅⋅−
⋅−= ∫
−
→II
wt
dwwwwwww
ewtG
sinhcoshlimRe
110
'
µ
will be calculated as done before.This integral has no singularity and the boundary 0→µ can be passed before integration; seeFigure 3.3–5-b.
The imaginary part of this integral:
( ) ( ) [ ] [ ]( )
⋅⋅−⋅⋅−
⋅−= ∫
−
→II
wt
dwwwwwww
ewtH
sinhcoshlimIm
110
'
µ
can be calculated numerically in a similar way.
It is also possible to solve this integral independently by using another integral path:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
151
( ) [ ] [ ]( )
( ) ( ) 210
0
110
ResidueResiduelim2
lim
sinhcoshlim
wwi
JJJ
dwwwwwww
ew
IVIIIII
wt
+⋅⋅⋅=
++=
⋅⋅−⋅⋅−
⋅
→
→
−
→ ∫
µ
µ
µ
π
IVJ disappears for ∞→R .
It can be found that: IIIII JJ ReRe −= and IIIII JJ ImIm +=
Then it follows:( )
( ) ( )
( ) [ ][ ] [ ] [ ]
⋅−⋅⋅⋅+⋅
⋅⋅⋅⋅=
+⋅−=
−=
−−
→
→
hhhh
hh
ww
JtH
tt
II
01
000
02
10
210
0
'
tanhcoshsinh
cosh
ResidueResiduelim
limIm
νννν
ννπ
πµ
µ
and the imaginary additional potential becomes:
( ) ( ) ( )
( ) ( )
[ ][ ] [ ] [ ]( )
( )( ) ( )
[ ][ ] [ ] [ ]
( ) ( ) [ ] ( ) ( ) ( )ye
yizt
hyizt
hhhh
tyiz
tyiz
hhhh
tyiz
tyiz
tH
z
t
tt
t
tt
ttt
tt
t
tt
jad
⋅⋅⋅−
⋅+⋅
+⋅⋅−⋅+⋅
⋅⋅⋅⋅+⋅
⋅⋅=
+⋅+
⋅−⋅+
⋅
−
⋅⋅⋅+⋅⋅⋅
⋅
=
+⋅+
⋅−⋅+
⋅+=
⋅−
∞
=
++∞
=
∞
=+
∞
=
+
∑∑
∑
∑
νπ
ννν
ννννπ
ν
νννν
ννπ
νφ
ν cos
Re!12
tanhRe!2
coshsinhcosh
!12Re
!2Re
coshsinhcosh
!12Re
!2Re
12
0
1212
00
0
22
0
000
02
0122
2
000
022
0
0
122
0
The same will be found as a difference between Equation 3.3–45 and Equation 3.3–29.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
152
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
153
3.4 Potential Theory of Frank
As a consequence of conformal mapping of a cross section to the unit circle, the cross sectionneeds to have a certain breadth at the water surface. Fully submersed cross sections - such asat the bulbous bow - cannot be mapped. Mapping problems can also appear for cross sectionswith a very high or low area coefficient. These cases require another approach: the pulsatingsource method of Frank or the so-called Frank Close-Fit Method.For explaining this method as it has been used in the computer code SEAWAY, relevant partsof the report of Frank [1967] have been copied to this Section, supplemented with somenumerical improvements.
Hydrodynamic research of horizontal cylinders oscillating in or below the free surface of adeep fluid has increased in importance in the last decades and has been studied by a number ofinvestigators. The history of this subject began with Ursell [1949], who formulated and solvedthe boundary-value problem for the semi-immersed heaving circular cylinder within theframework of linearised free-surface theory. He represented the velocity potential as the sumof an infinite set of multi-poles, each satisfying the linear free-surface condition and eachbeing multiplied by a coefficient determined by requiring the series to satisfy the kinematicboundary condition at a number of points on the cylinder.Grim [1953] used a variation of the Ursell method to solve the problem for two-parameterLewis form cylinders by conformal mapping onto a circle. Tasai [1959] and Porter [1960],using the Ursell approach obtained the added mass and damping for oscillating contoursmappable onto a circle by the more general Theodorsen transformation. Ogilvie [1963]calculated the hydrodynamic forces on completely submerged heaving circular cylinders.
Despite the success of the multi-pole expansion-mapping methods, Frank [1967] discussed theproblem from a different point of view. The velocity potential is represented by a distributionof sources over the submerged cross section. The density of the sources is an unknownfunction (of position along the contour) to be determined from integral equations found byapplying the kinematic boundary condition on the submerged part of the cylinder. Thehydrodynamic pressures are obtained from the velocity potential by means of the linearisedBernoulli equation. Integration of these pressures over the immersed portion of the cylinderyields the hydrodynamic forces or moments.
3.4.1 Notations of Frank
Frank’s notations have been maintained here as far as possible:
)(mA oscillation amplitude in the m -th modeB beam of cross section 0C
0C submerged part of cross sectional contour in rest positiong acceleration of gravity
)(mijI influence coefficient in-phase with displacement on the i -th midpoint
due to the j -th segment in the m -th mode of oscillation)(m
ijJ influence coefficient in-phase with velocity on the i -th midpoint due tothe j -th segment in the m -th mode of oscillation
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
154
( )( )mM ω added mass force or moment for the m -th mode of oscillation at
frequency ωN number of line segments defining submerged portion of half section in
rest position
( )( )mN ω damping force or moment for the m -th mode of oscillation at
frequency ω( )m
in direction cosine of the normal velocity at i -th midpoint for the m -thmode of oscillation
PV Cauchy pricipal value of integral( )m
ap hydrodynamic pressure in-phase with displacement for the m -th modeof oscillation
( )mvp hydrodynamic pressure in-phase with velocity for the m -th mode of
oscillation( )m
jQ source strength in-phase with displacement along the j -th segment forthe m -th mode of oscillation
( )mNjQ + source strength in-phase with velocity along the j -th segment for the
m -th mode of oscillations length variable along 0C
js j -th line segmentT draft of cross sectiont time
( )miv normal velocity component at the i -th midpoint for the m -th mode of
oscillation
1x abscissa of the i -th midpoint
iy ordinate of the i -th midpoint
0y ordinate of the center of rollyixz ⋅+= complex field point in region of fluid domain
iii yixz ⋅+= complex midpoint of i -th segment
iα angle between i -th segment and positive x -axis
ζ complex variable along 0C
jζ j -th complex input point along 0C
jη ordinate of j -th input point
g2ων = wave number
kν k -th irregular wave number for adjoint interior problem
jξ abscissa of the j -th input pointρ density of fluid
( )mΦ velocity potential for th m -th mode of oscillationω radian frequency of oscillation
kω k -th irregular frequency for adjoint interior problem(k -th eigen-frequency)
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
155
3.4.2 Formulation of the Problem
Consider a cylinder, whose cross section is a simply connected region, which is fully orpartially immersed horizontally in a previously undisturbed fluid of infinite depth. The body isforced into simple harmonic motion and it is assumed that steady state conditions have beenattained.The two-dimensional nature of the problem implies three degrees of freedom of motion.Therefore, consider the following three types of oscillatory motions: vertical or heave,horizontal or sway and rotational about a horizontal axis or roll.To use linearised free-surface theory, the following assumptions are made:1. the fluid is incompressible and inviscid,2. the effects of surface tension are negligible,3. the fluid is irrotational and4. the motion amplitudes and velocities are small enough that all but the linear terms of the
free-surface condition, the kinematic boundary condition on the cylinder and the Bernoulliequation may be neglected.
For complete discussions of linearised free-surface theory, Frank refers the reader to Stoker[1957] and Wehausen and Laitone [1960].
Given the above conditions and assumptions, the problem reduces to the following boundary-value problem of potential theory. The cylinder is forced into simple harmonic motion
( ) ( )tA m ⋅⋅ ωcos with a prescribed radian frequency of oscillation ω , where the superscript mmay take on the values 2, 3 and 4, denoting swaying, heaving and rolling motions,respectively.It is required to find a velocity potential:
( )( ) ( )( ) timm eyxtyx ⋅⋅−⋅=Φ ωφ ,Re,,
Equation 3.4–1
satisfying the following conditions:
1. The Laplace equation:
( )( ) ( )
02
2
2
22 =
∂Φ∂
+∂Φ∂
=Φ∇yx
mmm
Equation 3.4–2
in the fluid domain, i.e., for 0<y and outside the cylinder.
2. The free surface condition:( ) ( )
02
2
=∂Φ∂
⋅+∂Φ∂
yg
t
mm
Equation 3.4–3
on the free surface 0=y outside the cylinder, while g is the acceleration of gravity.
3. The seabed boundary condition for deep water:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
156
( )( )
0limlim =∂Φ∂
=Φ∇−∞→−∞→ y
m
y
m
y
Equation 3.4–4
4. The condition of the normal velocity component of the fluid at the surface of theoscillating cylinder being equal to the normal component of the forced velocity of thecylinder. i.e., if nv is the component of the forced velocity of the cylinder in the direction
of the outgoing unit normal vector n , then( )
nm vn =Φ∇⋅
Equation 3.4–5
This is the kinematic boundary condition on the oscillating body surface, being satisfied atthe mean (rest) position of the cylindrical surface.
5. The radiation condition that the disturbed surface of the fluid takes the form of regularprogressive outgoing gravity waves at large distances from the cylinder.
According to Wehausen and Laitone [1960], the complex potential at z of a pulsating pointsource of unit strength at the point ζ in the lower half plane is:
( ) ( ) ( )( )
( )
( ) ( )te
tdkk
ePVzztzG
zi
zki
⋅⋅−
⋅⋅
−⋅+−−−⋅
⋅=
−⋅⋅−
∞ −⋅⋅−
∫
ω
ων
ζζπ
ζ
ζν
ζ
sin
cos2lnln2
1,,
0
*
Equation 3.4–6
so that the real point-source potential is:
( ) ( ) tzGtyxH ,,Re,,,, * ζηξ =
Equation 3.4–7
where:yixz ⋅+= ηξζ ⋅+= i ηξζ ⋅−= i g2ων =
Letting:
( ) ( ) ( ) ( )
( ) ζν
ζ
νζζ
πζ
−⋅⋅−
∞ −⋅⋅−
⋅−
−⋅+−−−⋅
⋅= ∫
zi
zki
ei
dkk
ePVzzzG
Re
2lnlnRe2
1,
0
Equation 3.4–8
then:
( ) ( ) tietzGtyxH ⋅⋅−⋅= ωζηξ ,,Re,,,,
Equation 3.4–9
Equation 3.4–9 satisfies the radiation condition and also Equation 3.4–1 through Equation3.4–4.Another expression satisfying all these conditions is:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
157
( ) tiezGityxH ⋅⋅−⋅⋅=
⋅− ωζ
ωπηξ ,Re
2,,,,
Equation 3.4–10
Since the problem is linear, a superposition of Equation 3.4–9 and Equation 3.4–10 results inthe velocity potential:
( )( ) ( ) ( )
⋅⋅⋅=Φ ∫ ⋅⋅−
0
,Re,,C
tim dsezGsQtyx ωζ
Equation 3.4–11
where 0C is the submerged contour of the cylindrical cross section at its mean (rest) position
and ( )sQ represents the complex source density as a function of the position along 0C .Application of the kinematic boundary condition on the oscillating cylinder at z yields:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )mm
C
C
nAdszGsQn
dszGsQn
⋅⋅=
⋅⋅⋅∇⋅
=
⋅⋅⋅∇⋅
∫
∫
ωζ
ζ
0
0
,Im
0,Re
Equation 3.4–12
where ( )mA denotes the amplitude of oscillation and ( )mn the direction cosine of the normalvelocity at z on the cylinder. Both ( )mA and ( )mn depend on the mode of motion of thecylinder, as will be shown in the following section.The fact that ( )sQ is complex implies that Equation 3.4–12 represent a set of two coupledintegral equations for the real functions ( ) sQRe and ( ) sQIm . The solution of these integralequations and the evaluation of the kernel and potential integrals are described in thefollowing section and in Appendices II and III, respectively.
3.4.3 Solution of the Problem
Since ship sections are symmetrical, this investigation is confined to bodies with right and leftsymmetry.
Take the x -axis to be coincident with the undisturbed free surface of a conventional two-dimensional Cartesian co-ordinate system. Let the cross sectional contour 0C of thesubmerged portion of the cylinder be in the lower half plane, the y -axis being the axis of
symmetry of 0C ; see Figure 3.4–1.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
158
Figure 3.4–1: Axes system and notations, as used by Frank [1967]
Select 1+N points ( )ii ηξ , of 0C to lie in the fourth quadrant so that ( )11,ηξ is located on the
negative y -axis. For partially immersed cylinders, ( )11, ++ NN ηξ is on the positive x -axis. For
fully submerged bodies, 11 ξξ =+N and 01 <+Nη .Connecting these 1+N points by successive straight lines, N straight line segments areobtained which, together with their reflected images in the third quadrant, yield anapproximation to the given contour as shown in Figure 3.4–1.The co-ordinates, length and angle associated with the j -th segment are identified by thesubscript j , whereas the corresponding quantities for the reflected image in the third quadrant
are denoted by the subscript j− , so that by symmetry jj +− −= ξξ and jj +− += ηη for11 +≤≤ Nj .
Potentials and pressures are to be evaluated at the midpoint of each segment. The co-ordinatesof the midpoint of the i -th segment are:
21++
= iiix
ξξ and
21++
= iiiy
ηη for: Ni ≤≤1
Equation 3.4–13
The length of the i -th segment is:
( ) ( )21
21 iiiiis ηηξξ −+−= ++
Equation 3.4–14
while the angle made by the i -th segment with the positive x -axis is given by:
−−
=+
+
ii
iii ξξ
ηηα1
1arctan
Equation 3.4–15
The outgoing unit vector normal to the cross section at the i -th midpoint ( )ii yx , is:
iii jin αα cossin ⋅−⋅=
Equation 3.4–16
where i and j are unit vectors in the directions of increasing x and y , respectively.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
159
The cylinder is forced into simple harmonic motion with radian frequency ω , according to thedisplacement equation:
( ) ( ) ( )tAS mm ⋅⋅= ωcos
Equation 3.4–17
for 4,3,2=m corresponding to sway, heave or roll, respectively.
The rolling oscillations are about an axis through a point ( )0,0 y in the symmetry plane of thecylinder.In the translation modes, any point on the cylinder moves with the velocity:
( ) ( ) ( )tAiv ⋅⋅⋅⋅−= ωω sin :sway 22
Equation 3.4–18( ) ( ) ( )tAjv ⋅⋅⋅⋅−= ωω sin :heave 33
Equation 3.4–19
The rolling motion about ( )0,0 y is illustrated in Figure 3.4–1.
Considering a point ( )ii yx , on 0C , an inspection of this figure yields:
( )20
2 yyxR iii −+= and
=
−=
−=
i
i
i
i
i
ii
Rx
Ryy
xyy
arccos
arcsin
arctan
0
0θ
Therefore, by elementary two-dimensional kinematics, the unit vector in the direction ofincreasing θ is:
jRx
iR
yy
ji
i
i
i
i
iii
⋅+⋅−
−=
⋅+⋅−=
0
cossin θθτ
so that:( ) ( )
( ) ( ) ( )tjxiyyA
SRv
ii
ii
⋅⋅⋅−⋅−⋅⋅=
⋅⋅=
ωω
τ
sin
:roll
04
44
Equation 3.4–20
The normal components of the velocity ( ) ( )mi
mi vnv ⋅= at the midpoint of the i -th segment
( )ii yx , are:( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )txyyAv
tAv
tAv
iiiii
ii
ii
⋅⋅⋅+⋅−⋅⋅+=
⋅⋅⋅⋅+=
⋅⋅⋅⋅−=
ωααω
ωαω
ωαω
sincossin :roll
sincos :heave
sinsin :sway
044
33
22
Equation 3.4–21
Defining:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
160
( )( )
( ) ( )tAv
nm
mim
i ⋅⋅⋅=
ωω sinthen - consistent with the previously mentioned notation - the direction cosines for the threemodes of motion are:
( )
( )
( ) ( ) iiiii
ii
ii
xyyn
n
n
αα
α
α
cossin :roll
cos :heave
sin :sway
04
3
2
⋅+⋅−+=
+=
−=
Equation 3.4–22
Equation 3.4–22 illustrates that heaving is symmetrical, i.e., ( ) ( )33ii nn +− = . Swaying and
rolling, however, are anti-symmetrical modes, i.e., ( ) ( )22ii nn +− −= and ( ) ( )44
ii nn +− −= .Equation 3.4–12 is applied at the midpoints of each of the N segments and it is assumed thatover an individual segment the complex source strength ( )sQ remains constant, although itvaries from segment to segment. With these stipulations, the set of coupled integral equations(Equation 3.4–12) becomes a set of N2 linear algebraic equations in the unknowns:
( )( ) ( )mjj
m QsQ =Re and ( )( ) ( )mjNj
m QsQ +=Im
Thus, for Ni ,...,2,1= :
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )m
im
N
j
N
j
mij
mjN
mij
mj
N
j
N
j
mij
mjN
mij
mj
nAIQJQ
JQIQ
⋅⋅=⋅+⋅−
=⋅+⋅+
∑ ∑
∑ ∑
= =+
= =+
ω1 1
1 1
0
Equation 3.4–23
where the superscript ( )m denotes the mode of motion.
The ''influence coefficients'' ( )mijI and ( )m
ijJ and the potential ( )( )tyx iim ,,Φ are evaluated in
Appendix II. The resulting velocity potential consists of a term in-phase with the displacementand a term in-phase with the velocity.Note: Most authors refer to this first term as a component in phase with the acceleration.
However due to the displacement Equation 3.4–17, Frank deemed it more appropriateto refer to this term as being 180 degrees out-of-phase with the acceleration or in-phase with the displacement.
The hydrodynamic pressure at ( )ii yx , along the cylinder is obtained from the velocitypotential by means of the linearized Bernoulli equation:
( )( )( )
( )tyxt
tyxp ii
m
iim ,,,,,, ωρω
∂Φ∂
⋅−=
Equation 3.4–24
as:( )( ) ( )( ) ( ) ( )( ) ( )tyxptyxptyxp ii
mvii
maii
m ⋅⋅+⋅⋅= ωωωωω sin,,cos,,,,,
Equation 3.4–25
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
161
where ( )map and ( )m
vp are the hydrodynamic pressures in-phase with the displacement and in-phase with the velocity, respectively and ρ denotes the density of the fluid.As indicated by the notation of Equation 3.4–24 and Equation 3.4–25, the pressure as well asthe potential is a function of the oscillation frequency ω .The hydrodynamic force or moment (when 4=m ) per unit length along the cylinder,necessary to sustain the oscillations, is the integral of ( ) ( )mm np ⋅ over the submerged contourof the cross section 0C . It is assumed that the pressure at the i -th midpoint is the meanpressure for the i -th segment, so that the integration reduces to a summation, whence:
( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ∑
∑
=
=
⋅⋅⋅=
⋅⋅⋅=
N
ii
miii
mv
m
N
ii
miii
ma
m
snyxpN
snyxpM
1
1
,,2
,,2
ωω
ωω
Equation 3.4–26
for the added mass and damping forces or moments, respectively.The velocity potentials for very small and very large frequencies are derived and discussed inthe next section.
3.4.4 Low and High Frequencies
For very small frequencies, i.e., as 0→ω , the free-surface condition in Equation 3.4–3 of thesection formulating the problem degenerates into the wall-boundary condition:
0=∂Φ∂y
Equation 3.4–27
on the surface of the fluid outside the cylinder, whereas for extremely large frequencies, i.e.,when ∞→ω , the free-surface condition becomes the ''impulsive'' surface condition:
0=Φ
Equation 3.4–28
on 0=y and outside the cylinder.Equation 3.4–2, Equation 3.4–4 and Equation 3.4–5 remain valid for both asymptotic cases.The radiation condition is replaced by a condition of boundedness at infinity.Therefore, there is a Neumann problem for the case 0→ω and a mixed problem when
∞→ω . The appropriate complex potentials for a source of unit strength at a point ζ in thelower half plane are:
( ) ( ) ( ) 00 lnln2
1, KzzzG +−−−⋅
⋅= ζζ
πζ
Equation 3.4–29
and:
( ) ( ) ( ) ∞∞ +−−−⋅⋅
= KzzzG ζζπ
ζ lnln2
1,
Equation 3.4–30
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
162
for the Neuman and mixed problems, respectively, where 0K and ∞K are not yet specifiedconstants.Let:
( ) ( ) ζηξφ ,Re,,, zGyx aa =so that the velocity potentials for the m -th mode of motion are:
( )( ) ( )( ) ( )∫ ⋅⋅=Φ0
,,,,C
am
am
a dsyxsQyx ηξφ
Equation 3.4–31
for 0=a , and ∞=a , where ( )maQ is the expression for the source strength as a function of
position along the submerged contour of the cross section 0C .An analysis similar to the one in the section on formulating the problem leads to the integralequation:
( ) ( )( ) ( ) ( ) ( )mm
C
am
a nAdsyxsQn ⋅=⋅⋅⋅∇⋅ ∫0
,,, ηξφ
Equation 3.4–32
which - after application at the N segmental midpoint - yields a set of N linear algebraicequations in the N unknown source strengths jQ .It remains to be shown whether these two problems are, in the language of potential theory,well posed, i.e., whether the solutions to these problems lead to unique forces or moments.The mixed problem raises no difficulty, since as ∞→z , ( ) 0, →∞ ζzG . In fact 0=∞K , whichcan be inferred from the pulsating source-potential Equation 3.4–8 by letting ∞→ν .Considering the Neumann problem, note that the constant 0K in the Green's function equation(Equation 3.4–29) yields by integration an additive constant K to the potential. However, fora completely submerged cylinder the cross sectional contour 0C is a simply closed curve, sothat the contribution of K in integrating the product of the pressure with the direction cosineof the body-surface velocity vanishes. For partially submerged bodies 0C is no longer closed.
But since ( ) ( )mi
mi nn +− −= for m being even,
( ) 00
=⋅⋅∫C
m dsnK
so that the swaying force and rolling moment are unique.The heaving force on a partially submerged cylinder is not unique for, in this case,
( ) ( )33ii nn +− = , so that:
( ) 00
3 ≠⋅⋅∫C
dsnK
The constant 0K may be obtained by letting 0→ν in the pulsating source-potential Equation3.4–8.
3.4.5 Irregular Frequencies
John [1950] proved the existence and uniqueness of the solutions to the three- and two-dimensional potential problems pertaining to oscillations of rigid bodies in a free surface. The
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
163
solutions were subject to the provisions that no point of the immersed surface of the bodywould be outside a cylinder drawn vertically downward from the intersection of the body withthe free surface and that the free surface would be intersected orthogonally by the body in itsmean or rest position.John [1950] also showed that for a set of discrete ''irregular'' frequencies the Green's function-integral equation method failed to give a solution. He demonstrated that the irregularfrequencies occurred when the following adjoint interior-potential problem had eigen-frequencies.
Let ( )yx,ψ be such that:
1. 02
2
2
2
=∂∂
+∂∂
yxψψ
inside the cylinder in the region bounded by the immersed surface of the
body and the extension of the free surface inside the cylinder.
2. 0=⋅−∂∂ ψνψ
ky on the extension of the free surface inside the cylinder, kν being the wave
number corresponding to the irregular frequency kω , =k 1,2,3,…etc.
3. 0=ψ on the surface of the cylinder below the free surface.
For a rectangular cylinder with beam B and draft T , the irregular wave numbers may beeasily obtained by separation of variables in the Laplace equation. Separating variables givesthe eigen-functions:
⋅⋅⋅
⋅⋅⋅=
Byk
BxkBkk
ππψ sinhsin for: =k 1,2,3,…etc.
where kB are Fourier coefficients to be determined from an appropriate boundary condition.Applying the free surface condition (Equation 3.4–2) on Ty = for Bx <<0 , the eigen-wavenumbers (or irregular wave numbers):
⋅⋅⋅⋅=
BTk
Bk
kππν coth
Equation 3.4–33
are obtained for k = 1, 2, 3, ..., etc.In particular, the lowest such irregular wave number is given by:
⋅⋅=
BT
Bππν coth1
Equation 3.4–34
Keeping T fixed in Equation 3.4–34 but letting B vary and setting Bb π= , then from theTaylor expansion:
[ ] ( )
+⋅−⋅+
⋅⋅=⋅⋅ ......
4531coth
3TbTbTb
bTbb
it is seen that as 0→b , which is equivalent to ∞→B , T11 →ν .Therefore, for rectangular cylinders of draft T ,
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
164
T11 ≥ν
Equation 3.4–35
is a relation that John proved for general shapes complying with the restrictions previouslyoutlined.For a beam-to-draft ratio of 5.2=TB : 48.11 =ν , while for 0.2=TB : 71.11 =ν .
At an irregular frequency the matrix of influence coefficients of Equation 3.4–23 becomessingular as the number of defining points per cross section increases without limit, i,e., as
∞→N . In practice, with finite N , the determinant of this matrix becomes very small, notonly at the irregular frequency but also at an interval about this frequency. This interval can bereduced by increasing the number of defining points N for the cross section.
Most surface vessels have nearly constant draft over the length of the ship and the maximumbeam occurs at or near amidships, where the cross section is usually almost rectangular, sothat for most surface ships the first irregular frequency 1ω is less for the midsection than forany other cross section.As an example, for a ship with a 7:1 length-to-beam and a 5:2 beam-to-draft, the first irregularwave encounter frequency - in non-dimensional form with L denoting the ship length -occurs at 09.51 ≈⋅ gLω , which is beyond the range of practical interest for ship-motionanalysis.Therefore, for slender surface vessels, the phenomenon of the first irregular frequency ofwave encounter is not too important.
Increasing the number of contour line elements (or panels in 3-D) does not remove theirregular frequency, but tends to restrict the effects to a narrower band around it; see forinstance Huijsmans [1996]. It should be mentioned too that irregular frequencies appear forfree surface piercing bodies only; fully submerged bodies do not display these characteristics.An effective method to reduce the effects of irregular frequencies is ''closing'' the body bymeans of a discretisation of the free surface inside the body, i.e. putting a ''lid'' on the freesurface inside the body.See here the computed added mass and damping of a hemisphere in Figure 3.4–2. The solidline in this figure results from including the ''lid''.
0
5 00
10 00
15 00
20 00
0 1 2 3
h ea ve
su rge /swa y
Fre qu e ncy (ra d/ s)
Ma
ss (
ton
)
0
25 0
50 0
75 0
1 00 0
1 25 0
0 1 2 3
h ea ve
su rge /swa y
Fre qu en cy (ra d /s)
Da
mp
ing
(to
n/s
)
Figure 3.4–2: Effect of use of ''Lid-Method'' on irregular frequencies
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
165
3.4.6 Appendices
3.4.6.1 Appendix I: Evaluation of Principle Value Integrals
The real and imaginary parts of the principle value integral:( )
dkk
ePV
zki
⋅−∫
∞ −⋅⋅−
0 ν
ζ
are used in evaluating some of the kernel and potential integrals.The residue of the integrand at ν=k is ( )ζν −⋅⋅− zie , so that:
( ) ( ) ( )ζνζζ
πνν
−⋅⋅−∞ −⋅⋅−∞ −⋅⋅−
⋅−⋅−
=⋅− ∫∫ zi
zkizki
eidkk
edk
ke
PV00
Equation 3.4–36
where the path of integration is the positive real axis indented into the upper half plane aboutν=k .
Notice hereby that:( ) 02 >= gων 0Im <z 0Im ≤ζ
The transformation ( ) ( )ζνω −⋅−⋅= zki converts the contour integral on the right hand sideof Equation 3.4–36 to:
( ) ( )( )
( ) ( )( )
( )
( ) ( ) ( )
<−>−⋅
+
⋅−⋅⋅−⋅−
+−⋅⋅−+
⋅−=
<−>−⋅
+−⋅⋅−⋅−=
⋅⋅−=⋅−
∑
∫∫
∞
=
−⋅⋅−
−⋅⋅−
∞
−⋅⋅−
−−⋅⋅−
∞ −⋅⋅−
0
0 :for
0
2
!1
ln
0
0 :for
0
2
1
1
0
ξξπ
ζν
ζνγ
ξξπ
ζν
ν
ζν
ζν
ζν
ζνζ
x
xi
nnzi
zi
e
x
xiziEe
dww
eedk
ke
n
nnzi
zi
zi
wzi
zki
where 57722.0=γ is the well-known Euler-Mascheroni constant.The definition of 1E has been given by Abramovitz and Stegun [1964].Setting:
( )ζν −⋅−= zir and ( ) ( ) π
ζνζνθ +
−⋅⋅−−⋅⋅−
=zizi
ReIm
arctan
the following expression is obtained for Equation 3.4–36:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
166
( )( ) ( )( ) ( )( )
( ) ( )
( )
⋅⋅⋅
+
<−>−
−⋅
+
⋅⋅⋅
++
⋅−⋅⋅−−⋅⋅=⋅−
∑
∑
∫
∞
=
∞
=
+⋅∞ −⋅⋅−
1
1
0
!sin
0
0 :for
2
!cos
ln
sincos
n
n
n
n
yzki
nnnr
x
xi
nnnr
r
xixedkk
ePV
θξξ
πθθ
θγ
ξνξνν
ηνζ
Equation 3.4–37
Separating Equation 3.4–37 into its real and imaginary parts yields:( ) ( )( ) ( ) ( ) ( )( )
( ) ( )( )( ) ( )( ) ( ) ( ) ( )( )
( ) ( )( )
−⋅⋅−−⋅⋅
⋅=⋅−
−⋅⋅
−⋅⋅+−⋅⋅
⋅=⋅−
−⋅⋅
+⋅∞ +⋅
+⋅∞ +⋅
∫
∫
ξνθξνθ
νξ
ξνθξνθ
νξ
ηνη
ηνη
xrS
xrCedk
kxke
PV
xrS
xrCedk
kxke
PV
yyk
yyk
cos,
sin,sin
sin,
cos,cos
0
0
Equation 3.4–38
provided that:
( ) ( ) ( )
( ) ( )
<−>−
−+
⋅⋅⋅
+=
⋅⋅⋅
++=
∑
∑∞
=
∞
=
0
0 :for
2!sin
,
!cos
ln,
1
1
ξξ
πθθθθθ
θγθ
x
x
nnnr
rS
nnnr
rrC
n
n
n
n
3.4.6.2 Appendix II: Evaluation of Kernel Integrals
The influence coefficients of Equation 3.4–23 are:
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
⋅
⋅−
⋅
++−+⋅⋅
⋅−−⋅
⋅−
⋅
+−−−⋅⋅
⋅∇⋅=
=
∞ +⋅⋅−
∞ −⋅⋅−
∫∫
∫∫
−
+
i
j
j
zz
szki
m
szki
im
ij
dsdk
kePV
zz
dsdk
ke
PV
zz
nI
0
0
1
lnln2
1
11
lnln2
1
Re
νπ
ζζπ
νπ
ζζπ
ζ
ζ
Equation 3.4–39
and:
( ) ( ) ( ) ( ) ( )
⋅−−⋅⋅∇⋅=
=
+⋅⋅−−⋅⋅− ∫∫−+
ijj zz
s
zim
s
zii
mij dsedsenJ ζνζν 1Re
Equation 3.4–40
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
167
Notice that in the complex plane with iz on is :
( ) ( ) ( )i
i
izz
izzi
dzzdF
eizFn=
⋅=
⋅⋅−=⋅∇⋅ αReRe
Considering the term containing ( )ζ−zln , it is evident that the kernel integral is singularwhen ji = , so that the indicated differentiation cannot be performed under the integral sign.However, in that case one may proceed as follows.Since:
ηξζ ⋅+= iand
dse
dsids
didd
ji
jj
⋅=
⋅⋅+⋅=⋅+=
⋅α
ααηξζ
sincos
for ζ along the j -th segment.Therefore:
ζα deds ji ⋅= ⋅−
and:
( ) ( ) ( )
( )
⋅−
⋅−
=
−⋅⋅
⋅⋅−
=
⋅−⋅∇⋅
=
=
⋅−
⋅
=
∫
∫∫
+
+
j
j
j
j
j
j
j
j
jj
zz
zz
i
i
zzs
i
dzdzd
i
zdedzd
ei
dszn
1
1
lnRe
lnRelnRe
ζ
ζ
ζ
ζ
α
α
ζζ
ζζζ
Setting ζζ −= z' , the last integral becomes:
( ) ( ) ( )
π
ζζζζζ
ζ
=
−−−=
⋅⋅− +
=
−
−∫
+
1'' argarglnRe
1
jjjj
zz
z
z
zzddzd
i
j
j
j
Equation 3.4–41
If ji ≠ , differentiation under the integral sign may be performed, so that:
( ) ( ) ( )
( ) ( ) ( )( ) ( )
( )
−−
−
−−
⋅++
−+−
−+−⋅−=
⋅−⋅∇⋅=
+
+
++
=
∫
1
1
21
21
22
1
arctanarctancos
lnsin
lnRe
ji
ji
ji
jiji
jiji
jijiji
zzs
i
x
y
x
y
yx
yx
dsznL
ij
ξη
ξη
αα
ηξηξ
αα
ζ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
168
Equation 3.4–42
For the integral containing the ( )ζ−zln term, ζα deds ji ⋅= ⋅ , so that:
( ) ( ) ( )
( ) ( ) ( )( ) ( )
( )
−+
−
−+
⋅++
++−
++−⋅+=
⋅−⋅∇⋅=
+
+
++
=
∫
1
1
21
21
22
2
arctanarctancos
lnsin
lnRe
ji
ji
ji
jiji
jiji
jijiji
zzs
i
x
y
x
y
yx
yx
dsznL
ij
ξη
ξη
αα
ηξηξ
αα
ζ
Equation 3.4–43
The kernel integral, containing the principal value integrals, is:
( ) ( ) ( )
( ) ( )
( )
( ) ( )( )
( ) ( )( )
( )
( ) ( )( )
( ) ( )( )
⋅−
−⋅⋅−
⋅−
−⋅⋅+
⋅+−
⋅−
−⋅⋅−
⋅−
−⋅⋅+
⋅+=
⋅−
⋅⋅⋅−=
⋅
−⋅⋅∇⋅=
∫
∫
∫
∫
∫ ∫
∫ ∫
∞+
+⋅
∞ +⋅
∞+
+⋅
∞ +⋅
∞ −⋅⋅−+⋅
=
∞ −⋅⋅−
+
+
+
0
1
0
0
1
0
0
05
sin
sin
cos
cos
cos
sin
Re
Re
1
1
1
dkk
xkePV
dkk
xkePV
dkk
xkePV
dkk
xkePV
dkk
ePV
d
ddei
dkk
ePVdsnL
jiyk
jiyk
ji
jiyk
jiyk
ji
zkii
zzs
zki
i
ji
ji
ji
ji
j
j
iji
ij
νξ
νξ
αα
νξ
νξ
αα
νζζ
ν
η
η
η
η
ζ
ζ
ζαα
ζ
Equation 3.4–44
The first integral on the right hand side of Equation 3.4–40 becomes:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
169
( ) ( ) ( )
( )( ) ( )( )( ) ( )( )
( )( ) ( )( )( ) ( )( )
−⋅⋅−
−⋅⋅+⋅++
−⋅⋅−
−⋅⋅+⋅+−=
⋅⋅∇⋅=
++⋅
+⋅
++⋅
+⋅
=
−⋅⋅−
+
+
∫
1
1
7
sin
sincos
cos
cossin
Re
1
1
jiy
jiy
ji
jiy
jiy
ji
zzs
zii
xe
xe
xe
xe
dsenL
ji
ji
ji
ji
ij
ξν
ξναα
ξν
ξναα
ην
ην
ην
ην
ζν
Equation 3.4–45
The kernel integrals over the image segments are obtained from Equation 3.4–43 throughEquation 3.4–45 by replacing jξ , 1+jξ and jα with jj ξξ −=− , ( ) 11 ++− −= jj ξξ and jj αα −=− ,respectively.
3.4.6.3 Appendix III: Potential Integrals
The velocity potential of the m -th mode of oscillation at the i -th midpoint ( )ii yx , is:( )( )
( ) ( )( )
( )
( ) ( )( )
( ) ( ) ( ) ( )( )∑ ∫∫
∑
∫ ∫
∫ ∫
=
+⋅⋅−−⋅⋅−+
= ∞ +⋅⋅−
∞ −⋅⋅−
⋅⋅
⋅
⋅−−⋅⋅
⋅
⋅
−⋅++−+
⋅−−
⋅
⋅
−⋅+−−−
⋅⋅⋅
=Φ
−
N
j s
zim
s
zijN
N
j
s
zki
ii
m
s
zki
ii
j
iim
t
tdsedseQ
dsdkk
ePVzz
dsdkk
ePVzz
Q
tyx
j
i
j
i
j
i
j
i
1
1
0
0
sin
cos1Re
2lnln
1
2lnln
Re2
1
,,
ωω
νζζ
νζζ
π
ζνζν
ζ
ζ
m
Equation 3.4–46
The integration of the ( )ζ−izln term is straight forward, yielding:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
170
( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
−
−⋅−−
−
−⋅−+
−+−⋅−−
−+−+−⋅−+
⋅+
−
−⋅−+
−
−⋅−−
−+−⋅−−
−+−+−⋅−+
⋅+=
⋅−
+
++
+++
+
+
++
+++
+
∫
1
11
21
211
122
1
11
21
211
122
arctanarctan
ln
ln
sin
arctanarctan
ln
ln
coslnRe
ji
jiji
ji
jiji
jijiji
jjjijiji
j
ji
jiji
ji
jiji
jijiji
jjjijiji
js
i
x
yx
x
yx
yxy
yxy
x
yy
x
yy
yxx
yxx
dszj
ξη
ξξη
ξ
ηξη
ηηηξη
α
ξη
ηξη
η
ηξξ
ξξηξξ
αζ
Equation 3.4–47
In the integration of the ( )ζ−zln term, note that jη and 1+jη are replaced by jη− and 1+− jη ,
respectively.
To evaluate the potential integral containing the principal value integral, proceed in thefollowing manner.For an arbitrary z in the fluid domain:
( ) ( )
dkk
eek
ePVe
dedkk
ePVe
dek
dkPVedk
ke
PVds
jkikizki
i
kizki
i
zkiizki
jj
j
j
j
j
j
j
j
j
⋅−
⋅−
⋅−=
⋅⋅⋅−
⋅=
⋅⋅−
⋅=⋅−
⋅
⋅⋅⋅⋅∞ ⋅⋅−⋅
⋅⋅−∞ ⋅⋅−
⋅
−⋅⋅−∞
⋅∞ −⋅⋅−
+
+
++
∫
∫∫
∫∫∫ ∫
ζζα
ζ
ζ
ζα
ζ
ζ
ζαζ
ζ
ζ
ν
ζν
ζνν
1
1
11
0
0
00
where the change of integration is permissible since only one integral requires a principlevalue interpretation.
After dividing by ν and multiplying by kk +−ν under the integral sign, the last expressionbecomes:
( ) ( )
⋅−
−⋅−
+⋅−
⋅⋅⋅
− ∫∫∫∞ −⋅⋅−∞ −⋅⋅−⋅⋅⋅⋅∞
⋅⋅−⋅ ++
000
11
dkk
ePVdk
ke
PVdkk
eee
ei jjjj zkizkij
kikizki
i
ννν
ζζζζα
Equation 3.4–48
Regarding the first integral in Equation 3.4–48 as a function of z :
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
171
( ) dkk
eeezF j
kikizki
j
⋅−
⋅=⋅⋅⋅⋅∞
⋅⋅−+
∫ζζ 1
0
Equation 3.4–49
Differentiating Equation 3.4–49 with respect to z gives:
( ) ( ) ( )
11
00
11
' 1
+
∞−⋅⋅−
∞−⋅⋅−
−−
−=
⋅−⋅⋅−= ∫∫ +
j
zkizki
zz
dkedkeizF jj
ζζ
ζζ
So:( ) ( ) ( ) KzzzF jj +−−−= +1lnln ζζ
Equation 3.4–50
where K is a constant of integration to be determined presently. Since ( )zF is defined and
analytic for all z in the lower half plane and since by Equation 3.4–49, ( ) 0lim =−∞→
zFz
, it
follows from Equation 3.4–50 that 0=K .Therefore:
( )( )
( ) ( )( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
⋅−
−⋅⋅−⋅
−
−⋅⋅
−
+−
−
++
⋅+
⋅−
−⋅⋅−⋅
−
−⋅⋅
++−
++−
⋅+
⋅=
−⋅−
−⋅−
+
−−−
⋅⋅
−=
⋅−
⋅=
∫∫
∫∫
∫∫
∫ ∫
∞+
+⋅∞ +⋅
+
+
∞ +⋅∞+
+⋅
++
∞ −⋅⋅−∞ −⋅⋅−
+⋅
∞ −⋅⋅−
+
+
+
dkk
xkePVdk
k
xkePV
x
y
x
y
dkk
xkePVdk
k
xkePV
yx
yx
dkk
ePVdk
ke
PV
zzei
dkk
ePVdsK
jiyk
jiyk
ji
ji
ji
ji
j
jiyk
jiyk
jiji
jiji
j
zkizki
jijii
s
zki
jiji
jiji
jiji
j
j
i
0
1
0
1
1
00
1
21
21
22
00
1
0
5
sinsin
arctanarctan
cos
coscos
ln
sin
1
lnln
Re
Re
1
1
1
νξ
νξ
ξη
ξη
α
νξ
νξ
ηξ
ηξ
α
ν
νν
ζζ
ν
ν
ηη
ηη
ζζ
α
ζ
Equation 3.4–51
The integration of the potential component in-phase with the velocity over js gives:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
172
( ) ( )
( ) ( )( )( ) ( )( )
−−⋅⋅−
−−⋅⋅+⋅=
⋅=
++⋅
+⋅
−⋅⋅−
+
∫
jjiy
jjiy
s
zi
xe
xe
dseK
ji
ji
j
i
αξν
αξν
ν ην
ην
ζν
1
7
sin
sin1
Re
1
Equation 3.4–52
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
173
3.5 Comparisons between Calculated Potential Data
Figure 3.5–1 compares the calculated coefficients for an amidships cross section of acontainer vessel by the three previous methods:1. Ursell-Tasai's method with 2-parameter Lewis conformal mapping.2. Ursell-Tasai's method with 10-parameter close-fit conformal mapping.3. Frank's pulsating source method.
0
100
200
300
400
500
0 0.5 1.0 1.5 2.0 2 .5
H eave
M'
3 3
0
1 000
2 000
3 000
4 000
5 000
0 0.5 1.0 1.5 2 .0 2.5
R oll
M'
4 4
0
50
100
150
0 0 .5 1 .0 1.5 2.0 2.5
He ave
N'
33
f re qu ency (rad/s)
0
5 0
10 0
15 0
20 0
0 0. 5 1.0 1.5 2.0 2.5
Sway
N'
22
f requ ency (ra d/s)
Dam
ping
Coe
ffic
ient
0
100
200
300
0 0.5 1.0 1.5 2.0 2.5
M id ship s ectionof a c onta inersh ip
S way
M'
22
Mas
s C
oeff
icie
nt
0
5 0
10 0
15 0
20 0
0 0 .5 1.0 1.5 2.0 2 .5
Sw ay - Ro llRoll - Sway
N'
24 = N
'
42
f requ en cy (rad /s)
3000
3250
3500
3750
4000
4250
0 0.5 1.0 1.5 2.0 2.5
Sway - Ro llRoll - Sway
M'
2 4 = M
'
42
0
1 00
2 00
3 00
0 0 .5 1.0 1 .5 2.0 2 .5
C lo se-fit
L ew is
Fra nkRo ll
N'
44
f req ue ncy (ra d/s)
Figure 3.5–1: Comparison of various calculated potential coefficients
With the exception of the roll motions, the three results are very close. The roll motiondeviation, predicted with the Lewis conformal mapping method, is caused by a too muchrounded description of the ''bilge'' by the simple Lewis transformation.
A disadvantage of Frank's method could be the relatively large computing time, whencompared with Ursell-Tasai's method. However - because of the significantly increasedcomputing speed of nowadays computers - this should not be a problem anymore.
Generally, it is advised to use the very robust Ursell-Tasai's method with 10 parameter close-fit conformal mapping.For submerged sections, bulbous sections and sections with an area coefficient sσ less than0.4 however, Frank's pulsating source method should be used.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
174
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
175
3.6 Estimated Potential Surge Coefficients
An equivalent longitudinal section, being constant over the ship's breadth B , is defined by:sectional breadth xB = ship length L
sectional draught xd = amidships draught d
sectional area coefficient MxC = block coefficient BCBy using a Lewis transformation of this equivalent longitudinal section to the unit circle, thetwo-dimensional potential mass *
11M and damping *11N can be calculated in a similar manner
as has been described for the two-dimensional potential mass and damping of sway, '22M and
'22N .
With these two-dimensional values, the total potential mass and damping of surge are definedby:
*1111
*1111
NBN
MBM
⋅=
⋅=
Equation 3.6–1
in which B is the breadth of the ship.
These frequency-dependent hydrodynamic coefficients do not include three-dimensionaleffects. Only the hydrodynamic mass coefficient - of which a large three-dimensional effect isexpected - will be adapted here empirically.
According to Tasai [1961], the zero-frequency potential mass for sway can be expressed inLewis-coefficients:
( ) ( ) 23
21
2
31
'22 31
120 aa
aad
M x ⋅+−⋅
+−
⋅⋅=→πρω
Equation 3.6–2
When using this formula for surge, the total potential mass of surge is defined by:( ) ( )00 *
1111 →⋅=→ ωω MBM
Equation 3.6–3
A frequency-independent total hydrodynamic mass coefficient is estimated empirically bySargent and Kaplan [1974] as a proportion of the total mass of the ship ∇⋅ρ :
( ) ∇⋅⋅−
= ρa
aKSM
2&11
with:
⋅−
−+
⋅−
= bbb
bb
a 211
ln1
3
2
where 2
1
−=
LB
b
Equation 3.6–4
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
176
Figure 3.6–1: Hydrodynamic mass for surge
With this hydrodynamic mass value, a correction factor β for three-dimensional effects hasbeen determined:
( )( )0
&
11
11
==
ωβ
MKSM
Equation 3.6–5
The three-dimensional effects for the potential damping of surge are ignored.So, the potential mass and damping of surge are defined by:
*1111
*1111
NBN
MBM
⋅=
⋅⋅= β
Equation 3.6–6
To obtain a uniform approach during all ship motions calculations, the cross sectional two-dimensional values of the hydrodynamic mass and damping have to be obtained.Based on results of numerical 3-D studies with a Wigley hull form by Adegeest [1994], aproportionality of both the two-dimensional hydrodynamic mass and damping with theabsolute values of the derivatives of the cross sectional areas xA in the bx -direction has beenfound:
11'
11 Mdx
dxdA
dxdA
M
L
bb
x
b
x
⋅⋅
=
∫ and 11
'11 N
dxdxdA
dxdA
N
L
bb
x
b
x
⋅⋅
=
∫
Equation 3.6–7
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
177
4 Viscous Damping
The strip theory is based on the potential flow theory. This holds that viscous effects areneglected, which can deliver serious problems when predicting roll motions at resonancefrequencies. In practice, viscous roll damping effects can be accounted for by empiricalformulas. For surge and roll, additional damping coefficients have to be introduced. Becauseof these additional contributions to the damping are from a viscous origin mainly, it is notpossible to calculate the total damping in a pure theoretical way.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
178
4.1 Surge Damping
The total damping for surge vt BBB 111111 += consists of a potential part, 11B , and an
additional viscous part, vB11 . At forward ship speed V , the total damping coefficient, tB11 ,
can be determined simply from the resistance-speed curve of the ship in still water, ( )VRsw :( )
dVVRd
BBB swvt =+= 111111
Equation 4.1–1
4.1.1 Total Surge Damping
For a rough estimation of the still water resistance use can be made of a somewhat modifiedempirical formula of Troost [1955], in principle valid at the ship's service speed for hull formswith a block coefficient BC between about 0.60 and 0.80:
232 VCR tsw ⋅∇⋅⋅= ρ with: 60.0log0152.0
0036.010 +
+≈L
Ct
Equation 4.1–2
in which:∇ volume of displacement of the ship in m3,L length of the ship in m,V forward ship speed in m/s.
This total resistance coefficient tC is given in Figure 4.1–1 as a function of the ship length.
Figure 4.1–1: Total Still Water Resistance Coefficient of Troost
Then the total surge damping coefficient at forward ship speed V becomes:VCB tt ⋅∇⋅⋅⋅= 32
11 2 ρ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
179
4.1.2 Viscous Surge Damping
This total damping coefficient includes a viscous part, which can be derived from thefrictional part of the ship's resistance, defined by the 1957 ITTC-line:
( ) ( ) 22
2ln075.0
21
−⋅⋅⋅⋅=
RnSVVR f ρ with:
νLV
Rn⋅
=
Equation 4.1–3
in which:ν kinematic viscosity of seawaterS wetted surface of the hull of the shipRn Reynolds number
From this empirical formula follows the pure viscous part of the additional dampingcoefficient at forward ship speed V :
( ) dV
VRdB f
v =11
Equation 4.1–4
which can be obtained numerically.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
180
4.2 Roll Damping
In case of pure free rolling in still water (free decay test), the uncoupled linear equation of theroll motion about the centre of gravity G is given by:
( ) ( ) 044444444 =⋅+⋅++⋅+ φφφ CBBAI vxx&&&
with:
GMgC
bB
bOGbOGbOGbB
aOGaOGaOGaA
vv
⋅∇⋅⋅=
=⋅+⋅+⋅+=
⋅+⋅+⋅+=
ρ44
4444
22
2
24424444
22
2
24424444
Equation 4.2–1
For zero forward speed: 2442 aa = and 2442 bb = .
Equation 4.2–1 can be rewritten as:02 2
0 =⋅+⋅+ φωφνφ &&&
with:
44
44442AIBB
xx
v
++
=ν (quotient of damping and moment of inertia)
44
4420 AI
C
xx +=ω (natural roll frequency squared)
Equation 4.2–2
The non-dimensional roll damping coefficient, κ , is given by:
( ) 4444
4444
0
2 CAIBB
xx
v
⋅+⋅+=
=ωνκ
Equation 4.2–3
This damping coefficient is written as a fraction between the actual damping coefficient,
vBB 4444 + , and the critical damping coefficient, ( ) 444444 2 CAIB xxcr ⋅+⋅= ; so for critical
damping: 1=crκ .
Herewith, the equation of motion can be re-written as:02 2
00 =⋅+⋅⋅⋅+ φωφωκφ &&&
Equation 4.2–4
Suppose the vessel is deflected to an initial heel angle, aφ , in still water and then released. Thesolution of the equation of motion of this decay becomes:
( ) ( )
⋅⋅+⋅⋅⋅= ⋅− tte ta 0
00 sincos ω
ωνωφφ ν
Equation 4.2–5
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
181
Then, the logarithmic decrement of the motion is:
( )( )
+=
⋅⋅=⋅
φ
φφ
φφ
ωκν
Ttt
TT
ln
0
Equation 4.2–6
Because 220
2 νωωφ −= for the natural frequency oscillation and the damping is small so that2
02 ων << , one can neglect 2ν here and use 0ωωφ ≈ ; this leads to:
πωω φφφ 20 =⋅≈⋅ TT
Equation 4.2–7
The non-dimensional total roll damping is given now by:
( )( )
( )44
04444 2
ln2
1
CBB
Ttt
v ⋅⋅+=
+⋅
⋅=
ω
φφ
πκ
φ
Equation 4.2–8
The non-potential part of the total roll damping coefficient follows from the average value ofκ by:
440
4444
2B
CB v −
⋅⋅=
ωκ
Equation 4.2–9
4.2.1 Experimental Determination
The κ -values can easily been found when results of free rolling experiments with a model instill water are available, see Figure 4.2–1.
Figure 4.2–1: Time History of a Roll Decay Test
The results of free decay tests can be presented in different ways:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
182
1. Generally they are presented by plotting the non-dimensional damping coefficient,obtained from two successive positive or negative maximum roll angles
iaφ and 2+iaφ , by:
⋅⋅
=+2
ln2
1
i
i
a
a
φφ
πκ versus:
22+
+= ii aa
a
φφφ
Equation 4.2–10
2. To avoid spreading in the successively determined κ -values, caused by a possible zero-shift of the measuring signal, double amplitudes can be used instead:
−−
⋅⋅
=++
+
32
1ln2
1
ii
ii
aa
aa
φφφφ
πκ versus:
( ) ( )4
321 +++−+−
= iiii aaaaa
φφφφφ
Equation 4.2–11
3. Sometimes the results of free rolling tests are presented by:
a
a
φφ∆
versus: aφ
with the absolute value of the average of two successive positive or negativemaximum roll angles, given by:
21+
+= ii aa
a
φφφ
and the absolute value of the difference of the average of two successive positive ornegative maximum roll angles, given by:
1+−=∆
ii aaa φφφThen the total non-dimensional roll damping coefficient becomes:
∆−
∆+
⋅⋅
=
a
a
a
a
φφφφ
πκ
2
2ln
21
Equation 4.2–12
The decay coefficient κ can therefore be estimated from the decaying oscillation bydetermining the ratio between any pair of successive (double) amplitudes. When the dampingis very small and the oscillation decays very slowly, several estimates of the decay can beobtained from a single record. It is obvious that for a linear system a constant κ -value shouldbe found in relation to aφ .Note that these decay tests provide no information about the relation between the potentialcoefficients and the frequency of oscillation. Indeed, this is impossible since decay tests arecarried out at only one frequency: the natural frequency. These experiments deliver noinformation on the relation with the frequency of oscillation.The method is not really practical when ν is much greater than about 0.2 and is in any casestrictly valid for small values of ν only. Luckily, this is generally the case.Be aware that this damping coefficient is determined by assuming an uncoupled roll motion(no other motions involved). Strictly, this damping coefficient is not valid for the actualcoupled motions of a ship that will be moving in all directions simultaneously.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
183
The successively found values for κ , plotted on base of the average roll amplitude, will oftenhave a non-linear behaviour as illustrated in Figure 4.2–2.
0
0.01
0.02
0.03
0.04
0 1 2 3 4 5 6
mean linear and cubic dampingmean linear and quadrat ic dampingsecond experiment, negat ive anglessecond experiment, posit ive anglesfirst experiment, negative anglesfirst experiment, pos itive angles
Produc t carrier, V = 0 knots
mean roll amplitude (deg)
roll
dam
ping
coe
ffici
ent κ
(-)
Figure 4.2–2: Roll Damping Coefficients
For a behaviour like this, it will be found:
aφκκκ ⋅+= 21
Equation 4.2–13
while sometimes even a cubic roll damping coefficient, 23 aφκ ⋅ , has to be added to this
formula.This non-linear behaviour holds that during frequency domain calculations, the damping termis depending on the - so far unknown - solution for the transfer function of roll: aa ζφ . With a
known wave amplitude, aζ , this problem can be solved in an iterative manner. A less accurate
method is to use a fixed aφ .
4.2.2 Empirical Formula for Barges
From model experiments with rectangular barges - with its center of gravity, G , in the waterline - it has been found by Journee [1991]:
aφκκκ ⋅+= 21 with:
50.0
0013.0
2
2
1
=
⋅=
κ
κdB
Equation 4.2–14
in which B is the breadth and d is the draft of the barge.
4.2.3 Empirical Method of Miller
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
184
According to Miller [1974], the non-dimensional total roll damping coefficient, κ , can beobtained by:
aφκκκ ⋅+= 21
with:
b
b
b
bkbk
bbbV
CdBLr
BLrl
A
CFn
CFn
CFn
GML
BL
C
⋅⋅⋅⋅
⋅⋅+⋅⋅=
⋅+
+
⋅⋅⋅⋅=
3
3
2
32
1
0024.025.19
200085.0
κ
κ
Equation 4.2–15
where:
bkbkbk hlA ⋅= one sided area of bilge keel (m2)
bkl length of bilge keel (m)
bkh height of bilge keel (m)
br distance center line of water plane to turn of bilge (m)(first point at which turn of bilge starts, relative to water plane)
L length of ship (m)B breadth of ship (m)d draft of ship (m)
bC block coefficient (-)
GM initial metacentric height (m)Fn Froude number (-)
aφ amplitude of roll (rad)
VC correction factor on 1κ for speed effect (-)
(in the original formulation of Miller: 0.1=VC )
Generally 0.1=VC , but (according to an experienced user of computer code SEAWAY) for
slender ships, like frigates, a suitable value for VC seems to be:
GMCV ⋅−= 00.385.4
Equation 4.2–16
4.2.4 Semi-Empirical Method of Ikeda
Because the viscous part of the roll damping acts upon the viscosity of the fluid significantly,it is not possible to calculate the total roll damping coefficient in a pure theoretical way.Besides this, also experiments showed a non-linear behaviour of viscous parts of the rolldamping.Sometimes, for applications in frequency domain, an equivalent linear roll dampingcoefficient, ( )1
44vB , has to be determined. This coefficient can be obtained by stipulating that
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
185
an equivalent linear roll damping dissipates an identical amount of energy as the non-linearroll damping. This results for a linearised quadratic roll damping coefficient, ( )2
44vB , into:
( ) ( ) ∫∫ ⋅⋅⋅⋅=⋅⋅⋅φφ
φφφφφT
v
T
v dtBdtB0
244
0
144
&&&&&
or with some algebra:( ) ( )2
441
44 38
vav BB ⋅
⋅⋅
⋅= ωφ
π
Equation 4.2–17
For the estimation of the non-potential parts of the roll damping, use has been made of workpublished by Ikeda, Himeno and Tanaka [1978]. A few sub-ordinate parts have been modifiedand this empirical method is called here the ''Ikeda method''.
This Ikeda method estimates the following linear components of the roll damping coefficientof a ship:
SB44 a correction on the potential roll damping coefficient due to forward speed,
FB44 the frictional roll damping coefficient,
EB44 the eddy making roll damping coefficient,
LB44 the lift roll damping coefficient and
KB44 the bilge keel roll damping coefficient.So, the additional - mainly viscous - roll damping coefficient VB44 is given by:
KLEFSV BBBBBB 444444444444 ++++=
Equation 4.2–18
Ikeda, Himeno and Tanaka [1978] claim fairly good agreements between their predictionmethod and experimental results. They conclude that the method can be used safely forordinary ship forms, which conclusion has been confirmed by the author too. But for unusualship forms, for very full ship forms and for ships with a very large breadth to draft ratio themethod is not always accurate sufficiently.For numerical reasons three restrictions have to be made:• if, locally, 999.0>sσ then: 999.0=sσ .
• if, locally, ssDOG σ⋅−< then: ssDOG σ⋅−= .• if a calculated component of the viscous roll damping coefficient becomes less than zero,
this component has to be set to zero.
4.2.4.1 Notations of Ikeda et.al.
In this description of the Ikeda method, the notation of the authors (Ikeda, Himeno andTanaka) is maintained as far as could be possible here:
ρ density of waterν kinematic viscosity of waterg acceleration of gravityV forward ship speedRn Reynolds numberω circular roll frequency
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
186
aφ roll amplitudeL length of the shipB breadthD amidships draught
MC amidships section coefficient
BC block coefficientDLSL ⋅≈ lateral area
fS wetted hull surface area
OG distance of centre of gravity above still water level, positive upwards(this sign convention deviates from that in the paper of Ikeda)
sB sectional breadth water line
sD sectional draft
sA sectional area
sσ sectional area coefficient
0H sectional half breadth to draft ratio
1a sectional Lewis coefficient
3a sectional Lewis coefficient
sM sectional Lewis scale factor
fr average distance between roll axis and hull surface
OL distance point of taking representative angle of attack to roll axis,
approximated by DLO ⋅= 3.0
RL distance of centre of action of lift force in roll motion to roll axis,approximated by DLR ⋅= 5.0
kh height of the bilge keels
kL length of the bilge keels
kr distance between roll axis and bilge keel
kf correction for increase of flow velocity at the bilge
pC pressure coefficient
ml lever of the moment
br local radius of the bilge circle
4.2.4.2 Effect of Forward Speed, SB44
Ikeda obtained an empirical formula for the three-dimensional forward speed correction onthe zero speed potential damping by making use of the general characteristics of a doubletflow model. Two doublets have represented the rolling ship: one at the stern and one at thebow of the ship.With this, semi-theoretically the forward speed effect on the linear potential dampingcoefficient has been approximated as a fraction of the potential damping coefficient by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
187
( ) ( )[ ]( ) ( )
−
⋅−−⋅+
−Ω⋅⋅−++⋅⋅=
−⋅−0.1
12
3.020tanh115.0 225.0150
21
22
4444 ωeAA
AABB S
Equation 4.2–19
with:
44B potential roll damping coefficient of the ship (about G )gV⋅=Ω ω non-dimensional circular roll frequency
gDD ⋅= 2ωξ non-dimensional circular roll frequency squaredDeA D
ξξ ⋅−− ⋅+= 22.11 0.1 maximum value of 44B at 25.0=ω
DeA Dξξ ⋅−− ⋅+= 20.1
2 5.0 minimum value of 44B at large ω
4.2.4.3 Frictional Roll Damping, FB44
Kato deduced semi-empirical formulas for the frictional roll damping from experimentalresults of circular cylinders, wholly immersed in the fluid. An effective Reynolds number ofthe roll motion was defined by:
ν
ωφ
⋅
⋅
=
2
512.0a
fr
Rn
Equation 4.2–20
In here, for ship forms the average distance between the roll axis and the hull surface can beapproximated by:
( )π
OGLS
Cr
fB
f
⋅+⋅⋅+=
2145.0887.0
Equation 4.2–21
with a wetted hull surface area fS , approximated by:
( )BCDLS Bf ⋅+⋅⋅= 7.1
Equation 4.2–22
The relation between the density, kinematic viscosity and temperture of fresh water and seawater are given in Figure 4.2–3.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
188
990
1000
1010
1020
1030
0 10 20 30
Fresh Water
Sea Water
Temperature (0C)
De
nsi
ty
(kg
/m3)
0.5
1.0
1.5
2.0
0 10 20 30
Sea Water
Fresh Water
Temperature (0C)
Kin
em
ati
c V
isc
osi
ty
(m
2 s)
Figure 4.2–3: Relation between density, kinematic viscosity and temperature of water
When eliminating the temperature of water, the following relation can express the kinematicviscosity into the density of water in the kg-m-s system:
( ) ( )( ) ( )26
26
102502602.010251039.0063.110 : watersea
100007424.010003924.0442.110 :rfresh wate
−⋅+−⋅+=⋅
−⋅+−⋅+=⋅
ρρν
ρρν
Equation 4.2–23
as given in Figure 4.2–4.
0
5
10
15
20
25
997 998 999 1000
Viscosity Actual Viscosity Polynomial Temperature
Density Fresh Water (kg/m3)
Kin
em
ati
c V
isc
osi
ty *
10
7
(m
2 s)
Te
mp
era
ture
(
0 C)
Fresh Water
0
5
10
15
20
25
1023 1024 1025 1026 1027 1028
Viscosity Actual Viscosity Polynomial Temperature
Density Salt Water (kg/m3)
Kin
em
ati
c V
isc
osi
ty *
10
7
(m
2 s)
Te
mp
era
ture
(
0 C)
Salt Water
Figure 4.2–4: Kinematic viscosity as a function of density
Kato expressed the skin friction coefficient as:114.05.0 014.0328.1 −− ⋅+⋅= RnRnC f
Equation 4.2–24
The first part in this expression represents the laminar flow case. The second part has beenignored by Ikeda, but has been included here.Using this, the quadratic roll damping coefficient due to skin friction at zero forward shipspeed is expressed as:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
189
( )fffF CSrB ⋅⋅⋅⋅= 32
44 21
0ρ
Equation 4.2–25
This frictional roll damping component increases slightly with forward speed.Semi-theoretically, Tamiya deduced a modification coefficient for the effect of forward speedon the friction component. Accurate enough from a practical point of view, this results into thefollowing formula for the speed dependent frictional damping coefficient:
( )
⋅⋅+⋅⋅⋅⋅⋅=
LVCSrB fffF ω
ρ 1.40.121 32
44
Equation 4.2–26
Then, the equivalent linear roll damping coefficient due to skin friction is expressed as:
⋅⋅+⋅⋅⋅⋅⋅⋅
⋅=
LVCSrB fffaF ω
ρωφπ
1.40.121
38 3
44
Equation 4.2–27
Ikeda confirmed the use of his formula for the three-dimensional turbulent boundary layerover the hull of an oscillating ellipsoid in roll motion.
4.2.4.4 Eddy Making Damping, EB44
At zero forward speed the eddy making roll damping for the naked hull is mainly caused byvortices, generated by a two-dimensional separation. From a number of experiments with two-dimensional cylinders it was found that for a naked hull this component of the roll moment isproportional to the roll frequency squared and the roll amplitude squared. This means that thecorresponding quadratic roll damping coefficient does not depend on theperiod parameter but on the hull form only.When using a simple form for the pressure distribution on the hull surface it appears that thepressure coefficient pC is a function of the ratio γ of the maximum relative velocity maxU to
the mean velocity meanU on the hull surface:
meanUUmax=γ
Equation 4.2–28
The γ−pC relation was obtained from experimental roll damping data of two-dimensional
models. These experimental results are fitted by:50.10.235.0 187.0 +⋅−⋅= ⋅−− γγ eeCp
Equation 4.2–29
The value of γ around a cross section is approximated by the potential flow theory for arotating Lewis form cylinder in an infinite fluid.An estimation of the local maximum distance between the roll axis and the hull surface, maxr ,has to be made.Values of ( )ψmaxr have to be calculated for:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
190
0.01 ==ψψ and ( )
⋅+⋅
==
3
31
2
41
cos
5.0
aaa
ψψ
Equation 4.2–30
The values of ( )ψmaxr follow from:
( ) ( ) ( ) ( )( )( ) ( ) ( )( )
⋅⋅+⋅−
+⋅⋅−⋅+⋅=
231
231
max3coscos1
3sinsin1
ψψ
ψψψ
aa
aaMr s
Equation 4.2–31
With these two results a value maxr and a value ψ follow from the conditions:
• For ( ) ( )2max1max ψψ rr > : ( )1maxmax ψrr = and 1ψψ =• For ( ) ( )2max1max ψψ rr < : ( )2maxmax ψrr = and 2ψψ =
Equation 4.2–32
The relative velocity ratio γ on a cross section is obtained by:
+⋅⋅+⋅
+⋅⋅⋅
⋅= 22max
0
3 2
2
baHMr
DOG
HD
f s
sss σ
πγ
with:
( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( )25 11065.13
2131
21
231
313
2131
21
231
313
3312
32
1
3131
0
41
sin336
3sin15sin2
cos336
3cos15cos2
4cos62cos31291
112
2
sef
aaaaaa
aaab
aaaaaa
aaaa
aaaaaH
aaD
aaB
M
DBA
DB
H
sss
ss
ss
s
s
σ
ψ
ψψψ
ψψψψ
σ
−⋅⋅−⋅+=
⋅+⋅⋅++⋅⋅+
+⋅⋅−⋅+⋅⋅⋅−=⋅+⋅⋅−+⋅⋅−
+⋅⋅−⋅+⋅⋅⋅−=⋅⋅⋅−⋅⋅⋅−⋅⋅+⋅++=
+−=
++⋅=
⋅=
⋅=
Equation 4.2–33
With this a quadratic sectional eddy making damping coefficient for zero forward speedfollows from:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
191
( )
⋅−⋅+
⋅−+⋅
⋅−
⋅⋅
⋅⋅⋅=
2
102
11
2
max4'244
11
21
0
s
b
s
b
ss
b
ps
sE
Drf
HfD
rfDOG
Drf
CDr
DB ρ
with:[ ]
( )( ) ( ) ( )ss
s
sef
f
σπσπ
σσ ⋅⋅−⋅−⋅−⋅=
−⋅+⋅=⋅− 255
2
1
sin15.1cos15.0
1420tanh15.0
Equation 4.2–34
The approximations of the local radius of the bilge circle br are given as:
( )
2 : and 1for
: and 1for 4
12 :
2 and for
00
0
0
sbsb
sbsb
ssb
sbsb
BrDHrH
DrDrH
HDr
BrDr
=⋅><
=>>−
−⋅⋅⋅=<<
πσ
Equation 4.2–35
For three-dimensional ship forms the zero forward speed eddy making quadratic roll dampingcoefficient is found by an integration over the ship length:
( ) ( )∫ ⋅=L
bEE dxBB '244
244 00
Equation 4.2–36
This eddy making roll damping decreases rapidly with the forward speed to a non-linearcorrection for the lift force on a ship with a small angle of attack. Ikeda has analysed thisforward speed effect by experiments and the result has been given in an empirical formula.With this the equivalent linear eddy making damping coefficient at forward speed is given by:
( )2
24444 1
13
80 K
BB EaE +⋅⋅
⋅⋅
⋅= ωφ
π with:
LV
K⋅⋅
=ω04.0
Equation 4.2–37
4.2.4.5 Lift Damping, LB44
The roll damping coefficient due to the lift force is described by a modified formula ofYumuro:
⋅⋅+⋅+⋅⋅⋅⋅⋅⋅⋅=
RORRONLL LL
OGL
OGLLkVSB
2
44 7.04.10.121 ρ
Equation 4.2–38
The slope of the lift curve αLC is defined by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
192
−⋅⋅+
⋅⋅=
=
045.01.42
LB
LD
Ck L
N
χπα
Equation 4.2–39
in which the coefficient χ is given by Ikeda in relation to the amidships section coefficient
MC :
30.0 :99.097.0
10.0 :97.092.0
00.0 :92.0
=<<=<<=<
χχχ
M
M
M
C
C
C
Equation 4.2–40
These data are fitted here by:( ) ( )32 91.070091.0106 −⋅−−⋅= MM CCχ
with the restrictions:• if 91.0<MC then 00.0=χ• if 00.1>MC then 35.0=χ
Equation 4.2–41
4.2.4.6 Bilge Keel Damping, KB44
The quadratic bilge keel roll damping coefficient has been into two components:• a component ( )2
44 NKB due to the normal force on the bilge keels
• a component ( )244 SKB due to the pressure on the hull surface, created by the bilge keels.
The coefficient of the normal force component ( )244 NKB of the bilge keel damping can be
deduced from experimental results of oscillating flat plates. The drag coefficient DC dependson the period parameter or the Keulegan-Carpenter number. Ikeda measured this non-lineardrag also by carrying out free rolling experiments with an ellipsoid with and without bilgekeels. This resulted in a quadratic sectional damping coefficient:
( )DkkkK CfhrB
N⋅⋅⋅⋅= 23'2
44 ρ with: ( )sef
frh
C
k
kak
kD
σ
φπ−⋅−⋅+=
+⋅⋅⋅
⋅=
0.11603.00.1
40.25.22
Equation 4.2–42
The approximation of the local distance between the roll axis and the bilge keel kr is given as:22
0 293.00.1293.0
⋅−++
⋅−⋅=
s
b
ss
bsk D
rDOG
Dr
HDr
Equation 4.2–43
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
193
The approximation of the local radius of the bilge circle br in here is given before.Assuming a pressure distribution on the hull caused by the bilge keels, a quadratic sectionalroll damping coefficient can be defined:
( ) ∫ ⋅⋅⋅⋅⋅⋅=k
S
h
mpkkK dhlCfrB0
22'244 2
1 ρ
Equation 4.2–44
Ikeda carried out experiments to measure the pressure on the hull surface created by bilgekeels. He found that the coefficient +
pC the pressure on the front face of the bilge keel does
not depend on the period parameter, while the coefficient −pC of the pressure on the back face
of the bilge keel and the length of the negative pressure region depend on the periodparameter.Ikeda defines an equivalent length of a constant negative pressure region 0S over the height ofthe bilge keels, which is fitted to the following empirical formula:
kakk hrfS ⋅+⋅⋅⋅⋅= 95.13.00 φπ
Equation 4.2–45
The pressure coefficients on the front face of the bilge keel, +pC , and on the back face of the
bilge keel, −pC , are given by:
20.1=+pC and 20.15.22 −
⋅⋅⋅⋅−=−
akk
kp rf
hC
φπ
Equation 4.2–46
and the sectional pressure moment is given by:
( )∫ +− ⋅+⋅−⋅=⋅⋅kh
ppsmp CBCADdhlC0
2
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
194
( )
( )( ) ( )
( ) ( )
( )( ) ( )
( )( ) ( )
bb
b
b
bs
s
s
b
rSrS
mm
rSmmm
rS
rSmDS
m
mmHmHmH
m
mmHmHmH
m
mHm
mmm
DOG
m
Drm
mmmmmm
mmmmH
mB
mmmmA
⋅⋅<
−⋅⋅+=
⋅⋅>⋅+=
⋅⋅<=
⋅⋅>⋅⋅−=
⋅−⋅⋅−⋅⋅+−⋅+⋅
=
⋅−⋅⋅−⋅+⋅−⋅+⋅
=
−=
−−=
−=
=
⋅+⋅⋅+⋅−⋅−⋅⋅−
+⋅−⋅
=
−⋅+=
π
ππ
ππ
25.0 :for cos1414.1
25.0 :for 414.0
25.0 :for 0.0
25.0 :for 25.0
215.01215.00106.0382.00651.0414.0
215.01215.00106.0382.00651.0414.0
0.1
215.01621
215.03
00
17
0178
0
010
7
110
102
106
110
102
105
104
213
2
1
645311
232
1
10
34
27843
Equation 4.2–47
The equivalent linear total bilge keel damping coefficient can be obtained now by integratingthe two sectional roll damping coefficients over the length of the bilge keels and linearizingthe result:
( )∫ ⋅+⋅⋅⋅⋅
=k
SN
L
bKKaK dxBBB '44
'4444 3
8 ωφπ
Equation 4.2–48
Experiments of Ikeda have shown that the effect of forward ship speed on this roll dampingcoefficient can be ignored.
4.2.4.7 Calculated Roll Damping Components
In Figure 4.2–5 an example is given of the several roll damping components, as derived withIkeda's method, for the S-175 container ship design.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
195
Figure 4.2–5: Roll damping coefficients of Ikeda, Himeno and Tanaka
It may be noted that for full-scale ships, because of the higher Reynolds number, the frictionalpart of the roll damping is expected to be smaller than showed above.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
196
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
197
5 Hydromechanical Loads
With the approach as mentioned before, a description will be given here of the determinationof the hydromechanical forces and moments for all six modes of motions.
In the ''Ordinary Strip Theory'', as published by Korvin-Kroukovsky and Jacobs [1957] andothers, the uncoupled two-dimensional potential hydromechanical loads in the direction j aredefined by:
'*'*''RSjhjjjhjjjhj XNM
DtD
X +⋅+⋅= ζζ && (Ordinary Strip Theory, OST)
In the ''Modified Strip Theory'', as has been published later by for instance Tasai [1969] andothers, these loads become:
'*'''RSjhjjj
ejjhj XNiM
DtDX +
⋅
⋅−= ζ
ω& (Modified Strip Theory, MST)
In these definitions of the two-dimensional hydromechanical load, *hjζ& is the harmonic
oscillatory motion, 'jjM and '
jjN are the two-dimensional potential mass and damping and
the non-diffraction part 'RSjX is the two-dimensional quasi-static restoring spring term.
At all following pages, the hydromechanical load has been calculated in the ( )bbb zyxG ,, axes
system with the centre of gravity G in the still water level, so 0=OG .
Some of the terms in the hydromechanical loads have been outlined there:• the ''Modified Strip Theory'' (OST) includes these outlined terms, but• when ignoring these outlined terms, the ''Ordinary Strip Theory'' (MST) has been
presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
198
5.1 Hydromechanical Forces for Surge
The hydromechanical forces for surge are found by integration over the ship length of thetwo-dimensional values:
∫ ⋅=L
bhh dxXX '11
Equation 5.1–1
When assuming that the cross sectional hydromechanical force hold at a plane through thelocal centroid of the cross section, b , parallel to ( )bb yx , , equivalent longitudinal motions ofthe water particles, relative to the cross section of an oscillating ship in still water, are definedby:
θ
θθθζ
θ
θθζ
θζ
&&&&
&&&&&&&
&&
&&&
⋅+−≈
⋅∂
∂⋅+⋅
∂∂
⋅⋅−⋅+−=
⋅+−≈
⋅∂∂
⋅−⋅+−=
⋅+−=
bGx
xbG
VxbG
VbGx
bGx
xbG
VbGx
bGx
bbh
bh
h
2
22*
1
*1
*1
2
Equation 5.1–2In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a surging cross section in still water is defined by:
*1
'11'
11*
1'
11
*1
'11
*1
'11
'1
hb
h
hhh
dxdM
VNM
NMDtD
X
ζζ
ζζ
&&&
&&
⋅
⋅−+⋅=
⋅+⋅=
Equation 5.1–3
According to the ''Modified Strip Theory'' this hydromechanical force becomes:
*1
'11'
11*
1
'11
2'
11
*1
'11
'11
'1
hb
hbe
he
h
dxdM
VNdx
dNVM
Ni
MDtD
X
ζζω
ζω
&&&
&
⋅
⋅−+⋅
⋅+=
⋅
⋅−=
Equation 5.1–4
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
199
This results into the following coupled surge equation:
( )
1151515
131313
1111111
w
h
Xcba
zczbza
xcxbxaXx
=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+=−⋅∇⋅
θθθ
ρρ
&&&&&&
&&&&&
Equation 5.1–5
with:
0
0
0
0
0
15
11
'11'
1115
'11
2'
1115
13
13
13
11
11
'11'
1111
'11
2'
1111
=
⋅−⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−=
==
=
=
+⋅
−+=
⋅⋅+⋅+=
∫
∫∫
∫
∫∫
c
BGbdxbGdx
dMVNb
dxbGdx
dNVdxbGMa
c
b
a
c
bdxdx
dMNb
dxdx
dNVdxMa
VL
bb
Lb
beLb
VL
bb
Lb
beLb
ω
ω
Equation 5.1–6
The ''Modified Strip Theory'' includes the outlined terms. When ignoring these outlined termsthe ''Ordinary Strip Theory'' is presented.
A small viscous surge damping coefficient Vb11 , derived from the still water resistanceapproximation of Troost [1955], has been added here.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
200
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled surge equation become:
0
0
0
0
0
15
11
'11'
1115
'11
2'
1115
13
13
13
11
11'
1111
'1111
=
⋅−⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−=
==
=
=
+⋅+=
⋅+=
∫
∫∫
∫
∫
c
BGbdxbGdx
dMVNb
dxbGdx
dNVdxbGMa
c
b
a
c
bdxNb
dxMa
VL
bb
Lb
beLb
V
L
b
Lb
ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
201
5.2 Hydromechanical Forces for Sway
The hydromechanical forces for sway are found by integration over the ship length of the two-dimensional values:
∫ ⋅=L
bhh dxXX '22
Equation 5.2–1
The lateral and roll motions of the water particles, relative to the cross section of an oscillatingship in still water, are defined by:
φψψζ
φψψζ
φψζ
&&&&&&&&&
&&&&
⋅−⋅⋅+⋅−−=
⋅−⋅+⋅−−=
⋅−⋅−−=
OGVxy
OGVxy
OGxy
bh
bh
bh
2*2
*2
*2
φζ
φζ
φζ
&&&&
&&
−=
−=
−=
*4
*4
*4
h
h
h
Equation 5.2–2
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a swaying cross section in still water is defined by:
*4
'24'
24*
4'
24*
2
'22'
22*
2'
22
*4
'24
*4
'24
*2
'22
*2
'22
'2
hb
hhb
h
hhhhh
dxdM
VNMdx
dMVNM
NMDtD
NMDtD
X
ζζζζ
ζζζζ
&&&&&&
&&&&
⋅
⋅−+⋅+⋅
⋅−+⋅=
⋅+⋅+⋅+⋅=
Equation 5.2–3
According to the ''Modified Strip Theory'' this hydromechanical force becomes:
*4
'24'
24*
4
'24
2'
24
*2
'22'
22*
2
'22
2'
22
*2
'22
'22
*2
'22
'22
'2
hb
hbe
hb
hbe
he
he
h
dxdM
VNdx
dNVM
dxdMVN
dxdNVM
Ni
MDtD
Ni
MDtD
X
ζζω
ζζω
ζω
ζω
&&&
&&&
&&
⋅
⋅−+⋅
⋅++
⋅
⋅−+⋅
⋅+=
⋅
⋅−+
⋅
⋅−=
Equation 5.2–4
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
202
This results into the following coupled sway equation:
( )
2262626
242424
2222222
w
h
Xcba
cba
ycybyaXy
=⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+=−⋅∇⋅
ψψψφφφ
ρρ
&&&
&&&
&&&&&
with:
0
2
0
0
26
'22
2
2
'22
'22'
2226
'22
2
'222
'22'
222
'2226
24
'22'
22
'24'
2424
'22
2
'24
2
'22
'2424
22
'22'
2222
'22
2'
2222
=
⋅⋅+
⋅⋅⋅−⋅⋅
⋅−+=
⋅⋅⋅+⋅⋅+
⋅
⋅−⋅+⋅⋅+=
=
⋅
⋅−⋅+⋅
⋅−+=
⋅⋅⋅+⋅⋅+
⋅⋅+⋅+=
=
⋅
⋅−+=
⋅⋅+⋅+=
∫
∫∫
∫ ∫
∫∫
∫∫
∫∫
∫∫
∫
∫∫
c
dxdx
dNV
dxMVdxxdx
dMVNb
dxxdx
dNVdxN
V
dxdx
dMVN
VdxxMa
c
dxdx
dMVNOGdx
dxdM
VNb
dxdx
dNOG
Vdx
dxdNV
dxMOGdxMa
c
dxdx
dMVNb
dxdx
dNVdxMa
Lb
be
Lb
Lbb
b
L Lbb
beb
e
Lb
beLbb
Lb
bLb
b
Lb
beLb
be
L
b
L
b
Lb
b
Lb
beLb
ω
ωω
ω
ωω
ω
Equation 5.2–5
The ''Modified Strip Theory'' includes the outlined terms. When ignoring these terms the''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
203
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled sway equation become:
0
0
0
26
'22
'2226
'222
'2226
24
'22
'2424
'22
'2424
22
'2222
'2222
=
⋅⋅−⋅⋅+=
⋅⋅+⋅⋅+=
=
⋅⋅+⋅+=
⋅⋅+⋅+=
=
⋅+=
⋅+=
∫∫
∫∫
∫∫
∫∫
∫
∫
c
dxMVdxxNb
dxNV
dxxMa
c
dxNOGdxNb
dxMOGdxMa
c
dxNb
dxMa
Lb
Lbb
Lb
eLbb
L
b
L
b
Lb
Lb
Lb
Lb
ω
Equation 5.2–6
So no terms have been added for the ''Modified Strip Theory''.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
204
5.3 Hydromechanical Forces for Heave
The hydromechanical forces for heave are found by integration over the ship length of thetwo-dimensional values:
∫ ⋅=L
bhh dxXX '33
Equation 5.3–1
The vertical motions of the water particles, relative to the cross section of an oscillating shipin still water, are defined by:
θθζ
θθζ
θζ
&&&&&&&
&&&
⋅⋅−⋅+−=
⋅−⋅+−=
⋅+−=
Vxz
Vxz
xz
bh
bh
bh
2*3
*3
*3
Equation 5.3–2
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a heaving cross section in still water is defined by:
*3
*3
'33'
33*
3'
33
*3
*3
'33
*3
'33
'3
2
2
hwhb
h
hwhhh
ygdx
dMVNM
ygNMDtD
X
ζρζζ
ζρζζ
⋅⋅⋅⋅+⋅
⋅−+⋅=
⋅⋅⋅⋅+⋅+⋅=
&&&
&&
Equation 5.3–3
According to the ''Modified Strip Theory'' this hydromechanical force becomes:
*3
*3
'33'
33*
3
'33
2'
33
*3
*3
'33
'33
'3
2
2
hwhb
hbe
hwhe
h
ygdx
dMVN
dxdNV
M
ygNi
MDtD
X
ζρζζω
ζρζω
⋅⋅⋅⋅+⋅
⋅−+⋅
⋅+=
⋅⋅⋅⋅+
⋅
⋅−=
&&&
&
Equation 5.3–4
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
205
This results into the following coupled heave equation:
( )3353535
333333
3131313
w
h
Xcba
zczbza
xcxbxaXz
=⋅+⋅+⋅+⋅+⋅+⋅+∇⋅+⋅+⋅+⋅+=−⋅∇⋅
θθθρ
ρ
&&&&&&
&&&&&
Equation 5.3–5
with:
∫
∫
∫∫
∫ ∫
∫∫
∫
∫
∫∫
⋅⋅⋅⋅⋅−=
⋅⋅+
⋅⋅⋅+⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−
+
⋅
−⋅−⋅⋅−=
⋅⋅⋅⋅+=
⋅
⋅−+=
⋅⋅+⋅+=
===
Lbbw
Lb
be
Lb
Lbb
b
L Lbb
be
b
e
Lb
beLbb
Lbw
L
bb
L
bbeL
b
dxxygc
dxdx
dNV
dxMVdxxdx
dMVNb
dxxdx
dNVdxN
V
dxdx
dMN
VdxxMa
dxygc
dxdx
dMVNb
dxdx
dNVdxMa
c
b
a
ρ
ω
ωω
ω
ρ
ω
2
2
2
0
0
0
35
'33
2
2
'33
'33'
3335
'33
2'
332
'33'
332'
3335
33
'33'
3333
'33
2
'3333
31
31
31
Equation 5.3–6
The ''Modified Strip Theory Method'' includes the outlined terms. When ignoring these termsthe ''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
206
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled heave equation become:
∫
∫∫
∫∫
∫
∫
∫
⋅⋅⋅⋅⋅−=
⋅⋅+⋅⋅−=
⋅⋅−⋅⋅−=
⋅⋅⋅⋅+=
⋅+=
⋅+=
==
=
Lbbw
L
b
L
bb
L
b
eL
bb
Lbw
L
b
Lb
dxxygc
dxMVdxxNb
dxNV
dxxMa
dxygc
dxNb
dxMa
c
b
a
ρ
ω
ρ
2
2
0
0
0
35
'33
'3335
'332
'3335
33
'3333
'3333
31
31
31
Equation 5.3–7
So no terms have been added for the ''Modified Strip Theory''.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
207
5.4 Hydromechanical Moments for Roll
The hydromechanical moments for roll are found by integration over the ship length of thetwo-dimensional values:
∫ ⋅=L
bhh dxXX '44
Equation 5.4–1
The roll and lateral motions of the water particles, relative to the cross section of an oscillatingship in still water, are defined by:
φζ
φζ
φζ
&&&&
&&
−=
−=
−=
*4
*4
*4
h
h
h
φψψζ
φψψζ
φψζ
&&&&&&&&&
&&&&
⋅−⋅⋅+⋅−−=
⋅−⋅+⋅−−=
⋅−⋅−−=
OGVxy
OGVxy
OGxy
bh
bh
bh
2*2
*2
*2
Equation 5.4–2
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalmoment on a rolling cross section in still water is defined by:
*2
'42'
42*
2'
42
*4
3*
4
'44'
44*
4'
44
*2
'42
*2
'42
*4
3*
4'
44*
4'
44'
2
232
232
hb
h
hsw
hb
h
hh
hsw
hhh
dxdM
VNM
bGAy
gdx
dMVNM
NMDtD
bGAy
gNMDtD
X
ζζ
ζρζζ
ζζ
ζρζζ
&&&
&&&
&&
&&
⋅
⋅−+⋅+
⋅
⋅−⋅⋅⋅+⋅
⋅−+⋅=
⋅+⋅+
⋅
⋅−⋅⋅⋅+⋅+⋅=
Equation 5.4–3
According to the ''Modified Strip Theory'' this hydromechanical moment becomes:
*2
'42'
42*
2
'42
2
'42
*4
3
*4
'44'
44*
4
'44
2
'44
*2
'42
'42
*4
3*
4'
44'
44'
2
232
232
hb
hbe
hsw
hb
hbe
he
hsw
he
h
dxdM
VNdx
dNVM
bGAy
g
dxdM
VNdx
dNVM
Ni
MDtD
bGAy
gNi
MDtD
X
ζζω
ζρ
ζζω
ζω
ζρζω
&&&
&&&
&
&
⋅
⋅−+⋅
⋅++
+⋅
⋅−⋅⋅⋅+
⋅
⋅−+⋅
⋅+=
⋅
⋅−+
⋅
⋅−⋅⋅⋅+
⋅
⋅−=
Equation 5.4–4
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
208
This results into the following coupled roll equation:
( )( ) 4464646
444444
4242424
wxz
xx
hxzxx
XcbaI
cbaI
ycybyaXII
=⋅+⋅+⋅+−+⋅+⋅+⋅+++⋅+⋅+⋅+=−⋅−⋅
ψψψφφφ
ψφ
&&&
&&&
&&&&&&&
Equation 5.4–5
with:
2646
'42
2
2
26'
42
'42'
4246
'42
2
'422
26
'42'
422
'4246
24
3
44
2444
'42'
42
'44'
4444
'42
2
'44
2
24'
42'
4444
2242
22
'42'
4242
'24
222'
4242
0
2
232
0
cOGc
dxdx
dNV
bOGdxMVdxxdx
dMVNb
dxxdx
dNVdxN
V
aOGdxdx
dMVN
VdxxMa
GMg
cOGdxbGAy
gc
bOGbdxdx
dMVNOGdx
dxdM
VNb
dxdx
dNOG
Vdx
dxdNV
aOGdxMOGdxMa
cOGc
bOGdxdx
dMVNb
dxdx
dNVaOGdxMa
Lb
be
Lb
Lbb
b
L Lbb
beb
e
Lb
beLbb
Lb
sw
LVb
bLb
b
L Lb
beb
be
L
b
L
b
L
bb
L
bbeL
b
⋅+=
⋅⋅+
⋅+⋅⋅⋅−⋅⋅
⋅−+=
⋅⋅⋅+⋅⋅+
⋅+⋅
⋅−⋅+⋅⋅+=
⋅∇⋅⋅+=
⋅+⋅
⋅+⋅⋅⋅+=
⋅++⋅
⋅−⋅+⋅
⋅−+=
⋅⋅⋅+⋅⋅+
⋅+⋅⋅+⋅+=
⋅+=
⋅+⋅
⋅−+=
⋅⋅+⋅+⋅+=
∫
∫∫
∫ ∫
∫∫
∫
∫∫
∫ ∫
∫∫
∫
∫∫
ω
ωω
ω
ρ
ρ
ωω
ω
Equation 5.4–6
The ''Modified Strip Theory'' includes the outlined terms. When ignoring these terms the''Ordinary Strip Theory Method'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
209
A viscous roll damping coefficient Vb44 , derived for instance with the empirical method ofIkeda [1978], has been added here.
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled roll equation become:
0
0
46
26'
42'
4246
26'
422'
4246
44
2444'
42'
4444
24'
42'
4444
24
22'
4242
22'
4242
=
⋅+⋅⋅−⋅⋅+=
⋅+⋅⋅+⋅⋅+=
⋅∇⋅⋅+=
⋅++⋅⋅+⋅+=
⋅+⋅+⋅+=
=
⋅+⋅+=
⋅+⋅+=
∫∫
∫∫
∫∫
∫∫
∫
∫
c
bOGdxMVdxxNb
aOGdxNV
dxxMa
GMgc
bOGbdxNOGdxNb
aOGdxMdxMa
c
bOGdxNb
aOGdxMa
Lb
Lbb
Lb
eLbb
LVb
Lb
Lb
Lb
Lb
Lb
ω
ρ
Equation 5.4–7
So no terms have been added for the ''Modified Strip Theory''.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
210
5.5 Hydromechanical Moments for Pitch
The hydromechanical moments for pitch are found by integration over the ship length of thetwo-dimensional contributions of surge and heave into the pitch moment:
∫ ⋅=L
bhh dxXX '55 with: bhhh xXbGXX ⋅−⋅−= '
3'
1'
5
Equation 5.5–1
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalmoment on a pitching cross section in still water is defined by surge and heave contributions:
*3
*3
'33'
33*
3'
33
*1
'11'
11*
1'
11'
5
2 hbwhbb
hb
hb
hh
xygxdx
dMVNxM
bGdx
dMVNbGMX
ζρζζ
ζζ
⋅⋅⋅⋅⋅−⋅⋅
⋅−−⋅⋅−
⋅⋅
⋅−−⋅⋅−=
&&&
&&&
Equation 5.5–2
According to the ''Modified Strip Theory'' this hydromechanical moment becomes:
*3
*3
'33'
33*
3
'33
2'
33
*1
'11'
11*
1
'11
2'
11'
5
2 hbw
hbb
hbbe
hb
hbe
h
xyg
xdx
dMVNx
dxdNV
M
bGdx
dMVNbG
dxdNV
MX
ζρ
ζζω
ζζω
⋅⋅⋅⋅⋅−
⋅⋅
⋅−−⋅⋅
⋅+−
⋅⋅
⋅−−⋅⋅
⋅+−=
&&&
&&&
Equation 5.5–3
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
211
This results into the following coupled pitch equation:
( ) 5555555
535353
5151515
wyy
hyy
XcbaI
zczbza
xcxbxaXI
=⋅+⋅+⋅++
⋅+⋅+⋅+⋅+⋅+⋅+=−⋅
θθθ
θ
&&&&&&
&&&&&
Equation 5.5–4
with:
∫
∫
∫∫
∫
∫ ∫
∫∫
∫∫
∫
∫
∫∫
∫
∫∫
⋅⋅⋅⋅⋅+=
⋅⋅⋅−
+
⋅⋅⋅⋅−⋅⋅
⋅−+
⋅+⋅⋅
⋅−+=
⋅⋅⋅+⋅⋅⋅+
⋅⋅
−⋅+⋅⋅+
⋅⋅⋅+⋅⋅+=
⋅⋅⋅⋅⋅−=
⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−=
=
⋅−⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−=
Lbbw
Lbb
be
Lbb
Lbb
b
VL
bb
L Lbb
bebb
e
L
bbbeL
bb
Lb
beLb
Lbbw
Lbb
b
Lbb
beLbb
VL
bb
L
bbeL
b
dxxygc
dxxdx
dNV
dxxMVdxxdx
dMVN
BGbdxbGdx
dMVNb
dxxdx
dNVdxxN
V
dxxdx
dMN
VdxxM
dxbGdx
dNVdxbGMa
dxxygc
dxxdx
dMVNb
dxxdx
dNVdxxMa
c
BGbdxbGdx
dMVNb
dxbGdx
dNVdxbGMa
255
'33
2
2
'33
2'
33'33
2
11
2'
11'1155
2'
332
'332
'33'
3322'
33
2'
112
2'1155
53
'33'
3353
'33
2
'3353
51
11
'11'
1151
'11
2
'1151
2
2
2
0
ρ
ω
ωω
ω
ω
ρ
ω
ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
212
Equation 5.5–5
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled pitch equation become:
∫
∫∫
∫
∫
∫∫∫
∫∫
∫
∫∫
∫∫
∫
∫∫
⋅⋅⋅⋅⋅+=
⋅⋅+⋅⋅+
⋅+⋅⋅
⋅−+=
⋅⋅⋅−
+
⋅⋅+⋅⋅⋅+⋅⋅+
⋅⋅⋅+⋅⋅+=
⋅⋅⋅⋅⋅−=
⋅⋅−⋅⋅−=
⋅⋅+⋅⋅−=
=
⋅−⋅⋅
⋅−−=
⋅⋅⋅−
+⋅⋅−=
Lbbw
Lb
eLbb
V
L
bb
Lbb
e
L
b
eL
bb
eL
bb
L
bbeL
b
Lbbw
L
b
L
bb
L
b
eL
bb
VL
bb
L
bbeL
b
dxxygc
dxNV
dxxN
BGbdxbGdx
dMVNb
dxxNV
dxMV
dxxNV
dxxM
dxbGdx
dNVdxbGMa
dxxygc
dxMVdxxNb
dxNV
dxxMa
c
bGbdxbGdx
dMVNb
dxbGdx
dNVdxbGMa
255
'332
22'
33
2
11
2'
11'1155
'332
'332
'332
2'33
2'
112
2'1155
53
'33
'3353
'332
'3353
51
11
'11'
1151
'11
2'
1151
2
2
0
ρ
ω
ω
ωω
ω
ρ
ω
ω
Equation 5.5–6
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
213
5.6 Hydromechanical Moments for Yaw
The hydromechanical moments for yaw are found by integration over the ship length of thetwo-dimensional contributions of sway into the yaw moment:
∫ ⋅=L
bhh dxXX '66 with: bhh xXX ⋅+= '
2'
6
Equation 5.6–1
According to the ''Ordinary Strip Theory'' the two-dimensional potential hydromechanicalforce on a yawing cross section in still water is defined by sway contributions:
*4
'24'
24*
4'
24
*2
'22'
22*
2'
22'
2
hbb
hb
hbb
hbh
xdx
dMVNxM
xdx
dMVNxMX
ζζ
ζζ
&&&
&&&
⋅⋅
⋅−+⋅⋅+
⋅⋅
⋅−+⋅⋅=
Equation 5.6–2
According to the ''Modified Strip Theory'' this hydromechanical force becomes:
*4
'24'
24*
4
'24
2'
24
*2
'22'
22*
2
'22
2'
22'
2
hbb
hbbe
hbb
hbbe
h
xdx
dMVNx
dxdNV
M
xdx
dMVNx
dxdNV
MX
ζζω
ζζω
&&&
&&&
⋅⋅
⋅−+⋅⋅
⋅++
⋅⋅
⋅−+⋅⋅
⋅+=
Equation 5.6–3
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
214
This results into the following coupled yaw equation:
( )( ) 6666666
646464
6262626
wzz
zx
hzxzz
XcbaI
cbaI
ycybyaXII
=⋅+⋅+⋅+++⋅+⋅+⋅+−+⋅+⋅+⋅+=−⋅−⋅
ψψψφφφ
φψ
&&&
&&&
&&&&&&&
Equation 5.6–4
with:
0
2
0
0
66
'22
2
2
'22
2'
22'2266
2'
222
'222
'22'
2222'
2266
64
'22'
22
'24'
2464
'22
2
'24
2
'22
'2464
62
'22'
2262
'22
2'
2262
=
⋅⋅⋅+
⋅⋅⋅⋅−⋅⋅
⋅−+=
⋅⋅⋅+⋅⋅⋅+
⋅⋅
⋅−⋅+⋅⋅+=
=
⋅⋅
⋅−⋅+⋅⋅
⋅−+=
⋅⋅⋅⋅+⋅⋅⋅+
⋅⋅⋅+⋅⋅+=
=
⋅⋅
⋅−+=
⋅⋅⋅+⋅⋅+=
∫
∫∫
∫ ∫
∫∫
∫∫
∫∫
∫∫
∫
∫∫
c
dxxdx
dNV
dxxMVdxxdx
dMVNb
dxxdx
dNVdxxN
V
dxxdx
dMVN
VdxxMa
c
dxxdx
dMVNOGdxx
dxdM
VNb
dxxdx
dNOG
Vdxx
dxdNV
dxxMOGdxxMa
c
dxxdx
dMVNb
dxxdx
dNVdxxMa
Lbb
be
Lbb
Lbb
b
L Lbb
bebb
e
Lbb
beLbb
Lbb
bLbb
b
Lbb
beLbb
be
L
bb
L
bb
Lbb
b
Lbb
beLbb
ω
ωω
ω
ωω
ω
Equation 5.6–5
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
215
After simplification (see the end-terms in Section 2-5-3), the expressions for the totalhydromechanical coefficients in the coupled yaw equation become:
0
0
0
66
'222
22'
2266
'222
'222
2'
2222'
2266
64
'22
'22
'24
'2464
'222
'242
'22
'2464
62
'22
'2262
'222
'2262
=
⋅⋅−
+⋅⋅+=
⋅⋅⋅−
+
⋅⋅+⋅⋅⋅+⋅⋅+=
=
⋅⋅⋅+⋅⋅⋅+
⋅⋅+⋅⋅+=
⋅⋅⋅−
+⋅⋅−
+
⋅⋅⋅+⋅⋅+=
=
⋅⋅+⋅⋅+=
⋅⋅−
+⋅⋅+=
∫∫
∫
∫∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
c
dxNV
dxxNb
dxxNV
dxMV
dxxNV
dxxMa
c
dxMOGVdxxNOG
dxMVdxxNb
dxNOGV
dxNV
dxxMOGdxxMa
c
dxMVdxxNb
dxNV
dxxMa
Lb
eLbb
Lbb
e
Lb
eLbb
eLbb
Lb
Lbb
Lb
Lbb
Lb
eLb
e
L
bb
L
bb
Lb
Lbb
Lb
eLbb
ω
ω
ωω
ωω
ω
Equation 5.6–6
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
216
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
217
6 Exciting Wave Loads
6.1 Wave Potential
The first order wave potential in a fluid - with any arbitrary water depth h - is given by:( )[ ][ ] ( )µµωζ
ωsincossin
coshcosh
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅−
=Φ bbeab
w ykxkthk
zhkg
Equation 6.1–1
in an axes system with the centre of gravity in the waterline.The velocities and accelerations in the direction j of the water particles have to be defined.The local relative orbital velocities of the water particles in a certain direction follow from thederivative in that direction of the wave potential. The orbital accelerations of the waterparticles can be obtained from these velocities by:
''wjwj Dt
D ζζ &&& = with:
∂∂
⋅−∂∂
=bx
VtDt
D for: 4,3,2,1=j
Equation 6.1–2
With this, the relative velocities and accelerations in the different directions can be found:
• Surge direction:
( )[ ][ ] ( )
( )[ ][ ] ( )µµωζµζ
µµωζω
µ
ζ
sincossincosh
coshcos
sincoscoscosh
coshcos
'1
'1
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅+
=
∂Φ∂
=
bbeab
w
bbeab
b
ww
ykxkthk
zhkgk
ykxkthk
zhkgk
x
&&
&
Equation 6.1–3
• Sway direction:
( )[ ][ ] ( )
( )[ ][ ] ( )µµωζµζ
µµωζω
µ
ζ
sincossincosh
coshsin
sincoscoscosh
coshsin
'2
'2
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅+
=
∂Φ∂
=
bbeab
w
bbeab
b
ww
ykxkthk
zhkgk
ykxkthk
zhkgk
y
&&
&
Equation 6.1–4
• Heave direction:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
218
( )[ ][ ] ( )
( )[ ][ ] ( )µµωζζ
µµωζω
ζ
sincoscoscosh
sinh
sincossincosh
sinh
'3
'3
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅−
=
∂Φ∂
=
bbeab
w
bbeab
b
ww
ykxkthkzhk
gk
ykxkthkzhkgk
z
&&
&
Equation 6.1–5
• Roll direction:
0
0
'4
'3
'2'
4
=
=∂
∂−
∂∂
=
w
b
w
b
ww yz
ζ
ζζζ
&&
&&&
Equation 6.1–6
These zero solutions are obvious, because the potential fluid is free of rotation.
The pressure in the fluid follows from the linearised equation of Bernoulli:( )[ ][ ] ( )
bb
bb
bb
bbeab
b
dzzp
dyyp
dxxp
p
ykxkthk
zhkgzgp
⋅∂∂
+⋅∂∂
+⋅∂∂
+=
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅⋅⋅+⋅⋅−=
0
sincoscoscosh
cosh µµωζρρ
Equation 6.1–7
with the following expressions for the pressure gradients:( )[ ][ ] ( )
( )[ ][ ] ( )
( )[ ][ ] ( )µµωζρρ
µµωζµρ
µµωζµρ
sincoscoscosh
cosh
sincossincosh
coshsin
sincossincosh
coshcos
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅+⋅−=∂∂
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅⋅+=∂∂
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅⋅⋅+=∂∂
bbeab
b
bbeab
b
bbeab
b
ykxkthk
zhkkgg
zp
ykxkthk
zhkkg
yp
ykxkthk
zhkkgxp
Equation 6.1–8
These pressure gradients can be expressed in the orbital accelerations too:
( )'3
'2
'1
wb
wb
wb
gzp
yp
xp
ζρ
ζρ
ζρ
&&
&&
&&
+⋅−=∂∂
⋅+=∂∂
⋅−=∂∂
Equation 6.1–9
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
219
6.2 Classical Approach
First the classical approach to obtain the wave loads - according to the relative motionprinciple - is given here.
6.2.1 Exciting Wave Forces for Surge
The exciting wave forces for surge on a ship are found by integration over the ship length ofthe two-dimensional values:
∫ ⋅=L
bww dxXX '11
Equation 6.2–1
According to the ''Ordinary Strip Theory'' the exciting wave forces for surge on a restrainedcross section of a ship in waves are defined by:
'
1*
1
'11'
11*
1'
11
'1
*1
'11
*1
'11
'1
FKwb
w
FKwww
Xdx
dMVNM
XNMDtD
X
+⋅
⋅−+⋅=
+⋅+⋅=
ζζ
ζζ
&&&
&&
Equation 6.2–2
According to the ''Modified Strip Theory'' these forces become:
'1
*1
'11'
11*
1
'11
2'
11
'1
*1
'11
'11
'1
FKwb
wbe
FKwe
w
Xdx
dMVN
dxdNV
M
XNi
MDtD
X
+⋅
⋅−+⋅
⋅+=
+
⋅
⋅+=
ζζω
ζω
&&&
&
Equation 6.2–3
Figure 6.2–1: Wave pressure distribution on a cross section for surge
The Froude-Krilov force in the surge direction - so the longitudinal force due to the pressurein the undisturbed fluid, see Figure 6.2–1 - is given by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
220
∫ ∫
∫ ∫
−
+
−
−
+
−
⋅⋅⋅=
⋅⋅∂∂
−=
ζ
ζ
ζρT
y
ybbw
T
y
ybb
bFK
b
b
b
b
dzdy
dzdyxp
X
'1
1
&&
Equation 6.2–4
After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( )µωζµρ cossincos1 ⋅⋅−⋅⋅⋅⋅⋅−⋅⋅= beachFK xktgkAX
with:( ) ( )[ ]
[ ]∫−
⋅⋅⋅+⋅
⋅⋅⋅−
⋅⋅−⋅=
0
coshcosh
sinsinsin
2T
bbb
b
bch dzy
hkzhk
ykyk
Aµµ
Equation 6.2–5
When expanding the Froude-Krilov force in deep water with wy⋅⋅>> πλ 2 and T⋅⋅>> πλ 2in series, it is found:
( ) ( )µωζµρ cossincos...21 2
1 ⋅⋅−⋅⋅⋅⋅⋅−⋅
+⋅⋅+⋅+⋅= beayyFK xktgkIkSkAX
with:
∫−
⋅⋅=0
2T
bb dzyA ∫−
⋅⋅⋅=0
2T
bbby dzzyS ∫−
⋅⋅⋅=0
22T
bbby dzzyI
Equation 6.2–6
The acceleration term agk ζµ ⋅⋅⋅ cos in here is the amplitude of the longitudinal component
of the relative orbital acceleration in deep water at 0=bz .The dominating first term in this series consists of a mass and this acceleration. The massterm A⋅ρ is used to obtain from the total Froude-Krilov force an equivalent longitudinalcomponent of the orbital acceleration of the water particles:
*11 wFK AX ζρ &&⋅⋅=
Equation 6.2–7
This holds that the equivalent longitudinal components of the orbital acceleration and velocityare equal to the values at 0=bz in a wave with reduced amplitude *
1aζ :
( )
( )µωζω
µζ
µωζµζ
coscoscos
cossincos
*1
*1
*1
*1
⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅−⋅⋅⋅⋅⋅−=
beaw
beaw
xktgk
xktgk
&
&&
with:
ach
a AA ζζ ⋅=*
1
Equation 6.2–8
This equivalent acceleration and velocity will be used in the diffraction part of the wave forcefor surge.
From the foregoing follows the total wave loads for surge:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
221
∫
∫
∫∫
⋅+
⋅⋅
⋅−⋅+
⋅⋅⋅⋅
+⋅⋅+=
LbFK
bw
L be
Lbw
beLbww
dxX
dxdx
dMVN
dxdx
dNVdxMX
'1
*1
'11'
11
*1
'11*
1'
111
ζωω
ζωω
ζ
&
&&&&
Equation 6.2–9
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
6.2.2 Exciting Wave Forces for Sway
The exciting wave forces for sway on a ship are found by integration over the ship length ofthe two-dimensional values:
∫ ⋅=L
bww dxXX '22
Equation 6.2–10
According to the ''Ordinary Strip Theory'' the exciting wave forces for sway on a restrainedcross section of a ship in waves are defined by:
'
2*
2
'22'
22*
2'
22
'2
*2
'22
*2
'22
'2
FKwb
w
FKwww
Xdx
dMVNM
XNMDtD
X
+⋅
⋅−+⋅=
+⋅+⋅=
ζζ
ζζ
&&&
&&
Equation 6.2–11
According to the ''Modified Strip Theory'' these forces become:
'2
*2
'22'
22*
2
'22
2
'22
'2
*2
'22
'22
'2
FKwb
wbe
FKwe
w
Xdx
dMVN
dxdNV
M
XNi
MDtD
X
+⋅
⋅−+⋅
⋅+=
+
⋅
⋅+=
ζζω
ζω
&&&
&
Equation 6.2–12
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
222
Figure 6.2–2: Wave pressure distribution on a cross section for sway
The Froude-Krilov force in the sway direction - so the lateral force due to the pressure in theundisturbed fluid - is given by:
∫ ∫
∫ ∫
−
+
−
−
+
−
⋅⋅⋅=
⋅⋅∂∂
−=
ζ
ζ
ζρT
y
ybbw
T
y
ybb
bFK
b
b
b
b
dzdy
dzdyyp
X
'2
2
&&
Equation 6.2–13
After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( )µωζµρ cossinsin2 ⋅⋅−⋅⋅⋅⋅⋅−⋅⋅= beachFK xktgkAX
with:( ) ( )[ ]
[ ]∫−
⋅⋅⋅+⋅
⋅⋅⋅−
⋅⋅−⋅=
0
coshcosh
sinsinsin
2T
bbb
b
bch dzy
hkzhk
ykyk
Aµµ
Equation 6.2–14
When expanding the Froude-Krilov force in deep water with wy⋅⋅>> πλ 2 and T⋅⋅>> πλ 2in series, it is found:
( ) ( )µωζµρ cossinsin...21 2
2 ⋅⋅−⋅⋅⋅⋅⋅−⋅
+⋅⋅+⋅+⋅= beayyFK xktgkIkSkAX
with:
∫−
⋅⋅=0
2T
bb dzyA ∫−
⋅⋅⋅=0
2T
bbby dzzyS ∫−
⋅⋅⋅=0
22T
bbby dzzyI
Equation 6.2–15
The acceleration term agk ζµ⋅⋅⋅ sin in here is the amplitude of the lateral component of the
relative orbital acceleration in deep water at 0=bz .The dominating first term in this series consists of a mass and this acceleration.This mass term A⋅ρ is used to obtain from the total Froude-Krilov force an equivalent lateralcomponent of the orbital acceleration of the water particles:
*22 wFK AX ζρ &&⋅⋅=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
223
This holds that the equivalent lateral components of the orbital acceleration and velocity areequal to the values at 0=bz in a wave with reduced amplitude *
2aζ :
( )
( )µωζω
µζ
µωζµζ
coscossin
cossinsin
*2
*1
*2
*2
⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅−⋅⋅⋅⋅⋅−=
beaw
beaw
xktgk
xktgk
&
&&
with:
ach
a AA ζζ ⋅=*
2
Equation 6.2–16
This equivalent acceleration and velocity will be used in the diffraction part of the wave forcefor sway.
From the foregoing follows the total wave loads for sway:
∫
∫
∫∫
⋅+
⋅⋅
⋅−⋅+
⋅⋅⋅⋅
+⋅⋅+=
LbFK
bw
L be
Lbw
beLbww
dxX
dxdx
dMVN
dxdx
dNVdxMX
'2
*2
'22'
22
*2
'22*
2'
222
ζωω
ζωω
ζ
&
&&&&
Equation 6.2–17
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
6.2.3 Exciting Wave Forces for Heave
The exciting wave forces for heave on a ship are found by integration over the ship length ofthe two-dimensional values:
∫ ⋅=L
bww dxXX '33
Equation 6.2–18
According to the ''Ordinary Strip Theory'' the exciting wave forces for heave on a restrainedcross section of a ship in waves are defined by:
'
3*
3
'33'
33*
3'
33
'3
*3
'33
*3
'33
'3
FKwb
w
FKwww
Xdx
dMVNM
XNMDtD
X
+⋅
⋅−+⋅=
+⋅+⋅=
ζζ
ζζ
&&&
&&
Equation 6.2–19
According to the ''Modified Strip Theory'' these forces become:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
224
'3
*3
'33'
22*
3
'33
2
'33
'3
*3
'33
'33
'3
FKwb
wbe
FKwe
w
Xdx
dMVN
dxdNV
M
XNi
MDtD
X
+⋅
⋅−+⋅
⋅+=
+
⋅
⋅+=
ζζω
ζω
&&&
&
Equation 6.2–20
Figure 6.2–3: Wave pressure distribution on a cross section for heave
The Froude-Krilov force in the heave direction - so the vertical force due to the pressure in theundisturbed fluid - is given by:
( )∫ ∫
∫ ∫
−
+
−
−
+
−
⋅⋅+⋅=
⋅⋅∂∂
−=
ζ
ζ
ζρT
y
y
bbw
T
y
ybb
bFK
b
b
b
b
dzdyg
dzdyzp
X
'3
3
&&
Equation 6.2–21
After neglecting the second order terms, the Froude-Krilov force can be written as:( ) ( ) ( )µωζ
µµρ coscos
sinsinsin2
3 ⋅⋅−⋅⋅⋅⋅−⋅
+
⋅⋅−⋅⋅−
⋅⋅−
⋅= beashb
bwFK xktgkA
ykyk
ky
X
with:( ) ( )[ ]
[ ]∫−
⋅⋅⋅+⋅
⋅⋅⋅−
⋅⋅−⋅=
0
coshsinh
sinsinsin
2T
bbb
b
bsh dzy
hkzhk
ykyk
Aµµ
Equation 6.2–22
When expanding the Froude-Krilov force in deep water with wy⋅⋅>> πλ 2 and T⋅⋅>> πλ 2in series, it is found:
( ) ( )µωζρ coscos...212 2
3 ⋅⋅−⋅⋅⋅⋅−⋅
+⋅⋅+⋅++
⋅−⋅= beayy
wFK xktgkIkSkA
kyX
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
225
∫−
⋅⋅=0
2T
bb dzyA ∫−
⋅⋅⋅=0
2T
bbby dzzyS ∫−
⋅⋅⋅=0
22T
bbby dzzyI
Equation 6.2–23
??????*
3T can be considered as the draft at which the pressure in the vertical direction is equal to theaverage vertical pressure on the cross section in the fluid and can be obtained by.
???*3 =T
This holds that the equivalent vertical components of the orbital acceleration and velocity areequal to the values at *
3Tzb −= :???When expanding the Froude-Krilov force in shallow water with 0→⋅ hk and in long waveswith ??? in series, it is found:
???3 =Cwith:???So in shallow water, *
3T can be obtained by.
???*3 =T
This holds that the equivalent vertical components of the orbital acceleration and velocity areequal to the values at *
3Tzb −= :???It may be noted that this shallow water definition for *
3T is valid in deep water too, because:???These equivalent accelerations and velocities will be used to determine the diffraction part ofthe wave forces for heave.
From the foregoing follows the total wave loads for heave:
∫
∫
∫∫
⋅+
⋅⋅
⋅−⋅+
⋅⋅⋅⋅
+⋅⋅+=
LbFK
bw
L be
Lbw
beLbww
dxX
dxdx
dMVN
dxdx
dNVdxMX
'3
*3
'33'
33
*3
'33*
3'
333
ζωω
ζωω
ζ
&
&&&&
Equation 6.2–24
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
6.2.4 Exciting Wave Moments for Roll
The exciting wave moments for roll on a ship are found by integration over the ship length oftwo-dimensional values:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
226
∫ ⋅=L
bww dxXX '44
Equation 6.2–25
According to the ''Ordinary Strip Theory'' the exciting wave moments for roll on a restrainedcross section of a ship in waves are defined by:
'
2'
4*
2
'42'
42*
2'
42
'2
'4
*2
'42
*2
'42
'4
wFKwb
w
wFKwww
XOGXdx
dMVNM
XOGXNMDtD
X
⋅++⋅
⋅−+⋅=
⋅++⋅+⋅=
ζζ
ζζ
&&&
&&
Equation 6.2–26
According to the ''Modified Strip Theory'' these moments become:
'2
'4
*2
'42'
24*
2
'42
2
'42
'2
'4
*2
'42
'42
'4
wFKwb
wbe
wFKwe
w
XOGXdx
dMVN
dxdNV
M
XOGXNi
MDtD
X
⋅++⋅
⋅−+⋅
⋅+=
⋅++
⋅
⋅+=
ζζω
ζω
&&&
&
Equation 6.2–27
Figure 6.2–4: Wave pressure distribution on a cross section for roll
The Froude-Krilov moment in the roll direction - so the roll moment due to the pressure in theundisturbed fluid - is given by:
( ) ∫ ∫
∫ ∫
−
+
−
−
+
−
⋅⋅⋅++⋅−⋅=
⋅⋅
⋅
∂∂
+⋅∂∂
−−=
ζ
ζ
ζζρT
y
ybbbwbw
T
y
ybbb
bb
bFK
b
b
b
b
dzdyygz
dzdyyzp
zyp
X
'3
'2
4
&&&&
Equation 6.2–28
After neglecting the second order terms, the Froude-Krylov moment can be written as:
( ) ( )µωζµρ cossinsin24 ⋅⋅−⋅⋅⋅⋅⋅−⋅
++−⋅= beazsh
ychywFK xktgkI
k
S
k
CX
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
227
with:( ) ( )
( )( ) ( )[ ]
[ ]( ) ( )
( )( )[ ][ ]∫
∫
−
−
⋅⋅⋅+⋅
⋅⋅⋅−
⋅⋅−−⋅⋅−
⋅⋅−
⋅=
⋅⋅⋅⋅+⋅⋅
⋅⋅−⋅⋅−⋅=
⋅⋅⋅−
⋅⋅−−⋅⋅−
⋅⋅−
⋅=
03
2
0
32
coshcosh
sin
sincossin
sinsin
2
coshcosh
sinsinsin2
sin
sincossin
sinsin
2
Tbb
b
b
bb
b
zsh
T
bbbb
b
bych
wb
ww
w
yw
dzyhk
zhkyk
ykyk
yk
I
dzzyhk
zhkyk
ykS
yyk
ykyk
yk
C
µ
µµµ
µµ
µ
µµµ
Equation 6.2–29
For deep water, the cosine-hyperbolic expressions in here reduce to exponential expressions.
From the foregoing follows the total wave loads for roll:
2'
4
*2
'42'
42
*2
'42*
2'
424
wL
bFK
bwL be
Lbw
beLbww
XOGdxX
dxdx
dMVN
dxdx
dNVdxMX
⋅+⋅+
⋅⋅
⋅−⋅+
⋅⋅⋅⋅
+⋅⋅+=
∫
∫
∫∫
ζωω
ζωω
ζ
&
&&&&
Equation 6.2–30
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
6.2.5 Exciting Wave Moments for Pitch
The exciting wave moments for pitch are found by integration over the ship length of the two-dimensional contributions of surge and heave into the pitch moment:
∫ ⋅=L
bww dxXX '55
with:
bwww xXbGXX ⋅−⋅−= '3
'1
'5
Equation 6.2–31
In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.From this follows the total wave loads for pitch:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
228
∫
∫
∫∫
∫
∫
∫∫
⋅⋅−
⋅⋅⋅
⋅−⋅−
⋅⋅⋅⋅⋅
−+⋅⋅⋅−
⋅⋅−
⋅⋅⋅
⋅−⋅−
⋅⋅⋅⋅⋅
−+⋅⋅⋅−=
LbbFK
bwbL be
Lbwb
beLbwb
L
bFK
bw
L be
L
bwbeL
bww
dxxX
dxxdx
dMVN
dxxdx
dNVdxxM
dxbGX
dxbGdx
dMVN
dxbGdx
dNVdxbGMX
'3
*3
'33'
33
*3
'33*
3'
33
'1
*1
'11'
11
*1
'11*
1'
115
ζωω
ζωω
ζ
ζωω
ζωω
ζ
&
&&&&
&
&&&&
Equation 6.2–32
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
6.2.6 Exciting Wave Moments for Yaw
The exciting wave moments for yaw are found by integration over the ship length of the two-dimensional contributions of sway into the yaw moment:
∫ ⋅=L
bww dxXX '66
with:
bww xXX ⋅+= '2
'6
Equation 6.2–33
From this follows the total wave loads for yaw:
∫
∫
∫∫
⋅⋅+
⋅⋅⋅
⋅−⋅+
⋅⋅⋅⋅⋅
+⋅⋅⋅+=
LbbFK
bwb
L be
Lbwb
beLbwbw
dxxX
dxxdx
dMVN
dxxdx
dNVdxxMX
'2
*2
'22'
22
*2
'22*
2'
226
ζωω
ζωω
ζ
&
&&&&
Equation 6.2–34
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
229
6.3 Approximating 2-D Diffraction Approach
In the classic relative motion theory, the average (or equivalent) motions of the water particlesaround the cross section are calculated from the pressure distribution in the undisturbed waveson this cross section. An alternative approach - based on diffraction of waves - to determinethe equivalent accelerations and velocities of the water particles around the cross section, asgiven by Journee and van ‘t Veer [1995], is described now.
6.3.1 Hydromechanical Loads
Suppose an infinite long cylinder in the still water surface of a fluid. The cylinder is forced tocarry out a simple harmonic oscillation about its initial position with frequency of oscillationω and small amplitude of displacement jax :
( )txx jaj ⋅⋅= ωcos for: 4,3,2=j
Equation 6.3–1
The 2-D hydrodynamic loads 'hiX in the sway, heave and roll directions i , exercised by the
fluid on a cross section of the cylinder, can be obtained from the 2-D velocity potentials andthe linearised equations of Bernoulli. The velocity potentials have been obtained by using thework of Ursell [1949] and N -parameter conformal mapping. These hydrodynamic loads are:
( ) ( ) jj xijxij
jawlhi tBtA
gygX ΦΦ +⋅⋅++⋅⋅⋅
⋅⋅⋅⋅⋅= εωεω
πζ
ρ sincos2'
Equation 6.3–2
in where j is the mode of oscillation and i is the direction of the load. The phase lag jxΦε is
defined as the phase lag between the velocity potential of the fluid Φ and the forced motion
jx . The radiated damping waves have an amplitude jaζ and wly is half the breadth of the
cross section at the waterline. The potential coefficients ijA and ijB and the phase lags jxΦε ,
expressed in terms of conformal mapping coefficients, are given in a foregoing chapter.
These loads 'hiX can be expressed in terms of in-phase and out-phase components with the
harmonic oscillations:
( ) ( ) ( ) ( ) tQBPAtPBQA
g
x
aX
jijjijjijjij
ja
ja
ijhi
⋅⋅⋅−⋅+⋅⋅⋅+⋅
⋅
⋅⋅
⋅⋅
=
ωω
πζ
ωρ
sincos 0000
2
2'
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
230
j
j
xwlja
jaj
xwlja
jaj
wl
wl
yg
xQ
yg
xP
a
ya
a
ya
a
Φ
Φ
⋅⋅⋅⋅+=
⋅⋅⋅⋅−=
=⋅=
=
==
εωπζ
εωπζ
cos
sin
2
4
2
4
2
2
0
2
0
44
42
33
24
22
Equation 6.3–3
The phase lag jxΦε between he velocity potentials and the forced motion is incorporated in the
coefficients jP0 and jQ0 and can be obtained by using:
+−
=Φj
jx Q
Pj
0
0arctanε
Equation 6.3–4
This equation will be used further on for obtaining wave load phases.Generally, these hydrodynamic loads are expressed in terms of potential mass and dampingcoefficients:
( ) ( )txNtxM
xNxMX
jaijjaij
jijjijhi
⋅⋅⋅⋅+⋅⋅⋅⋅=
⋅−⋅−=
ωωωω sincos2
' &&&
with:
444
342
233
324
222
20
20
20
20
2
2
2
2
wl
wl
wl
wl
wl
jj
ojijojijijij
jj
ojijojijijij
yb
yb
yb
yb
yb
QP
PBQAbN
QP
PBQAbM
⋅=
⋅=
⋅=
=
⋅=
⋅+
⋅−⋅⋅⋅=
+
⋅+⋅⋅⋅=
ωρ
ρ
Equation 6.3–5
Note that the phase lag information jxΦε is vanished here.
Tasai [1965] has used the following potential damping coupling coefficients in his formulationof the hydrodynamic loads for roll:
'
'44'
42wl
NN = and ''
22'
24 wlNN ⋅=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
231
Equation 6.3–6
in which 'wl is the lever of the rolling moment.
Because '24
'42 NN = , one may write for the roll damping coefficient:
( ) ( )'
22
2'42
'22
2'24'
44 NN
NN
N ==
Equation 6.3–7
This relation - which has been confirmed by numerical calculations with SEAWAY - will beused further on for obtaining the wave loads for roll from those for sway.
6.3.2 Energy Considerations
The wave velocity, wavec , and the group velocity, groupc , of regular waves are defined by:
kcwave
ω= and [ ]hk
hkcc wave
group ⋅⋅⋅⋅
==2sinh
22
Equation 6.3–8
Consider a cross section which is harmonic oscillating with a frequency Tπω ⋅= 2 and an
amplitude jax in the direction j in previously still water by an oscillatory force 'hjX in the
same direction j :
( )( )
( ) ( )tXtX
tXX
txx
hjhjahjhja
hjhjahj
jaj
⋅⋅⋅−⋅⋅⋅=
+⋅⋅=
⋅⋅=
ωεωε
εω
ω
sinsincoscos
cos
cos
''''
''' for: 4,3,2=j
Equation 6.3–9
The energy required for this oscillation should be equal to the energy radiated by the dampingwaves:
groupa
T
jjjj
T
jhj
cg
dtxxNT
dtxXT
⋅⋅⋅⋅⋅=
⋅⋅⋅⋅=⋅⋅⋅ ∫∫2
0
'
0
'
21
2
11
ζρ
&&&
or:
groupa
jajjhjjahja
cg
xNxX
⋅⋅⋅=
⋅⋅⋅=⋅⋅⋅⋅
2
22'''
21
sin21
ζρ
ωεω
Equation 6.3–10
From the first part of Equation 6.3–10 follows:
a
jajj
a
hjhja xN
X
ζω
ζε
⋅⋅=⋅ '
'' sin
Equation 6.3–11
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
232
From the second part of Equation 6.3–10 follows the amplitude ratio of the oscillatorymotions and the radiated waves:
'
21
jj
group
a
ja
N
cgx ⋅⋅⋅⋅=
ρωζ
Equation 6.3–12
Combining these last two equations provides for the out-phase part - so the damping part - ofthe oscillatory force:
'''
2sin
jjgroupa
hjhja NcgX
⋅⋅⋅⋅=⋅
ρζ
ε for: 4,3,2=j
In here, '' sin hjhjaX ε⋅ is the in-phase with the velocity part of the exciting force or moment.
6.3.3 Wave Loads
Consider now the opposite case: the cross section is restrained and is subject to regularincoming beam waves with amplitude aζ . Let wjx represent the equivalent (or average)oscillation of the water particles with respect to the restrained cross section. The resultingwave force, '
wjX , is caused by these motions, which will be in phase with its velocity(damping waves). Then the energy consumed by this oscillation is equal to the energysupplied by the incoming waves.
( )( )';
'
cos
cos
wjwjawj
wjwjawj
tXX
txx
εω
εω
+⋅⋅=
+⋅⋅= for: 4,3,2=j
Equation 6.3–13
in which 'wjε is the phase lag with respect to the wave surface elevation at the center of the
cross section.This leads for the amplitude of the exciting wave force to:
''
2 jjgroupa
wja NcgX
⋅⋅⋅⋅= ρζ
for: 4,3,2=j
Equation 6.3–14
which is in principle the same equation as the previous one for the out-phase part of theoscillatory force in still water.However, for the phase lag of the wave force, '
wjε , an approximation has to be found.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
233
Figure 6.3–1 Vector diagrams of wave components for sway and heave
6.3.3.1 Heave Mode
The vertical wave force on a restrained cross section in waves is:( )
( ) ( )tXtX
tXX
wawwaw
waww
⋅⋅⋅−⋅⋅⋅=
+⋅⋅=
ωεωε
εω
sinsincoscos
cos'
3'
3'
3'
3
'3
'3
'3
Equation 6.3–15
of which the amplitude is equal to:'
33'
3 2 NcgX groupaaw ⋅⋅⋅⋅⋅= ρζ
Equation 6.3–16
For the phase lag of this wave force, '3wε , an approximation has to be found.
The phase lag of a radiated wave, '3wRε , at the intersection of the ship's hull with the waterline,
wlb yy = , is wlwR yk ⋅='3ε . The phase lag of the wave force, '
3wε , has been approximated bythis phase:
wlwRw yk ⋅== '3
'3 εε
Equation 6.3–17
Then, the in-phase and out-phase parts of the wave loads are:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
234
'3
'33
'3
'3
'32
'3
'33
'3
'3
'31
'3
sin2
sin
cos2
cos
wgroupa
waww
wgroupa
wawwFK
Ncg
XX
Ncg
XXX
ερζ
ε
ερζ
ε
⋅
⋅⋅⋅⋅⋅−=
⋅−=
⋅
⋅⋅⋅⋅⋅+=
⋅+=+
Equation 6.3–18
from which the diffraction terms, '31wX and '
32wX follow.These diffraction terms can also be written as:
'3
'33
'32
'3
'33
'31
vNX
aMX
w
w
⋅=
⋅=
Equation 6.3–19
in which '3a and '
3v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Herewith, the equivalent acceleration and velocity amplitudes of the water particles are:
'33
'32'
3
'33
'31'
3
NX
v
MX
a
w
w
=
=
Equation 6.3–20
6.3.3.2 Sway Mode
The horizontal wave force on a restrained cross section in beam waves is:( )
( ) ( )tXtX
tXX
wawwaw
waww
⋅⋅⋅−⋅⋅⋅=
+⋅⋅=
ωεωε
εω
sinsincoscos
cos'
2'
2'
2'
2
'2
'2
'2
Equation 6.3–21
of which the amplitude is equal to:'
22'
2 2 NcgX groupaaw ⋅⋅⋅⋅⋅= ρζ
Equation 6.3–22
For the phase lag of this wave force, '2wε , an approximation has to be found.
The phase lag of an incoming undisturbed wave, '2wIε , at the intersection of the ship's hull
with the waterline, wlb yy = , is:
πεεµ
µε
+=<
⋅⋅−='
2'
2
'2
: then0sin if
sin
wIwI
wlwI yk
Equation 6.3–23
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
235
In very short waves - so at high wave frequencies ∞→ω - the ship's hull behaves like a
vertical wall and all waves will be diffracted. Then, the phase lag of the wave force, '2wε , is
equal to:( ) '
2'
2 wIw εωε −=∞→
Equation 6.3–24
The acceleration and velocity amplitudes of the water particles in the undisturbed surface ofthe incoming waves are:
( )( )
ωµ
ω
µ
sin
sin'
2surface water still
'2
surface water still'
2
⋅⋅=
−=
⋅⋅−=
gkav
gka
Equation 6.3–25
In very long waves - so at low wave frequencies 0→ω - the wave force is dominated by theFroude-Krylov force and the amplitudes of the water particle motions do not change verymuch over the draft of the section. Apparently, the phase lag of the wave force, '
2wε , can beapproximated by:
( ) ( )
⋅⋅
⋅
⋅⋅−⋅+−=→
ωµ
µωεsin
sinarctan0
'22
'22
'2'
2 gkN
gkMXFKw
Equation 6.3–26
When plotted against ω , the two curves ( )0'2 →ωεw and ( )∞→ωε '
2w will intersect each
other. The phase lag of the wave force, '2wε , can now be approximated by the lowest of these
two values:( )
( ) ( ) ( )0 : then0 if '2
'2
'2
'2
'2
'2
→=∞→>→
∞→=
ωεεωεωε
ωεε
wwww
ww
Equation 6.3–27
Because ( )∞→ωε '2w goes to zero in the low frequency region and ( )0'
2 →ωεw can havevalues between 0 and π⋅2 , one simple precaution has to be taken:
( ) ( ) ( ) πωεωεπωε ⋅−→=∞→>→ 20 : then20 if '2
'2
'2 www
Equation 6.3–28
Now the in-phase and out-phase terms of the wave force in beam waves are:
'2
'22
'2
'2
'22
'2
'22
'2
'2
'21
'2
cos2
sin
sin2
sin
wgroupa
waww
wgroupa
wawwFK
Ncg
XX
Ncg
XXX
ερζ
ε
ερζ
ε
⋅
⋅⋅⋅⋅⋅+=
⋅+=
⋅
⋅⋅⋅⋅⋅−=
⋅−=+
Equation 6.3–29
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
236
from which the diffraction terms, '21wX and '
22wX follow.These terms can also be written as:
'2
'22
'22
'2
'22
'21
vNX
aMX
w
w
⋅=
⋅=
Equation 6.3–30
in which '2a and '
2v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Then - when using an approximation for the influence of the wave direction - the equivalentacceleration and velocity amplitudes of the water particles are:
µ
µ
sin
sin
'22
'22'
2
'22
'21'
2
⋅=
⋅=
NX
v
MX
a
w
w
Equation 6.3–31
6.3.3.3 Roll Mode
The fluid is free of rotation; so the wave moment for roll consists of sway contributions only.However, the equivalent amplitudes of the acceleration and the velocity of the water particleswill differ from those of sway.From a study on potential coefficients, the following relation between sway and roll dampingcoefficients has been found:
( ) ( )'
22
2'42
'22
2'24'
44 NN
NN
N ==
The horizontal wave moment on a restrained cross section in beam waves is:( )'4
'4
'4 cos waww tXX εω +⋅⋅=
Equation 6.3–32
of which the amplitude is equal to:
( )
'22
'24'
2
'22
'24'
22
'22
2'24
'44
'4
2
2
2
N
NX
N
NNcg
NN
cg
NcgX
aw
groupa
groupa
groupaaw
⋅=
⋅⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅=
ρζ
ρζ
ρζ
Equation 6.3–33
The in-phase and out-phase parts of the wave moment in beam waves are:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
237
( )
'22
'24'
22'
42
'22
'24'
21'
2'
41'
4
N
NXX
N
NXXXX
ww
wFKwFK
⋅=
⋅+=+
Equation 6.3–34
from which the diffraction terms, '41wX and '
42wX follow.These terms can also be written as:
'24
'24
'42
'24
'24
'41
vNX
aMX
w
w
⋅=
⋅=
Equation 6.3–35
in which '24a and '
24v are the equivalent amplitudes of the acceleration and the velocity of thewater particles around the cross section.Then - when using an approximation for the influence of the wave direction - the equivalentacceleration and velocity amplitudes of the water particles are:
µ
µ
sin
sin
'24
'42'
24
'24
'41'
24
⋅=
⋅=
NX
v
MX
a
w
w
Equation 6.3–36
6.3.3.4 Surge Mode
The equivalent acceleration and velocity amplitudes of the water particles around the crosssection for surge have been found from:
ω
µ'
1'1
'2'
1tan
av
aa
−=
=
Equation 6.3–37
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
238
6.4 Numerical Comparisons
Figure 6.4–1 and Figure 6.4–2 give a comparison between these sway, heave and roll waveloads on a crude oil carrier in oblique waves - obtained by the classic approach and the simplediffraction approach, respectively - with the 3-D zero speed ship motions program DELFRACof Pinkster; see Dimitrieva [1017].
Figure 6.4–1: Comparison of classic wave loads with DELFRAC data
Figure 6.4–2: Comparison of simple diffraction wave loads with DELFRAC data
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
239
7 Transfer Functions of Motions
After dividing the left and right hand terms by the wave amplitude aζ , two sets of six coupledequations of motion are available.
The 6 variables in the coupled equations for the vertical plane motions are:
θζθζ
ζζ
ζζ
εζθε
ζθ
εζ
εζ
εζ
εζ
sinandcos:Pitch
sinz
andcosz
:Heave
sinx
andcosx
:Surge
a
a
a
a
a
a
a
a
a
a
a
a
⋅⋅
⋅⋅
⋅⋅
zz
xx
The 6 variables in the coupled equations for the horizontal plane motions are:
ψζψζ
φζφζ
ζζ
εζψε
ζψ
εζφε
ζφ
εζ
εζ
sinandcos:Yaw
sinandcos:Roll
siny
andcosy
:Sway
a
a
a
a
a
a
a
a
a
a
a
a
⋅⋅
⋅⋅
⋅⋅ yy
These sets of motions have to be solved by a numerical method. A method that providescontinuous good results, given by de Zwaan [1977], has been used in the strip theory programSEAWAY.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
240
7.1 Centre of Gravity Motions
From the solutions of these in and out of phase terms follow the transfer functions of themotions (or Reponse Amplitude Operators, RAO’s), which is the motion amplitude to waveamplitude ratio:
a
axζ a
ayζ a
azζ a
a
ζφ
a
a
ζθ
a
a
ζψ
The associated phase shifts of these motions relative to the wave elevation are:
ζεx ζεy ζεz φζε θζε ψζε
The transfer functions of the translations are non-dimensional. The transfer functions of therotations can be made non-dimensional by dividing the amplitude of the rotations by theamplitude of the wave slope ak ζ⋅ in lieu of the wave amplitude aζ :
a
axζ a
ayζ a
azζ a
a
k ζφ⋅ a
a
k ζθ⋅ a
a
k ζψ⋅
Some examples of calculated transfer functions of a crude oil carrier and a containership aregiven in Figure 7.1–1, Figure 7.1–2 and Figure 7.1–3.
-720
-630
-540
-450
-360
-270
-180
-90
0
0 0.25 0.50 0.75 1.00
µ = 900
µ = 1800
Wave Frequency (rad/s )
Pha
se ε
z ζ (d
eg)
0
0 .5
1 .0
1 .5
0 0 .25 0 .5 0 0 .7 5 1 .0 0
Cru de O il Carr ierV = 0 k nHe a ve
µ = 900
µ = 1 8 00
W ave Fre qu e ncy (ra d/ s)
RA
O H
eav
e (
-)
0
0 .5
1 .0
1 .5
0 0 .2 5 0. 50 0 .7 5 1. 00
Cru de O il Ca rr ie r
V = 0 knP it ch
µ = 900
µ = 18 00
W a ve Freq u en cy (rad /s )
RA
O P
itch
(-)
-720
-630
-540
-450
-360
-270
-180
-90
0
0 0.25 0.50 0.75 1.00
µ = 900
µ = 1800
Wave F requency (rad/s )
Phas
e ε θ
ζ
Figure 7.1–1: Heave and Pitch Motions of a Crude Oil Carrier, V = 0 kn
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
241
0
0 .5
1 .0
1 .5
0 0 .25 0 .5 0 0 .7 5 1 .0 0
Cru de O il Carr ierV = 1 6 knHe a ve
µ = 900
µ = 18 00
W ave Fre qu e ncy (ra d/ s)
RA
O H
eav
e (
-)
-720
-630
-540
-450
-360
-270
-180
-90
0
0 0.25 0.50 0.75 1.00
µ = 900
µ = 1800
Wave Frequency (rad/s )
Pha
se ε
z ζ (d
eg)
0
0 .5
1 .0
1 .5
0 0 .2 5 0. 50 0 .7 5 1. 00
Cru de O il Ca rr ie r
V = 1 6 knP it ch
µ = 900
µ = 1 800
W a ve Freq u en cy (rad /s )
RA
O P
itch
(-)
-720
-630
-540
-450
-360
-270
-180
-90
0
0 0.25 0.50 0.75 1.00
µ = 900
µ = 1800
Wave F requency (rad/s )
Phas
e ε θ
ζ
Figure 7.1–2: Heave and Pitch Motions of a Crude Oil Carrier, V = 16 kn
0
0.25
0.50
0.75
1.00
1.25
1.50
0 0.2 0.4 0.6 0.8 1.0
RAO of pitch Head waves
ContainershipL
pp = 175 metre
V = 0 knots
V = 10 knots
V = 20 knots
wave frequency (rad/s)
Non
-dim
. RA
O o
f pitc
h (-
)
0
5
10
15
0 0.2 0.4 0.6 0.8 1.0
RAO of roll Beam waves
ContainershipL
pp = 175 metre
V = 20 knots
V = 10 knots
V = 0 knots
wave frequency (rad/s)
Non
-dim
. RAO
of r
oll (
-)
Figure 7.1–3: Roll and Pitch Motions of a Containership
Notice the different speed effects on the motions in these figures.
For motions with a spring term in the equation of motion, three frequency regions can bedistinguished:• the low frequency region ( ( )amc +<<2ω ), with motions dominated by the restoring
spring term,
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
242
• the natural frequency region ( ( )amc +≈2ω ), with motions dominated by the dampingterm and
• the high frequency region ( ac>>2ω ), with motions dominated by the mass term.
An example for heave motions is given in Figure 7.1–4.
Figure 7.1–4: Frequency Regions and Motional Behaviour
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
243
7.2 Local Absolute Displacements
Consider a point ( )bbb zyxP ,, on the ship in the ( )bbb zyxG ,, ship-bound axes system. The
harmonic displacements in the ship-bound bx , by and bz directions - or in the earth bound x ,
y and z directions - in any point ( )bbb zyxP ,, on the ship can be obtained from the six centreof gravity motions as presented below.
The harmonic longitudinal displacement is given by:
( )ζεωθψ
PxePa
bbP
tx
zyxx
+⋅⋅=
⋅+⋅−=
cos
The harmonic lateral displacement is given by:
( )ζεωφψ
PyePa
bbP
ty
zxyy
+⋅⋅=
⋅−⋅+=
cos
The harmonic vertical displacement is given by:
( )ζεωφθ
PzePa
bbP
tz
yxzz
+⋅⋅=
⋅−⋅−=
cos
With the six motions of the centre of gravity, the harmonic motions of any point ( )bbb zyxP ,,
on the ship in the ship-bound bx , by and bz directions - or in the earth bound system in x , yand z directions - can be calculated by using the previous equations.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
244
7.3 Local Absolute Velocities
The harmonic velocities in the ship-bound bx , by and bz directions - or in the earth bound x ,
y and z directions - in any point ( )bbb zyxP ,, on the ship can be obtained by taking thederivative of the three harmonic displacements.
The harmonic longitudinal velocity is given by:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωωθψ
P
P
P
xePa
xePae
xePae
bbP
tx
tx
tx
zyxx
&&
&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅+⋅−=
cos
2cos
sin
The harmonic lateral velocity is given by:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωωφψ
P
P
P
yePa
yePae
yePae
bbP
ty
ty
ty
zxyy
&&
&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅−⋅+=
cos
2cos
sin
The harmonic vertical velocity is given by:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωωφθ
P
P
P
zePa
zePae
zePae
bbP
tz
tz
tz
yxzz
&&
&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅−⋅−=
cos
2cos
sin
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
245
7.4 Local Absolute Accelerations
In the earth-bound axes system, the harmonic accelerations on the ship are obtained by takingthe second derivative of the displacements. In the ship-bound axes system, a component of theacceleration of gravity has to be added to the accelerations in the horizontal plane direction.
7.4.1 Accelerations in the Earth-Bound Axes System
In the earth-bound axes system, ( )zyxO ,, , the harmonic accelerations in a point ( )bbb zyxP ,,on the ship in the x , y and z directions can be obtained by taking the second derivative ofthe three harmonic displacements.
Thus:
• Longitudinal acceleration:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωω
θψ
P
P
P
xePa
xePae
xePae
bbP
tx
tx
tx
zyxx
&&&&
&&&&&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅+⋅−=
cos
cos
cos2
2
• Lateral acceleration:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωω
φψ
P
P
P
yePa
yePae
yePae
bbP
ty
ty
ty
zxyy
&&&&
&&&&&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅−⋅+=
cos
cos
cos2
• Vertical acceleration:
( )( )
( )ζ
ζ
ζ
εω
πεωω
εωω
φθ
P
P
P
zePa
zePae
zePae
bbP
tz
tz
tz
yxzz
&&&&
&&&&&&&&
+⋅⋅=
−+⋅⋅⋅=
+⋅⋅⋅−=
⋅−⋅−=
cos
cos
cos2
2
7.4.2 Accelerations in the Ship-Bound Axes System
In the ship-bound axes system, ( )bbb zyxG ,, , a component of the acceleration of gravity ghas to be added to the accelerations in the longitudinal and lateral direction in the earth-boundaxes system. The vertical acceleration does not change.These accelerations are the accelerations that will be ''felt'' by for instance the cargo or sea-fastenings on the ship.
Thus:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
246
• Longitudinal acceleration:
( ) ( )( )ζ
θζζ
εω
εωθεωω
θθψ
P
P
xePa
eaxePae
bbP
tx
tgtx
gzyxx
&&&&
&&&&&&&&
+⋅⋅=
+⋅⋅⋅−+⋅⋅⋅−=
⋅−⋅+⋅−=
cos
coscos2
• Lateral acceleration:
( ) ( )( )ζ
φζζ
εω
εωφεωω
φφψ
P
P
yePa
eayePae
bbP
ty
tgty
gzxyy
&&&&
&&&&&&&&
+⋅⋅=
+⋅⋅⋅++⋅⋅⋅−=
⋅+⋅−⋅+=
cos
coscos2
• Vertical acceleration:
( )( )ζ
ζ
εω
εωω
φθ
P
P
zePa
zePae
bbP
tz
tz
yxzz
&&&&
&&&&&&&&
+⋅⋅=
+⋅⋅⋅−=
⋅−⋅−=
cos
cos2
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
247
7.5 Local Vertical Relative Displacements
The harmonic vertical relative displacement with respect to the wave surface of a point( )bbb zyxP ,, connected to the ship can be obtained too:
( )ζεωφθζ
PsePa
bbPP
ts
yxzs
+⋅⋅=
⋅+⋅+−=
cos
with:( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbeaP ykxkt
It may be noted that the sign of the relative motion is chosen here in such a way that a positiverelative displacement implies a decrease of the freeboard.An oscillating ship will produce waves and these phenomena will change the relative motion.A dynamical swell up should be taken into account, which is not included in the previousformulation.Notice the different behaviours of the absolute and relative vertical motions, as given inFigure 7.5–1.
0
1
2
3
4
5
0 0.5 1 .0 1 .5
R A O te nd sto 0 .0
R A O te nd sto 1 .0
C o nta ine rsh ipH e ad w ave s
V = 20 kno ts
V = 10 kno ts
V = 0 kn ots
w a ve fre qu en cy (ra d/s )
RA
O o
f ve
rtic
al a
bso
lute
bo
w m
otio
ns
(m/m
)
0
1
2
3
4
5
0 0 .5 1.0 1 .5
R A O te nd sto 1 .0
R AO ten d sto 0.0
C on tai ne rsh ip
H ea d w ave s
V = 0 kn ots
V = 1 0 kn ots
V = 2 0 kn o ts
w a ve fre q ue ncy (r ad /s )
RA
O o
f ve
rtic
al r
ela
tive
bo
w m
otio
ns
(m/m
)
Figure 7.5–1: Absolute and Relative Vertical Motions at the Bow
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
248
7.6 Local Vertical Relative Velocities
The harmonic vertical relative velocity with respect to the wave surface of a certain point( )bbb zyxP ,, , connected to the ship, can be obtained by:
( )ζεωφθθζ
φθζ
PsePa
bbP
bbPP
ts
yVxz
yxzDtD
s
&&
&&&&
&
+⋅⋅=
⋅+⋅−⋅+−=
⋅+⋅+−=
cos
in which for the vertical velocity of the water surface itself:( )µµωζωζ sincossin ⋅⋅−⋅⋅−⋅⋅⋅−= bbeaP ykxkt&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
249
8 Anti-Rolling Devices
Since the disappearance of sails on ocean-going ships, with their stabilising wind effect on therolling motions, naval architects have been concerned in reducing the rolling of ships amongwaves. With bilge keels they performed a first successful attack on the problem of rolling, butin several cases these bilge keels did not prove to be sufficient. Since 1880, numerous othermore or less successful ideas have been tested and used.
Four types of anti-rolling devices and its contribution to the equations of motion are describedhere:• bilge keels• passive free-surface tanks• active fin stabilisers• active rudder stabilisers.
The active fin and rudder stabilisers are not built into the program SEAWAY yet.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
250
8.1 Bilge Keels
Bilge keels can deliver an important contribution to an increase the damping of the rollingmotions of ships. A reliable method to determine this contribution is given by Ikeda, Himenoand Tanaka [1978], as described before.Ikeda divides the two-dimensional quadratic bilge keel roll damping into a component due tothe normal force on the bilge keels and a component due to the pressure on the hull surface,created by the bilge keels.The normal force component of the bilge keel damping has been deduced from experimentalresults of oscillating flat plates. The drag coefficient DC depends on the period parameter orthe Keulegan-Carpenter number. Ikeda measured the quadratic two-dimensional drag bycarrying out free rolling experiments with an ellipsoid with and without bilge keels.Assuming a pressure distribution on the hull caused by the bilge keels, a quadratic two-dimensional roll damping can be defined. Ikeda carried out experiments to measure thepressure on the hull surface created by bilge keels. He found that the coefficient +
pC of the
pressure on the front face of the bilge keel does not depend on the period parameter, while thecoefficient −
pC of the pressure on the back face of the bilge keel and the length of thenegative pressure region depend on the period parameter. Ikeda defines an equivalent lengthof a constant negative pressure region 0S over the height of the bilge keels and a two-dimensional roll-damping component can be found.The total bilge keel damping has been obtained by integrating these two two-dimensional roll-damping components over the length of the bilge keels.Experiments of Ikeda showed that the effect of forward speed on the roll damping due to thebilge keels could be ignored.The equivalent linear total bilge keel damping has been obtained by linearising the result, ashas been shown in a separate Chapter.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
251
8.2 Passive Free-Surface Tanks
The roll damping, caused by a passive free-surface tank, is essentially based on the existenceof a hydraulic jump or bore in the tank. Verhagen and van Wijngaarden [1965] give atheoretical approach to determine the counteracting moments by free-surface anti-rollingtanks. Van den Bosch and Vughts [1966] give extended quantitative information on thesemoments.
8.2.1 Theoretical Approach
When a tank that contains a fluid with a free surface is forced to carry out roll oscillations,resonance frequencies can be obtained with high wave amplitudes at lower water depths.Under these circumstances a hydraulic jump or bore is formed, which travels periodicallyback and forth between the walls of the tank. This hydraulic jump can be a strongly non-linearphenomenon. A theory, based on gas-dynamics for the shock wave in a gas flow under similarresonance circumstances, as given by Verhagen and van Wijngaarden [1965], has beenadapted and used to describe the motions of the fluid. For low and high frequencies and thefrequencies near to the natural frequency, different approaches have been used.Observe a rectangular tank with a length l and a breadth b , which has been filled until awater level h with a fluid with a mass density ρ . The distance of the tank bottom above thecentre of gravity of the vessel is s . Figure 8.2–1 shows a 2-D sketch of this tank with the axissystem and notations.
Figure 8.2–1: Axes System and Notations of an Oscillating Tank
The natural frequency of the surface waves in a harmonic rolling tank appears as the wavelength λ in the tank equals twice the breadth b , so: b⋅= 20λ .With the wave number and the dispersion relation:
λπ⋅
=2
k and [ ]hkgk ⋅⋅⋅= tanhω
it follows for the natural frequency of surface waves in the tank:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
252
⋅
⋅⋅
=b
hb
g ππω tanh0
Verhagen and van Wijngaarden [1965] have investigated the shallow water wave loads in arolling rectangular container, with the centre of rotation at the bottom of the container. Theirexpressions for the internal wave loads are rewritten and modified to be useful for anyarbitrary vertical position of the centre of rotation by Journée [1997]. For low and highfrequencies and the frequencies close to the natural frequency, different approaches have beenused. A calculation routine has been made to connect these regions.
8.2.1.1 Low and High Frequencies
The harmonic roll motion of the tank is defined by:( )ta ⋅⋅= ωφφ sin
In the axis-system of Figure 8.2–1 and after linearisation, the vertical displacement of the tankbottom is described by:
φ⋅+= ysz
and the surface elevation of the fluid is described by:ζ++= hsz
Relative to the bottom of the tank, the linearised surface elevation of the fluid is described by:φζξ ⋅−+= yh
Using the shallow water theory, the continuity and momentum equations are:
0
0
=⋅+∂∂
⋅+∂∂
⋅+∂∂
=∂∂
⋅+∂∂
⋅+∂∂
φξ
ξξξ
gy
gyv
vtv
yv
yv
t
In these formulations, $v$ denotes the velocity of the fluid in the y -direction and the verticalpressure distribution is assumed to be hydrostatic. Therefore, the acceleration in the z -direction, introduced by the excitation, must be small with respect to the acceleration ofgravity g , so:
gba <<⋅⋅ 2ωφ
The boundary conditions for v have been determined by the velocity produced in thehorizontal direction by the excitation. Between the surface of the fluid and the bottom of thetank, the velocity of the fluid v varies between sv and [ ]hkvs ⋅cosh with a mean velocity:
( )hkvs ⋅ . However, in very shallow water v does not vary between the bottom and thesurface. When taking the value at the surface, it is required that:
( ) φ&⋅+−= hsv at: 2b
y ±=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
253
For small values of aφ , the continuity equation and the momentum equation can be given in alinearised form:
0
0
=⋅+∂∂
⋅+∂∂
=∂∂
⋅+∂∂
φξ
ξξ
gy
gtv
yh
t
The solution of the surface elevation ξ in these equations, satisfying the boundary values forv , is:
( )
⋅
⋅⋅⋅
⋅
⋅⋅⋅⋅
⋅++⋅⋅
−= φωωπ
ωωπωπ
ωωξ
0
0
2
0
sin
2cos
1
byg
hsb
h
Now, the roll moment follows from the quasi-static moment of the mass of the frozen liquidhbl ⋅⋅⋅ρ and an integration of ξ over the breadth of the tank:
∫+
−
⋅⋅⋅⋅⋅+⋅
+⋅⋅⋅⋅⋅=
2
22
b
b
dyylgh
shblgM ξρφρφ
This delivers the roll moment amplitude for low and high frequencies at small water depths:
( )a
aa
ghs
blg
hshblgM
φωπ
ωωωπ
ωπωωρ
φρφ
⋅
⋅−
⋅⋅
⋅
⋅⋅⋅
⋅+
+⋅⋅⋅⋅
+⋅
+⋅⋅⋅⋅⋅=
2
0
0
3
02
3
2tanh21
2
For very low frequencies, so for the limit value 0→ω , this will result into the static moment:
a
bhshblgM φρφ ⋅
+
+⋅⋅⋅⋅⋅=
122
3
The phase lags between the roll moments and the roll motions have not been obtained here.However, they can be set to zero for low frequencies and to π− for high frequencies:
πε
ε
φ
φ
φ
φ
−=
=
M
M 0 for:
0
0
ωωωω
>><<
8.2.1.2 Natural Frequency Region
For frequencies near to the natural frequency 0ω , the expression for the surface elevation ofthe fluid ξ goes to infinity. Experiments showed the appearance of a hydraulic jump or a boreat these frequencies. Obviously, then the linearised equations are not valid anymore.Verhagen and van Wijngaarden [1965] solved the problem by using the approach in gasdynamics, when a column of gas has been oscillated at small amplitude, e.g. by a piston. At
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
254
frequencies near to the natural frequency at small water depths, they found a roll momentamplitude, defined by:
( )
⋅⋅−⋅⋅−⋅
⋅⋅⋅⋅
⋅⋅⋅⋅=
a
aa g
bb
hblgMφωωπφ
πρφ 32
13
2412
20
243
The phase shifs between the roll moment and the roll motion at small water depths are givenby:
απε
απε
φ
φ
φ
φ
−−=
+−=
2
2
M
M for:
0
0
ωωωω
>><<
with:
( )
( )( )
−⋅⋅⋅−⋅⋅−⋅⋅
−
⋅⋅−⋅⋅
⋅=
20
2
20
2
20
2
396arcsin
24arcsin2
ωωπφωωπ
φωωπα
bgb
gb
a
a
Because that the arguments of the square roots in the expression for φφεM have to be positive,
the limits for the frequency ω are at least:
2020
2424π
φωωπ
φω⋅
⋅⋅+<<
⋅⋅⋅
−b
gb
g aa
8.2.1.3 Comparison with Experimental Data
An example of the results of this theory with experimental data of an oscillating free-surfacetank by Verhagen and van Wijngaarden [1965] is given in Figure 8.2–2.
Figure 8.2–2: Comparison between Theoretical and Experimental Data
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
255
The roll moments have been calculated here for low and high frequencies and for frequenciesnear to the natural frequency of the tank. A calculation routine connects these three regions.
8.2.2 Experimental Approach
Van den Bosch and Vugts [1966] have described the physical behaviour of passive free-surface tanks, used as an anti-rolling device. Extended quantitative information on thecounteracting moments, caused by the water transfer in the tank, has been provided.With their symbols, the roll motions and the exciting moments of an oscillating rectangularfree-surface tank, are defined by:
( )( )ϕεω
ωϕϕ
ttat
a
tKK
t
+⋅⋅=⋅⋅=
cos
cos
and the dimensions of the rectangular free-surface tank are given by:l length of the tankb breadth of the tanks distance of tank bottom above rotation pointh water depth in the tank at rest
*ρ mass density of the fluid in the tank\endtabular
A non-dimensional frequency range is defined by:
60.100.0 <⋅<gbω
In this frequency range, van den Bosch and Vugts have presented extended experimental dataof:
3* blgKta
a ⋅⋅⋅=
ρµ and ϕεt
for:=aϕ 0.0333, 0.0667 and 0.1000 radians=bs -0.40, -0.20, 0.00 and +0.20=bh 0.02, 0.04, 0.06, 0.08 and 0.10
An example of a part of these experimental data has been shown for 40.0−=bs and1000.0=aϕ radians in Figure 8.2–3, taken from the report of van den Bosch and Vugts
[1966].
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
256
Figure 8.2–3: Experimental Data on Anti-Rolling Free-Surface Tanks
When using these experimental data, the external roll moment due to an, with a frequency ω ,oscillating free surface tank can be written as:
ϕϕϕ ϕϕϕ ⋅+⋅+⋅= 444 cbaKt &&&
with:
ϕϕ
ϕ
ϕ
ϕ
εϕ
ω
εϕ
ta
ta
ta
ta
Kc
K
b
a
cos
sin
0
4
4
4
⋅=
⋅=
=
It is obvious that for an anti-rolling free-surface tank, built into a ship, it holds:
aa ϕφ = and ωω =e
So it can be written:( )
( )ϕφξ
φζ
εεω
εωφφ
ttat
a
tKK
t
++⋅⋅=
+⋅⋅=
cos
cos
Then, an additional moment has to be added to the right-hand side of the equations of motionfor roll:
φφφ ⋅+⋅+⋅= tank44tank44tank44tank4 cbaX &&&
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
257
ϕ
ϕ
εφ
ω
εφ
ta
ta
ta
ta
Kc
K
b
a
cos
sin
0
tank44
tank44
tank44
⋅=
⋅=
=
This holds that the anti-rolling coefficients tank44a , tank44b and tank44c have to be subtracted
from the coefficients 44a , 44b and 44c in the left-hand side of the equations of motion for roll.
8.2.3 Effect of Free-Surface Tanks
Figure 8.2–4 shows the significant reduction of the roll transfer functions and the significantroll amplitude of a trawler, being obtained by a free-surface tank.
0
1 0
2 0
3 0
4 0
0 0.5 1.0 1 .5 2 .0 2.5
T ra w ler L = 2 3.9 0 m e tre
W ith tan k
W itho ut ta nk
c ircu lar w ave fre qu en cy (1 /s)
Tra
nsfe
r fu
nctio
n ro
ll (d
eg/m
)
0
5
1 0
1 5
2 0
2 5
3 0
0 1 2 3 4 5 6
T r aw l er L = 2 3.9 0 m e tre
W ith tan k
W i th ou t ta nk
S ig nif ica nt w a ve he ig ht (m )
Sig
nifi
cant
ro
ll a
mp
litu
de (
deg
)
Figure 8.2–4: Effect of a Free-Surface Tank on Roll Motions in Beam Waves
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
258
8.3 Active Fin Stabilisers
To determine the effect of active fin stabilisers on ship motions, use has been made here ofreports published by Schmitke [1978] and Lloyd [1989].
The oscillatory angle of the portside fin is given by:( )βφεωββ +⋅⋅= ta cos
The exciting forces and moments, caused by an oscillating fin pair are given by:
βββ
βββ
βββ
βββ
βββ
βββ
⋅+⋅+⋅=
⋅+⋅+⋅=
⋅+⋅+⋅=
666fin6
444fin4
222fin2
cbaX
cbaX
cbaX
&&&
&&&
&&&
with:
( )( )( )
ββ
ββ
ββ
ββ
ββ
ββ
ββ
ββ
ββ
γ
γ
γ
γγ
γγ
γγ
γ
γ
γ
cxc
bxb
axa
czyb
bzyb
azya
cc
bb
aa
b
b
b
bb
bb
bb
⋅⋅⋅−=
⋅⋅⋅−=
⋅⋅⋅−=
⋅⋅+⋅⋅+=
⋅⋅+⋅⋅+=
⋅⋅+⋅⋅+=
⋅⋅−=
⋅⋅−=
⋅⋅−=
sin2
sin2
sin2
sincos2
sincos2
sincos2
sin2
sin2
sin2
fin6
fin6
fin6
finfin4
finfin4
finfin4
2
2
2
and:
( )
( )kCC
AVc
kCCc
AVb
csa
fin
Lfin
fin
Lfinfin
finfin
⋅
∂∂
⋅⋅⋅⋅=
⋅
∂∂
+⋅⋅⋅⋅⋅=
⋅
⋅⋅⋅=
αρ
απρ
πρ
β
β
β
2
3
21
221
221
In here:γ angle of port fin
fin
LC
∂∂
αlift curve slope of fin
( )kC circulation delay function
Vc
k re
⋅⋅
=2
ωreduced frequency
finA projected fin area
fins span of fin
finc mean chord of fin
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
259
finbx bx -co-ordinate of the centroid of fin forces
finby by -co-ordinate of the centroid of fin forces
finbz bz -co-ordinate of the centroid of fin forces
The nominal lift curve slope of a fin profile in a uniform flow is approximated by:( )( )
0.4cos
cos80.1
80.1
4
2
+Λ
⋅Λ+
⋅⋅=
∂∂
E
EL
AR
ARC πα
with:Λ sweep angle of fin profile( )EAR effective aspect ratio of fin profile
Of normal fins, the sweep angle of the fin profile is zero, so 0=Λ or 1cos =Λ .The fin acts in the boundary layer of the ship, which will reduce the lift. This effect istranslated into a reduced lift curve slope of the fin.
The velocity distribution in the hull boundary layer is estimated by the following twoequations:
( ) τδδ
δBL
VV ⋅= with: BLδδ <
2.0377.0 −⋅⋅= xfinBL Rxδ with: ν
xVRx
⋅=
in which:( )δV flow velocity inside boundary layer
V forward ship speedδ normal distance from hull
BLδ thickness of boundary layer
finx distance aft of forward perpendicular of fin
xR local Reynolds numberν kinematic density of fluid
The kinematic viscosity of seawater can be found from the water temperature T in degreescentigrade by:
26
000221.00336.00.178.1
10TT ⋅+⋅+
=⋅ν m2/s
It is assumed here that the total lift of the fin can be found from:
( ) ( ) finLfin
s
L AVCdcVCfin
⋅⋅⋅⋅=⋅⋅⋅⋅⋅ ∫ 2
0
2
21
21 ρδδδρ
where ( )δc is the chord at span-wise location δ .For rectangular fins, this is simply an assumption of a uniform loading.Because:
( ) ( )fin
tfinrfinrfin scccc δδ ⋅−−=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
260
in which:
rfinc root chord of fin
tfinc tip chord of fin
2tfinrfin
fin
ccc
+= mean chord of fin
the correction to the lift curve slope is:
⋅−⋅
−−
⋅⋅
−⋅=fin
BLtfinrfinBL
fin
rfinBL s
cc
sc
cE
fin8
129
21
2δδ
Then the corrected lift curve slope of the fin is:( )
( ) 0.480.1
80.12 ++
⋅⋅⋅=
∂∂
finE
finEBL
fin
L
AR
ARE
C πα
Generally a fin is mounted close to the hull, so the effective aspect ratio is about twice thegeometric aspect ratio:
( ) ( )fin
finfinfinE c
sARAR ⋅=⋅= 22
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
261
8.4 Active Rudder Stabilisers
To determine the effect of rudder stabilisers on ship motions, use has been made of reportspublished by Lloyd [1989] and Schmitke [1978].
The oscillatory rudder angle is given by:( )δφεωδδ +⋅⋅= tea cos
with δ is positive in a counter-clockwise rotation of the rudder.So, a positive δ results in a positive side force, a positive roll moment and a negative yawmoment.The exciting forces and moments, caused by this oscillating rudder are given by:
δδδ
δδδ
δδδ
δδδ
δδδ
δδδ
⋅+⋅+⋅=
⋅+⋅+⋅=
⋅+⋅+⋅=
666rud6
444rud4
222rud2
cbaX
cbaX
cbaX
&&&
&&&
&&&
with:
δδ
δδ
δδ
δδ
δδ
δδ
δδ
δδ
δδ
cxc
bxb
axa
czb
bzb
aza
cc
bb
aa
b
b
b
b
b
b
⋅+=⋅+=
⋅+=⋅−=⋅−=
⋅−=+=
+=+=
rud6
rud6
rud6
rud4
rud4
rud4
2
2
2
and:
( )
( )kCC
AVc
kCCc
AVb
csa
rud
Lrud
rud
Lrudrud
rudrud
⋅
∂∂
⋅⋅⋅⋅=
⋅
∂∂
+⋅⋅⋅⋅⋅=
⋅
⋅⋅⋅=
αρ
απρ
πρ
δ
δ
δ
2
3
21
221
221
In here:VVrud ⋅≈ 125.1 equivalent flow velocity at rudder
rud
LC
∂∂
αlift curve slope of rudder
Vc
k rude
⋅⋅
=2
ωcirculation delay function
rudA projected area of rudder
ruds span of rudder
rudc mean chord of rudder
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
262
rudbx bx -co-ordinate of centroid of rudder forces
rudbz bz -co-ordinate of centroid of rudder forces
The lift curve slope of the rudder is approximated by:( )
( ) 0.480.1
80.12 ++
⋅⋅=
∂∂
rudE
rudE
rud
L
AR
ARC πα
Generally a rudder is not mounted close to the hull, so the effective aspect ratio is equal to thegeometric aspect ratio:
( ) ( )rud
rudrudrudE c
sARAR ==
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
263
9 External Linear Springs
Suppose a linear spring connected to point P on the ship, see Figure 8.4–1.
Figure 8.4–1: Co-ordinate System of Springs
The harmonic longitudinal, lateral and vertical displacements of a certain point P on the shipare given by:
( )( )( ) φθ
φψ
θψ
⋅+⋅−=
⋅−⋅+=
⋅+⋅−=
pp
pp
pp
yxzPz
zxyPy
zyxPx
The linear spring coefficients in the three directions in a certain point P are defined by( )pzpypx CCC ,, . The units of these coefficients are N/m or kN/m.
9.1 External Loads
The external forces and moments, caused by these linear springs, acting on the ship are givenby:
( )( )( )
pspss
pspss
pspss
pppzs
pppys
pppxs
xXyXX
xXzXX
yXzXX
yxzCX
zxyCX
zyxCX
⋅+⋅−=
⋅−⋅+=
⋅+⋅−=
⋅+⋅−⋅−=
⋅−⋅+⋅−=
⋅+⋅−⋅−=
216
315
324
3
2
1
φθ
φψ
θψ
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
264
9.2 Additional Coefficients
After a change of sign, this results into the following coefficients ijc∆ , which have to be
added to the restoring spring coefficients ijc of the hydromechanical loads in the left-handside of the equations of motions:
• Surge:
ppx
ppx
px
yCc
zCc
c
c
c
Cc
⋅−=∆
⋅+=∆=∆
=∆=∆
+=∆
16
15
14
13
12
11
0
0
0
• Sway:
ppy
ppy
py
xCc
c
zCc
c
Cc
c
⋅+=∆=∆
⋅−=∆
=∆
+=∆=∆
26
25
24
23
22
21
0
0
0
• Heave:
0
0
0
36
35
34
33
32
31
=∆
⋅−=∆
⋅+=∆
+=∆
=∆=∆
c
xCc
yCc
Cc
c
c
ppz
ppz
pz
• Roll:
pppy
pppz
ppzppy
ppz
ppy
zxCc
yxCc
yCzCc
yCc
zCc
c
⋅⋅−=∆
⋅⋅−=∆
⋅+⋅+=∆
⋅+=∆
⋅−=∆
=∆
46
45
2244
43
42
41 0
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
265
• Pitch:
pppx
ppzppx
pppz
ppz
ppx
zyCc
xCzCc
yxCc
xCc
c
zCc
⋅⋅−=∆
⋅+⋅+=∆
⋅⋅−=∆
⋅−=∆=∆
⋅+=∆
56
2255
54
53
52
51
0
• Yaw:
2266
65
64
63
62
61
0
ppyppx
pppx
pppy
ppy
ppx
xCyCc
zyCc
zxCc
c
xCc
yCc
⋅+⋅+=∆
⋅⋅−=∆
⋅⋅−=∆=∆
⋅+=∆
⋅−=∆
It is obvious that in case of several springs, a linear superposition of the coefficients can beused.
When using linear springs, generally 12 sets of coupled equations with the in and out of phaseterms of the motions have to be solved. Because of these springs, the surge, heave and pitchmotions will be coupled then with the sway, roll and yaw motions.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
266
9.3 Linearised Mooring Coefficients
Figure Figure 9.3–1 shows an example of results of static catenary line calculations, see forinstance Korkut and Hebert [1970], for an anchored platform.
Figure 9.3–1: Horizontal Forces on a Floating Structure as Function of Surge Displacements
Figure 9.3–1-a shows the platform anchored by two anchor lines of chain at 100 m waterdepth. Figure 9.3–1-b shows the horizontal forces at the suspension points of both anchorlines as a function of the horizontal displacement of the platform. Finally, Figure 9.3–1-cshows the relation between the total horizontal force on the platform and its horizontaldisplacement.This figure shows clearly the non-linear relation between the horizontal force on the platformand its horizontal displacement.
A linearised spring coefficient, to be used in frequency domain computations, can be obtainedfrom Figure 9.3–1-c by determining an average restoring spring coefficient, pxC , in the surgedisplacement region:
=ntDisplaceme
Force TotalMEANCpx
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
267
10 Added Resistance due to Waves
A ship moving forward in a wave field will generate ''two sets of waves'': waves associatedwith forward speed through still water and waves associated with its vertical relative motionresponse to waves. Since both wave patterns dissipate energy, it is logical to conclude that aship moving through still water will dissipate less energy than one moving through waves.The extra wave-induced loss of energy can be treated as an added propulsion resistance.
Figure 9.3–1 shows the resistance in regular waves as a function of the time: a constant partdue the calm water resistance and an oscillating part due to the motions of the ship, relative tothe incoming regular waves. The time-averaged part of the increase of resistance is called: theadded resistance due to waves, awR .
0
50 0
1 00 0
1 50 0
2 00 0
2 50 0
0 10 20 3 0
Re sistan ce
S til l water re sista n ce RSW
+M e an a dd ed re sista nce R
AW
Stil l water re sista n ce RSW
T im e (s)
Re
sist
ance
(kN
)
Figure 9.3–1: Increase of Resistance in Regular Waves
Two theoretical methods have been used for the estimation of the time-averaged addedresistance of a ship due to the waves and the resulting ship motions:
• a radiated wave energy method, as introduced by Gerritsma and Beukelman [1972],suitable for head to beam waves.
• an integrated pressure method, as introduced by Boese [1970], suitable for all wavedirections.
Because of the added resistance of a ship due to the waves is proportional to the relativemotions squared, its inaccuracy will be gained strongly by inaccuracies in the predictedmotions.The transfer function of the mean added resistance is presented as:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
268
2"
a
awaw
RR
ζ=
In a non-dimensional way the transfer function of the mean added resistance is presented as:
LBgR
Ra
awaw 22
"
⋅⋅⋅=
ζρin which:
L length between perpendicularsB maximum breadth of the waterline
Both methods will be described here.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
269
10.1 Radiated Energy Method
The wave energy - radiated during one period of oscillation of a ship in regular waves - hasbeen defined by Gerritsma and Beukelman [1972] as:
∫ ∫ ⋅⋅⋅=eT
Lbz dtdxVbP
0
2*'33
in which:'
33b hydrodynamic damping coefficient of the vertical motion of the cross section*
zV vertical average velocity of the water particles, relative to the cross sections
eT period of vertical oscillation of the cross section
The speed dependent hydrodynamic damping coefficient for the vertical motion of a crosssection is defined here as showed before:
bdxdM
VNb'
33'33
'33 ⋅−=
The harmonic vertical relative velocity of a point on the ship with respect to the waterparticles is defined by:
( )
( )φθθζ
φθζ
&&&&
&
⋅+⋅+⋅−−=
⋅+⋅−−=
bbw
bbwz
yVxz
yxzDtD
V
'3
'3
For a cross section of the ship, an equivalent harmonic vertical relative velocity has to befound, defined here by:
( )( )
ζεω
θθζ
*cos*
*3
*
zVeza
bwz
tV
VxzV
+⋅⋅=
⋅+⋅−−= &&&
With this the radiated energy during one period of oscillation is given by:
∫ ⋅⋅
⋅−⋅=
L
bzabe
dxVdx
dMVNP2*
'33'
33ωπ
To maintain a constant forward ship speed, this energy should be delivered by the ship'spropulsion plant. A mean added resistance awR has to be gained.The energy delivered to the surrounding water is given by:
µπ
µ
cos2
cos
⋅−⋅
⋅=
⋅
−⋅=
kR
Tc
VRP
aw
eaw
From this the transfer function of the mean added resistance according to Gerritsma andBeukelman can be found:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
270
∫ ⋅
⋅
⋅−⋅
⋅⋅−
=L
ba
za
bea
aw dxV
dxdM
VNkR
2*'33'
332 2cos
ζωµ
ζ
Equation 10.1–1
This method gives good results in head to beam waves. However, in following waves thismethod could fail.When the wave speed in following waves approaches the ship speed the frequency ofencounter in the denominator tends to zero, 0→eω . At these low frequencies, the potentialsectional mass is very high and the potential sectional damping is almost zero. The dampingmultiplied with the relative velocity squared in the nominator does not tend to zero, as fast asthe frequency of encounter. This is caused by the presence of a natural frequency for heaveand pitch at this low eω , so a high motion peak can be expected. This results into an extremepositive or negative added resistance.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
271
10.2 Integrated Pressure Method
Boese [1970] calculates the added resistance by integrating the longitudinal components ofthe oscillating pressures on the wetted surface of the hull. A second small contribution of thelongitudinal component of the vertical hydrodynamic and wave forces has been added.
The wave elevation is given by:( )µµωζζ sincoscos ⋅⋅−⋅⋅−⋅⋅= bbea ykxkt
The pressure in the undisturbed waves is given by:( )[ ][ ]( )[ ][ ] ( )µµωζρρ
ζρρ
sincoscoscosh
cosh
coshcosh
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅⋅+⋅⋅−=
⋅⋅+⋅
⋅⋅+⋅⋅−=
bbeab
b
ykxkthk
zhkgzg
hkzhk
gzgp
The horizontal force on an oscillating cross section is given by:
( )
( )[ ] ( )
−+⋅
⋅+
+−+−⋅⋅=
⋅= ∫+−
xsxs
zDbb
zDhk
zDg
dzptxfxs
ζζζρ
ζ
tanh2
,
22
with: θ⋅−= bx xzz .
As the mean added resistance during one period will be calculated, the constant term and thefirst harmonic term can be ignored. So:
( ) ( )[ ]
⋅
−⋅++−⋅⋅=hkzzgtxf xx
b tanh2,
22* ζζζρ
The vertical relative motion is defined by xzs −=ζ , so:
( ) [ ]
⋅
⋅++−⋅⋅=hk
szgtxf xb tanh2,
22* ζζρ
The average horizontal force on a cross section follows from:
( ) ( )
( )[ ]
⋅⋅
−⋅⋅−⋅⋅++−⋅
⋅⋅=
⋅= ∫
hk
xkszg
dttxfxf
a
sba
a
xaa
T
bb
e
tanh
coscos21
2
,
2
22
0
**
ζεµ
ζζρ ζ
The added resistance due to this force is:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
272
( )
( )[ ] b
b
w
L a
sba
a
xaa
b
L b
wbaw
dxdxdy
hk
xkszg
dxdxdy
xfR
⋅
−⋅
⋅⋅
−⋅⋅−⋅⋅−−−⋅
⋅⋅=
⋅
−⋅⋅=
∫
∫
tanh
coscos21
2
2
2
22
*1
ζεµ
ζζρ ζ
where wy is the still water line.
For deep water, this part of the mean added resistance reduces to:
∫ ⋅⋅⋅⋅−
=L
bb
waaw dx
dxdy
sg
R 21 2
ρ (as given by Boese for deep water)
The integrated vertical hydromechanical and wave forces in the ship-bounded system vary notonly in time but also in direction with the pitch angle.From this follows a second contribution to the mean added resistance:
( ) ( ) ( )
( ) ( ) dtttzT
dtttZtZT
R
e
e
T
e
T
whe
aw
⋅⋅⋅∇⋅⋅−
=
⋅⋅+⋅−
=
∫
∫
0
02
1
1
θρ
θ
&&
For this second contribution can be written:
( )θζζ εεθωρ −⋅⋅⋅⋅∇⋅⋅= zaaeaw zR cos21 2
2
So the transfer function of the total mean added resistance according to Boese is given by:( )
[ ]
( )θζζ
ζ
εεθωρ
ζεµ
ζρ
ζ
−⋅⋅⋅⋅∇⋅⋅+
⋅
−⋅
⋅⋅−⋅⋅−⋅⋅
−−−⋅⋅⋅= ∫
zaae
bb
w
L a
sba
a
xa
a
aw
z
dxdxdy
hk
xkszg
R
cos21
tanh
coscos21
21
2
2
2
2
Equation 10.2–1
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
273
10.3 Comparison of Results
Figure 10.3–1 shows an example of a comparison between computed and experimental data.
Figure 10.3–1: Added Resistance of the S-175 Containership Design
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
274
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
275
11 Bending and Torsion Moments
The axes system (of which the hydrodynamic sign convention differs from that commonlyused in structural engineering) and the internal load definitions are given in Figure 10.3–1.
Figure 10.3–1: Axis System and Internal Load Definitions
To obtain the vertical and lateral shear forces and bending moments and the torsion momentsthe following information over a length mL on the solid mass distribution of the shipincluding its cargo is required:
( )bxm' distribution over the ship length of the solid mass of the ship per unitlength, see Figure 10.3–2
( )bm xz ' distribution over the ship length of the vertical bz -values of the centreof gravity of the solid mass of the ship per unit length
( )bxx xk ' distribution over the ship length of the radius of inertia of the solidmass of the ship per unit length, about a horizontal longitudinal axisthrough the centre of gravity
Figure 10.3–2: Distribution of Solid Mass
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
276
The input values for the calculation of shear forces and bending and torsion moments areoften more or less inaccurate. Mostly small adaptations are necessary, for instance to avoid aremaining calculated bending moment at the forward end of the ship.The total mass of the ship is found by an integration of the mass per unit length:
( )∫ ⋅=mL
bb dxxmm '
It is obvious that this integrated mass should be equal to the mass of displacement, calculatedfrom the underwater hull form:
∇⋅= ρm
Both terms will be calculated from independently derived data, so small deviations arepossible. A proportional correction of the masses per unit length ( )bxm' can be used, seeFigure 10.3–3.Then ( )bxm' will be replaced by:
( )m
xm b
∇⋅⋅ρ'
Figure 10.3–3: Mass Correction for Buoyancy
The longitudinal position of the centre of gravity is found from the distribution of the massper unit length:
( )∫ ⋅⋅⋅=mL
bbbG dxxxmm
x '1
An equal longitudinal position of the ship's centre of buoyancy Bx is required, so:
BG xx =
Again, because of independently derived data, a small deviation is possible.Then, for instance, ( )bxm' can be replaced by ( ) ( )bb xcxm +' , with:
( ) ( )01 −−⋅−= Abb xxcxc for: 40 mAb Lxx <−<( ) ( )21 mAbb Lxxcxc −−⋅+= for: 434 mAbm LxxL ⋅<−<( ) ( )mAbb Lxxcxc −−⋅−= 1 for: mAbm LxxL <−<⋅ 43
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
277
with:( )31
32
m
GB
Lxx
c−⋅∇⋅⋅
=ρ
In here:
Ax bx -co-ordinate of hindmost part of mass distribution
mL total length of mass distribution
Figure 10.3–4: Mass Correction for Cent re of Buoyancy
For relative slender bodies, the longitudinal radius of inertia of the mass can be found fromthe distribution of the mass per unit length:
( )∫ ⋅⋅⋅=mL
bbbyy dxxxmm
k 2'2 1
It can be desirable to change the mass distribution in such a way that a certain requiredlongitudinal radius of inertia ( )newk yy or ( )newkzz will be achieved, without changing thetotal mass or the position of its centre of gravity.Then, for instance, ( )bxm' can be replaced by ( ) ( )bb xcxm +' , see Figure 10.3–5, with:
( ) ( )02 −−⋅+= Abb xxcxc for: 80 mAb Lxx <−<( ) ( )822 mAbb Lxxcxc ⋅−−⋅−= for: 838 mAbm LxxL ⋅<−<( ) ( )842 mAbb Lxxcxc ⋅−−⋅+= for: 8483 mAbm LxxL ⋅<−<⋅( ) ( )842 mAbb Lxxcxc ⋅−−⋅−= for: 8584 mAbm LxxL ⋅<−<⋅( ) ( )862 mAbb Lxxcxc ⋅−−⋅+= for: 8785 mAbm LxxL ⋅<−<⋅( ) ( )mAbb Lxxcxc −−⋅−= 2 for: mAbm LxxL <−<⋅ 87
with:( ) ( )
3
22
2 9
3204
m
yyyy
L
oldknewkc
⋅
−⋅∇⋅⋅=
ρ
In here:
Ax bx -co-ordinate of hindmost part of mass distribution
mL total length of mass distribution
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
278
Figure 10.3–5: Mass Correction for Radius of Inertia
The position in height of the centre of gravity is found from the distribution of the heights ofthe centre of gravity of the masses per unit length:
( ) ( )∫ ⋅⋅⋅=mL
bbmbG dxxzxmm
z ''1
It is obvious that this value should be zero. If not so, this value has to be subtracted from( )bm xz ' .
So, ( )bm xz ' will be replaced by ( ) Gbm zxz −' .
The transverse radius of inertia xxk is found from the distribution of the radii of inertia of themasses per unit length:
( )∫ ⋅⋅⋅=mL
bxxbyy dxkxmm
k2''2 1
If this value of xxk differs from a required value ( )newkxx of the radius of inertia, aproportional correction of the longitudinal distribution of the radii of inertia can be used:
( ) ( ) ( )( )oldknewk
oldxknewxkxx
xxbxxbxx '
''' ,, ⋅=
Consider a section of the ship with a length bdx to calculate the shear forces and the bendingand the torsion moments.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
279
Figure 10.3–6: Loads on a Cross Section
When a load ( )bxq loads the disk, this implies for the disk:
( ) ( )bbb xdQdxxq −=⋅ so: ( ) ( )b
b
b xqdx
xdQ−=
( ) ( )bbb xdMdxxQ +=⋅ so: ( ) ( )bb
b xQdx
xdM+=
in which:( )bxQ shear force
( )bxM bending moment
The shear force and the bending moment in a cross section 1x follows from an integration ofthe loads from the hindmost part of the ship 0x to this cross section 1x :
( ) ( )
( ) ( )
( )b
x
x
x
xb
b
b
x
xbb
x
x
bb
b
dxdxdx
xdQ
dxxQxM
dxdx
xdQxQ
b
⋅
⋅+=
⋅−=
⋅−=
∫ ∫
∫
∫
1
0 0
1
0
1
0
1
1
So, the shear force ( )1xQ and the bending moment ( )1xM in a cross section can be expressedin the load ( )bxq by the following integrals:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
280
( ) ( )
( ) ( ) ( )
( ) ( )∫∫
∫
∫
⋅⋅−⋅⋅+=
⋅−⋅+=
⋅−=
1
0
1
0
1
0
1
0
1
11
1
x
x
bb
x
x
bbb
x
xbbb
x
x
bb
dxxqxdxxxq
dxxxxqxM
dxxqxQ
For the torsion moment an approach similar to the approach for the shear force can be used.The load ( )bxq consists of solid mass and hydromechanical terms. The ordinates of theseterms will differ generally, so numerical integrations of these two terms have to be carried outseparately.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
281
11.1 Still Water Loads
Consider the forces acting on a section of the ship with a length bdx .
Figure 11.1–1: Still Water Loads on a Cross Section
According to Newton's second law of dynamics, the vertical forces on the unfastened disk of aship in still water are given by:
( ) ( ) ( ) bbswb dxxqgdxm ⋅=−⋅⋅ 3'
with:gmgAq ssw ⋅−⋅⋅= '
3 ρ
So, the vertical shear force ( )13 xQ sw and the bending moment ( )15 xQ sw in still water in a cross
section can be obtained from the vertical load ( )13 xq sw by the following integrals:
( ) ( )
( ) ( ) ( )∫∫
∫
⋅⋅−⋅⋅+=
⋅−=
1
0
1
0
1
0
31315
313
x
x
bbsw
x
x
bbbswsw
x
xbbswsw
dxxqxdxxxqxQ
dxxqxQ
For obtaining the dynamic parts of the vertical shear forces and the vertical bending momentsin regular waves, reference is given to Fukuda [1962]. For the lateral mode and the roll modea similar procedure can be followed. This will be showed in the following Sections.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
282
11.2 Dynamical Lateral Loads
Consider the forces acting on a section of the ship with a length bdx .
Figure 11.2–1: Lateral Loads on a Cross Section in Waves
According to Newton's second law of dynamics, the harmonic lateral dynamic load per unitlength on the unfastened disk is given by:
( ) ( ) ( )( ) ( )φφψφρ ⋅+⋅−⋅+⋅−⋅⋅⋅+
++=
gzxyxmAg
xXxXxq
mbbs
bwbhb
&&&&&& ''
'2
'22
The sectional hydromechanical loads for sway are given by:
ψψψφφφ
⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=
'26
'26
'26
'24
'24
'24
'22
'22
'22
'2
cba
cba
ycybyaXh
&&&
&&&&&&
with:
0
2
0
0
'26
'22
2
2'
22
'22'
22'
26
'22
2'
222
'22'
222'
22'
26
'24
'22'
22
'24'
24'
24
'22
2
'24
2'
22'
24'
24
'22
'22'
22'
22
'22
2'
22'
22
=
⋅+⋅⋅−⋅
⋅−+=
⋅⋅+⋅+
⋅−⋅+⋅+=
=
⋅−⋅+⋅−+=
⋅⋅+⋅+⋅++=
=
⋅−+=
⋅++=
c
dxdNV
MVxdx
dMVNb
xdx
dNVN
Vdx
dMVN
VxMa
c
dxdM
VNOGdx
dMVNb
dxdN
OGV
dxdNV
MOGMa
c
dxdM
VNb
dxdNV
Ma
beb
b
bbeebe
b
bb
bebe
b
be
ω
ωωω
ωω
ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
283
The sectional wave loads for sway are given by:
'2
*2
'22'
22
*2
'22*
2'
22'
2
FK
wbe
wbe
ww
X
dxdM
VN
dxdNV
MX
+
⋅
⋅−⋅+
⋅⋅⋅
+⋅+=
ζωω
ζωω
ζ
&
&&&&
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.Then the harmonic lateral shear forces ( )12 xQ and the bending moments ( )16 xQ in waves in
cross section 1x can be obtained from the horizontal load ( )bxq2 by the following integrals:( ) ( )
( )
( ) ( )
( ) ( )∫∫
∫
⋅⋅−⋅⋅+
+⋅⋅=
⋅−
+⋅⋅=
1
0
1
0
6
1
0
2
212
616
2
212
cos
cos
x
x
bb
x
x
bbb
Qea
x
x
bb
Qea
dxxqxdxxxq
tQxQ
dxxq
tQxQ
ζ
ζ
εω
εω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
284
11.3 Dynamical Vertical Loads
Consider the forces acting on a section of the ship with a length bdx .
Figure 11.3–1: Vertical Loads on a Cross Section in Waves
According to Newton's second law of dynamics, the harmonic longitudinal and verticaldynamic loads per unit length on the unfastened disk are given by:
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )θ
θ&&&&
&&&&
⋅−⋅−++=
⋅−⋅−++=
bbbwbhb
bbwbhb
xzxmxXxXxq
bGxxmxXxXxq''
3'
33
''1
'11
The sectional hydromechanical loads for surge are given by:
θθθ ⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=
'15
'15
'15
'13
'13
'13
'11
'11
'11
'1
cba
zczbza
xcxbxaXh
&&&&&&
&&&
with:
0
0
0
0
0
'15
11
'11'
11'
15
'11
2'
11'
15
'13
'13
'13
'11
11
'11'
11'
11
'11
2'
11'
11
=
⋅−⋅
⋅−−=
⋅⋅+⋅−=
=
=
=
=
+−+=
⋅++=
c
bGbbGdx
dMVNb
bGdx
dNVbGMa
c
b
a
c
bdx
dMNb
dxdNV
Ma
Vb
be
Vb
be
ω
ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
285
The sectional hydromechanical loads for heave are given by:
θθθ ⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=
'35
'35
'35
'33
'33
'33
'31
'31
'31
'3
cba
zczbza
xcxbxaXh
&&&&&&
&&&
with:
bw
beLb
Lbb
b
bbeebe
b
w
b
be
xygc
dxdNV
dxMVdxxdx
dMVNb
xdx
dNVN
Vdx
dMN
VxMa
ygc
dxdM
VNb
dxdNV
Ma
c
b
a
⋅⋅⋅⋅−=
⋅+⋅⋅⋅+⋅⋅
⋅−−=
⋅⋅−
+⋅−
+
−⋅−⋅−=
⋅⋅⋅+=
⋅−+=
⋅++=
=
=
=
∫∫
ρ
ω
ωωω
ρ
ω
2
2
2
0
0
0
'35
'33
2
2'
33
'33'
33'
35
'33
2'
332
'33'
332'
33'
35
'33
'33'
33'
33
'33
2'
33'
33
'31
'31
'31
The sectional wave loads for surge and heave are given by:
'1
*1
'11'
11
*1
'11*
1'
11'
1
FK
wbe
wbe
ww
X
dxdM
VN
dxdNV
MX
+
⋅
⋅−⋅+
⋅⋅⋅
+⋅+=
ζωω
ζωω
ζ
&
&&&&
'3
*3
'33'
33
*3
'33*
3'
33'
3
FK
wbe
wbe
ww
X
dxdM
VN
dxdNV
MX
+
⋅
⋅−⋅+
⋅⋅⋅
+⋅+=
ζωω
ζωω
ζ
&
&&&&
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
286
Then the harmonic vertical shear forces ( )13 xQ and the bending moments ( )15 xQ in waves in
cross section 1x can be obtained from the longitudinal and vertical load ( )bxq1 and ( )bxq3 bythe following integrals:
Figure 11.3–2 shows a comparison between measured and calculated distributions of thevertical wave bending moment amplitudes over the length of the ship.
Figure 11.3–2: Distribution of Vertical Bending Moment Amplitudes
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
287
11.4 Dynamical Torsion Loads
Consider the forces acting on a section of the ship with a length bdx .
Figure 11.4–1: Torsion Loads on a Cross Section in Waves
According to Newton's second law of dynamics, the harmonic torsion dynamic load per unitlength on the unfastened disk about a longitudinal axis at a distance 1z above the ship's centreof gravity is given by:
( ) ( ) ( )( ) ( )
( )b
bmxxb
bwbhb
xqz
gxyzkxm
xXxXzxq
21
'2''
'4
'414 ,
⋅+⋅+⋅+−⋅⋅−
++=
φψφ &&&&&&
The sectional hydromechanical load for roll is given by:
ψψψφφφ
⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−⋅−=
'46
'46
'46
'44
'44
'44
'42
'42
'42
'4
cba
cba
ycybyaXh
&&&
&&&&&&
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
288
'26
'46
'42
2
2'
26'
42
'42'
42'
46
'42
2
'422
'26
'42'
422
'42
'46
3'
44
'24
'44
'42'
42
'44'
44'
44
'42
2
'44
2
'24
'42
'44
'44
'24
'22
'42'
42'
42
'24
2
'22
'42
'42
2
232
0
cOGc
dxdNV
bOGMVxdx
dMVNb
xdx
dNVN
VaOG
dxdM
VNV
xMa
bGAy
gc
bOGbdx
dMVNOG
dxdM
VNb
dxdN
OGV
dxdNV
aOGMOGMa
c
bOGdx
dMVNb
dxdNV
aOGMa
beb
b
bbeebe
b
sw
Vbb
bebe
b
be
⋅+=
⋅+⋅+⋅⋅−⋅
⋅−+=
⋅⋅+⋅+⋅+
⋅−⋅+⋅+=
⋅+⋅⋅⋅+=
⋅++
⋅−⋅+⋅−+=
⋅⋅+⋅+⋅+⋅++=
=
⋅+⋅−+=
⋅+⋅++=
ω
ωωω
ρ
ωω
ω
In here, bG is the vertical distance of the centre of gravity of the ship G above the centroidb of the local submerged sectional area.
The sectional wave load for roll is given by:
'2
'4
*2
'42'
42
*2
'42*
2'
42'
4
wFK
wbe
wbe
ww
XOGX
dxdM
VN
dxdNV
MX
⋅++
⋅
⋅−⋅+
⋅⋅⋅
+⋅+=
ζωω
ζωω
ζ
&
&&&&
The ''Modified Strip Theory'' includes the outlined terms. When ignoring the outlined termsthe ''Ordinary Strip Theory'' is presented.
Then the harmonic torsion moments ( )114 , zxQ in waves in cross section 1x at a distance 1zabove the centre of gravity can be obtained from the torsion load ( )14 , zxq b by the followingintegral:
( ) ( )
( ) b
x
xb
Qea
dxzxq
tQzxQ
⋅−=
+⋅⋅=
∫1
0
4
14
4114
,
cos, ζεω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
289
12 Statistics in Irregular Waves
To compare the calculated behaviour of different ship designs or to get an impression of thebehaviour of a specific ship design in a seaway, standard representations of the wave energydistributions are necessary.
Three well-known types of normalised wave energy spectra are described here:• the Neumann wave spectrum, a somewhat wide wave spectrum, which is sometimes used
for open sea areas• the Bretschneider wave spectrum, an average wave spectrum, frequently used in open sea
areas• the Mean JONSWAP wave spectrum, a narrow wave spectrum, frequently used in North
Sea areas.
The mathematical formulations of these normalised uni-directional wave energy spectra arebased on two parameters:• the significant wave height 3/1H
• the average wave period 1T , based on the centroid of the spectral area curve.To obtain the average zero-crossing period 2T or the spectral peak period pT , a fixed relation
with 1T can be used not-truncated spectra.
From these wave energy spectra and the transfer functions of the responses, the responseenergy spectra can be obtained.Generally the frequency ranges of the energy spectra of the waves and the responses of theship on these waves are not very wide. Then the Rayleigh distribution can be used to obtain aprobability density function of the maximum and minimum values of the waves and theresponses. With this function, the probabilities on exceeding threshold values by the shipmotions can be calculated.Bow slamming phenomena are defined by a relative bow velocity criterion and a peak bottomimpact pressure criterion.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
290
12.1 Normalised Wave Energy Spectra
Three mathematical definitions with two parameters of normalized spectra of irregular uni-directional waves have been described:• the Neumann wave spectrum, a somewhat wide spectrum• the Bretschneider wave spectrum, an average spectrum• the mean JONSWAP wave spectrum, a narrow spectrumA comparison of the Neumann, the Bretschneider and the mean JONSWAP wave spectra isgiven here for a sea state with a significant wave height of 4 meters and an average waveperiod of 8 seconds.
Figure 12.1–1: Comparison of Three Spectral Formulations
12.1.1 Neumann Wave Spectrum
In some cases in literature the Neumann definition of a wave spectrum for open sea areas isused:
( )
⋅−
⋅⋅⋅
= −− 22
1
65
1
23/1 8.69
exp3832 ωωωζ TT
HS
12.1.2 Bretschneider Wave Spectrum
A very well known two-parameter wave spectrum of open seas is defined by Bretschneider as:
( )
⋅−
⋅⋅⋅
= −− 42
1
54
1
23/1 2.691
exp8.172 ωωωζ TT
HS
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
291
Another name of this wave spectrum is the Modified Two-Parameter Pierson-MoskowitzWave Spectrum.
This formulation is accepted by the 2nd International Ship Structures Congress in 1967 andthe 12th International Towing Tank Conference in 1969 as a standard for seakeepingcalculations and model experiments. This is reason why this spectrum is also called I.S.S.C.or I.T.T.C. Wave Spectrum.The original One-Parameter Pierson-Moskowitz Wave Spectrum for fully developed seas canbe obtained from this definition by using a fixed relation between the significant wave heightand the average wave period in this Bretschneider definition: 3/11 861.3 HT ⋅= .
12.1.3 Mean JONSWAP Wave Spectrum
In 1968 and 1969 an extensive wave measurement program, known as the Joint North SeaWave Project (JONSWAP) was carried out along a line extending over 100 miles into theNorth Sea from Sylt Island. From analysis of the measured spectra, a spectral formulation ofwind generated seas with a fetch limitation was found.The following definition of a Mean JONSWAP wave spectrum is advised by the 15th ITTC in1978 for fetch limited situations:
( ) BATT
HS γωωωζ ⋅⋅
⋅−
⋅⋅⋅
= −− 42
1
54
1
23/1 2.691
exp8.172
with:658.0=A
⋅
−−=
2
2
0.1
expσ
ωω
pB
3.3=γ (peakedness factor)
pp T
πω ⋅= 2
(circular frequency at spectral peak)
=σ a step function of ω: if pωω < then: 07.0=σif pωω > then: 09.0=σ
The JONSWAP expression is equal to the Bretschneider definition multiplied by thefrequency function BA γ⋅ .Sometimes, a third free parameter is introduced in the JONSWAP wave spectrum by varyingthe peakedness factor γ .
12.1.4 Definition of Parameters
The nth order spectral moments of the wave spectrum, defined as a function of the circularwave frequency ω , are:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
292
( )∫∞
⋅⋅=0
ωωωζζ dSm nn
The breadth of a wave spectrum is defined by:
ζζ
ζε40
221mm
m
⋅−=
The significant wave height is defined by:
ζ03/1 4 mH ⋅=
The several definitions of the average wave period are:
pT peak or modal wave period, corresponding to peak of spectralcurve
ζ
ζπ1
01 2
m
mT ⋅⋅= average wave period, corresponding to centroid of spectral
curve
ζ
ζπ2
02 2
m
mT ⋅⋅= average zero-crossing wave period, corresponding to radius of
inertia of spectral curve
For not-truncated mathematically defined spectra, the theoretical relations between the periodsare tabled below:
=⋅=⋅⋅==⋅⋅=⋅=
p
p
p
TTTTTT
TTT
21
21
21
407.1296.1711.0921.0
772.0086.1 for Bretschneider wave spectra
=⋅=⋅⋅==⋅⋅=⋅=
p
p
p
TTTTTT
TTT
21
21
21
287.1199.1777.0932.0
834.0073.1 for JONSWAP wave spectra
Truncation of wave spectra during numerical calculations can cause differences between inputand calculated wave periods. Generally, the wave heights will not differ much.
In Figure 12.1–2 and Table 12.1–1 - for ''Open Ocean Areas'' and ''North Sea Areas'' - anindication is given of a possible average relation between the scale of Beaufort or the windvelocity at 19.5 meters above the sea level and the significant wave height 3/1H and theaverage wave periods 1T or 2T .Notice that these data are an indication only. A generally applicable fixed relation betweenwave heights and wave periods does not exist.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
293
Figure 12.1–2: Wave Spectra Parameter Indications
Table 12.1–1: Wave Spectra Parameter Indications
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
294
Other open ocean definitions for the North Atlantic and the North Pacific, obtained fromBales [1983] and adopted by the 17th ITTC (1984), are given in Table 12.1–2. The modal orcentral periods in these tables correspond with the peak period pT . For not-truncated spectra,
the relations with 1T and 2T are defined before.
Table 12.1–2: Sea State Parameters
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
295
12.2 Response Spectra and Statistics
The energy spectrum of the responses ( )tr of a sailing ship in the irregular waves followsfrom the transfer function of the response and the wave energy spectrum by:
( ) ( )ωζ
ω ζSr
Sa
ar ⋅
=
2
or ( ) ( )ea
aer S
rS ω
ζω ζ⋅
=
2
This has been visualized for a heave motion in Figure 12.2–1and Figure 12.2–2.
Figure 12.2–1: Principle of Transfer of Waves into Responses
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
296
0
0 .5
1 .0
1 .5
2 .0
0 0.5 1.0 1.5 2.0
C on tain ersh ipL = 1 75 m etre H ea d w ave sV = 2 0 kn ots
T ran s ferfu nc t io nhe ave
0
1
2
3
4
5
0 0.5 1.0 1.5 2.0
W avesp ec trum
H1/3
= 5.00 m
T2 = 8 .0 0 s
0
1
2
3
4
5
0 0.5 1.0 1 .5 2 .0
W avespe c tru m
H1/3
= 5 .0 0 m
T2 = 8.00 s
Sp
ectr
al d
ens
ity w
ave
(m2 s)
0
0.5
1.0
1.5
2.0
0 0.5 1.0 1 .5 2 .0
C onta ine rshi pL = 17 5 m e treH ead w ave sV = 20 kno ts
T ra ns ferfun c ti onh eave
Tra
nsfe
r fu
nctio
n h
eave
(m
/m)
0
2
4
6
8
0 0.5 1.0 1 .5 2 .0
H e avesp ec tru m
za
1 /3
= 1.92 m
Tz
2
= 7.74 s
w a ve fre qu en cy (ra d/s )
Sp
ectr
al d
ens
ity h
eav
e (
m2s)
0
2
4
6
8
0 0.5 1 .0 1.5 2 .0
za
1/3
= 1 .92 m
Tz
2
= 7 .74 s
fr eq ue ncy of en cou nter ( rad /s )
Figure 12.2–2: Heave Spectra in the Wave and Encounter Frequency Domain
The moments of the response spectrum are given by:
( )∫∞
⋅⋅=0
en
eernr dSm ωωω with: ,...2,1,0=n
From the spectral density function of a response the significant amplitude can be calculated.The significant amplitude is defined to be the mean value of the highest one-third part of thehighest wave heights, so:
ra mr 03/1 2 ⋅=
A mean period can be found from the centroid of the spectrum by:
r
rr m
mT
1
01 2 ⋅⋅= π
Another definition, which is equivalent to the average zero-crossing period, is found from thespectral radius of inertia by:
r
rr m
mT
2
02 2 ⋅⋅= π
The probability density function of the maximum and minimum values, in case of a spectrumwith a frequency range that is not too wide, is given by the Rayleigh distribution:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
297
( )
⋅−⋅=
r
a
r
aa m
rmrrf
0
2
0 2exp
This implies that the probability of exceeding a threshold value a by the response amplitude
ar becomes:
⋅−
=
⋅
⋅−
⋅=> ∫∞
r
a
a r
a
r
aa
ma
drmr
mr
arP
0
2
0
2
0
2exp
2exp
The number of times per hour that this happens follows from:
arPT
N ar
hour >⋅=2
3600
The spectral value of the waves ( )eS ωζ , based on eω , is not equal to the spectral value
( )ωζS , based on ω . Because of the requirement of an equal amount of energy in the
frequency bands eω∆ and ω∆ , it follows:
( ) ( ) ωωωω ζζ dSdS ee ⋅=⋅
From this the following relation is found:
( ) ( )ωω
ωω ζ
ζ dd
SS
ee =
The relation between the frequency of encounter and the wave frequency, of which anexample is illustrated in Figure 12.2–3, is given by:
µωω cos⋅⋅−= Vke
Figure 12.2–3: Example of Relation Between eω and ω
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
298
From the relation between eω and ω follows:
dkdV
dd e
ωµ
ωω cos
0.1⋅
−=
The derivative dkdω follows from the relation between ω and k :
[ ]hkgk ⋅⋅⋅= tanhω
So:
[ ] [ ][ ]hkgk
hkhgk
hkg
dkd
⋅⋅⋅⋅⋅⋅
⋅+⋅⋅
=tanh2
coshtanh 2ω
As can be seen in Figure 12.2–3, in following waves the derivative ωω dd e can approachfrom both sides, a positive or a negative side, to zero. As a result of this, around a wave speedequal to twice the forward ship speed component in the direction of the wave propagation, thetransformed spectral values will range from plus infinite to minus infinite. This implies thatnumerical problems will arise in the numerical integration routine.
This is the reason why the spectral moments have to be written in the following format:
( ) ( )
( ) ( )
( ) ( )∫∫
∫∫
∫∫
∞∞
∞∞
∞∞
⋅⋅=⋅⋅=
⋅⋅=⋅⋅=
⋅=⋅=
0
2
0
22
00
1
000
ωωωωωω
ωωωωωω
ωωωω
dSdSm
dSdSm
dSdSm
ereeerr
ereeerr
reerr
with:
( ) ( )ωζ
ω ζSr
Sa
ar ⋅
=
2
If ( )erS ω has to be known, for instance for a comparison of the calculated response spectra
with measured response spectra, these values can be obtained from this ( )ωrS and thederivative ωω dd e . So an integration of ( )erS ω over eω has to be avoided.
Because of the linearities, the calculated significant values can be presented by:
3/1
3/1
Hra versus 1T or 2T
with:
3/1H significant wave height
21,TT average wave periods
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
299
The mean added resistance in a seaway follows from:
( ) ωωζ ζ dSR
Ra
awAW ⋅⋅⋅= ∫
∞
022
Because of the linearities of the motions, the calculated mean added resistance values can bepresented by:
23/1H
RAW versus 1T or 2T
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
300
12.3 Shipping Green Water
The effective dynamic freeboard will differ from the results obtained from the geometricfreeboard at zero forward speed in still water and the calculated vertical relative motions of asailing ship in waves.When sailing in still water, sinkage, trim and the ship's wave system will effect the localgeometric freeboard. A static swell up should be taken into account.An empirical formula, based on model experiments, for the static swell up at the forwardperpendicular is given by Tasaki [1963]:
275.0 FnLL
BffE
e ⋅⋅⋅−=
with:
ef effective freeboard at the forward perpendicularf geometric freeboard at the forward perpendicularL length of the shipB breadth of the ship
EL length of entrance of the waterlineFn Froude number
An oscillating ship will produce waves and these dynamic phenomena will influence theamplitude of the relative motion. A dynamic swell up should be taken into account.Tasaki [1963] carried out forced oscillation tests with ship models in still water and obtainedan empirical formula for the dynamic swell-up at the forward perpendicular in head waves:
gLC
ss eB
a
a ⋅⋅
−=
∆ 2
345.0 ω
with the restrictions:block coefficient: 80.060.0 << BCFroude number: 29.016.0 << Fn
In this formula as is the amplitude of the relative motion at the forward perpendicular asobtained in head waves, calculated from the heave, the pitch and the wave motions.Then the actual amplitude of the relative motions becomes:
aaa sss ∆+=*
Then, shipping green water is defined by:
ea fs >* at the forward perpendicular
The spectral density of the vertical relative motion at the forward perpendicular is given by:
( ) ( )ωζ
ω ζSs
Sa
as
⋅
=
2*
*
The spectral moments are given by:
( )∫∞
⋅⋅=0
** ωωω dSm nesns
with: ,...2,1,0=n
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
301
When using the Rayleigh distribution the probability of shipping green water is given by:
⋅−=>
*0
2*
2exp
s
eea m
ffsP
The average zero-crossing period of the relative motion is found from the spectral radius ofinertia by:
*
*
*
2
02
2s
ss m
mT ⋅⋅= π
The number of times per hour that green water will be shipped follows from:
ea
s
hour fsPT
N >⋅= *
2 *
3600
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
302
12.4 Bow Slamming
Slamming is a two-node vibration of the ship caused by suddenly pushing the ship by thewaves. A complete prediction of slamming phenomena is a complex task, which is beyond thescope of any existing theory.Slamming impact pressures are affected by the local hull section shape, the relative velocitybetween ship and waves at impact, the relative angle between the keel and the water surface,the local flexibility of the ship's bottom plating and the overall flexibility of the ship'sstructure.
12.4.1 Criterium of Ochi
Ochi [1964] translated slamming phenomena into requirements for the vertical relativemotions of the ship.He defined slamming by:• an emergence of the bow of the ship at 10 percentile of the length aft of the forward
perpendiculars• an exceeding of a certain critical value at the instance of impact by the vertical relative
velocity, without forward speed effect, between the wave surface and the bow of the ship
Ochi defines the vertical relative displacement and velocity of the water particles with respectto the keel point of the ship by:
θζ
θζ&&&& ⋅+−=
⋅+−=
bx
bx
xzs
xzs
b
b
with:( )
( )µωζωζ
µωζζ
cossin
coscos
⋅⋅−⋅⋅⋅−=
⋅⋅−⋅⋅=
beaex
beax
xkt
xkt
b
b
&
So a forward speed effect ( θ⋅V -term) is not included in this vertical relative velocity. Thespectral moments of the vertical relative displacements and velocities are defined by sm0 and
sm &0 .Emergence of the bow of the ship happens when the vertical relative displacement amplitude
as at L⋅90.0 is larger than the ship's draft sD at this location.The probability of emergence of the bow follows from:
⋅−=>
s
ssa m
DDsP0
2
2exp
The second requirement states that the vertical relative velocity exceeds a threshold value.According to Ochi, 12 feet per second can be taken as a threshold value for a ship with alength of 520 feet.Scaling results into:
Lgscr ⋅⋅= 0928.0&
The probability of exceeding this threshold value is:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
303
⋅−=>
s
crcra m
sssP&
&&&
0
2
2exp
Both occurrences, emergence of the bow and exceeding the threshold velocity, are statisticallyindependent. In case of slamming both occurrences have to appear at the same time.So the probability on a slam is the product of the both independent probabilities:
⋅−
+⋅
−=
>⋅>=
s
cr
s
s
crasa
ms
mD
ssPDsPslamP
&
&
&&
0
2
0
2
22exp
12.4.2 Criterium of Conolly
Conolly [1974] translated slamming phenomena into requirements for the peak impactpressure of the ship.He defined slamming by:• an emergence of the bow of the ship• an exceeding of a certain critical value by the peak impact pressure at this location.
The peak impact pressure is defined by:2
21
crp sCp &⋅⋅⋅= ρ
The coefficient pC has been taken from experimental data of slamming drop tests withwedges and cones, as given in literature.Some of these data, as for instance presented by Lloyd [1989] as a function of the deadriseangle β , are illustrated in Figure 12.4–1.
Figure 12.4–1: Peak Impact Pressure Coefficients
An equivalent deadrise angle β is defined here by the determination of an equivalent wedge.The contour of the cross section inside 10 percentile of the half breadth 2B of the ship hasbeen used to define an equivalent wedge with a half breadth: 210.0 Bb ⋅= .
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
304
The accessory draught t of the wedge follows from the section contour. In the fore body ofthe ship, this draught can be larger than 10 percentile of the amidships draught T . If so, thesection contour below T⋅10.0 has been used to define an equivalent wedge: Tt ⋅= 10.0 . Ifthis draught is larger than the local draught, the local draught has been used.The accessory half breadth b of the wedge follows from the section contour.
Figure 12.4–2: Definition of an Equivalent Wedge
Then the sectional area sA below local draught t has to be calculated.Now the equivalent deadrise angle β follows from:
=
baarctanβ 20 πβ ≤≤
( )b
Atba s−⋅⋅
=2
Critical peak impact pressures crp have been taken from Conolly [1974]. He gives measuredimpact pressures at a ship with a length of 112 meter over 30 per cent of the ship length fromforward. From this, a lower limit of crp has been assumed. This lower limit is presented inFigure 12.4–3.
Figure 12.4–3: Measured Impact Pressures of a 112 Meter Ship
These values have to be scaled to the actual ship size. Bow emergence and exceeding of thislimit is supposed to cause slamming.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
305
This approach can be translated into local hull-shape-depending threshold values of thevertical relative velocity too:
p
crcr C
ps
⋅⋅
=ρ2
&
The vertical relative velocity, including a forward speed effect, of the water particles withrespect to the keel point of the ship is defined by:
θθζ
θζ
⋅−⋅+−=
⋅+−=
Vxz
xzDtD
s
bx
bx
b
b
&&&
&
with:( )
( )µωζωζ
µωζζ
cossin
coscos
⋅⋅−⋅⋅⋅−=
⋅⋅−⋅⋅=
beaex
beax
xkt
xkt
b
b
&
Then:
⋅−+
⋅−=
s
cr
s
s
ms
mDslamP
&
&
0
2
0
2
22exp
Notice that, because of including the forward speed effect, the spectral moment of thevelocities does not follow from the spectral density of the relative displacement as showed inthe definition of Ochi.The average period of the relative displacement is found by:
s
s
s
ss m
mmm
T&0
0
2
02 22 ⋅⋅=⋅⋅= ππ
Then the number of times per hour that a slam will occur follows from:
slamPT
Ns
hour ⋅=2
3600
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
306
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
307
13 Twin-Hull Ships
When not taking into account the interaction effects between the two individual hulls, thewave loads and motions of twin-hull ships can be calculated easily. Each individual hull has tobe symmetric with respect to its centre plane. The distance between the two centre planes ofthe single hulls ( Ty⋅2 ) should be constant. The co-ordinate system for the equations ofmotion of a twin-hull ship is given in Figure 12.4–1.
Figure 12.4–1: Co-ordinate System of Twin-Hull Ships
13.1 Hydromechanical Coefficients
The hydromechanical coefficients ija , ijb and ijc in this section are those of one individualhull, defined in the co-ordinate system of the single hull, as given and discussed before.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
308
13.2 Equations of Motion
The equations of motion for six degrees of freedom of a twin-hull ship are defined by:
66
55
44
33
22
11
:Yaw
:Pitch
:Roll
Heave
:Sway
:Surge
TwThTzxTzz
TwThTyy
TwThTxzTxx
TwThT
TwThT
TwThT
XXII
XXI
XXII
XXz
XXy
XXx
=−⋅−⋅=−⋅=−⋅−⋅=−⋅∇⋅=−⋅∇⋅=−⋅∇⋅
φψθ
ψφρρρ
&&&&
&&
&&&&&&
&&
&&
in which:
T∇ volume of displacement of the twin-hull ship
TijI solid mass moment of inertia of the twin-hull ship
321 ,, ThThTh XXX hydromechanical forces in the x -, y - and z -directions
654 ,, ThThTh XXX hydromechanical moments about the x -, y - and z -axes
321 ,, TwTwTw XXX exciting wave forces in the x -, y - and z -directions
654 ,, TwTwTw XXX exciting wave moments about the x -, y - and z -axes
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
309
13.3 Hydromechanical Forces and Moments
The equations of motion for six degrees of freedom and the hydromechanical forces andmoments in here, are defined by:
θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
151515
131313
1111111
222
222
222
cba
zczbza
xcxbxaXTh
&&&&&&
&&&
ψψψφφφ
⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
262626
242424
2222222
222
222
222
cba
cba
ycybyaXTh
&&&
&&&
&&&
θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
353535
333333
3131313
222
222
222
cba
zczbza
xcxbxaXTh
&&&&&&
&&&
ψψψφφφ
φφφ
⋅⋅+⋅⋅+⋅⋅+⋅⋅⋅+⋅⋅⋅+⋅⋅⋅+
⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
464646
332
332
332
444444
4242424
222
222
222
222
cba
cybyay
cba
ycybyaX
TTT
Th
&&&
&&&
&&&
&&&
θθθ ⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
555555
535353
5151515
222
222
222
cba
zczbza
xcxbxaXTh
&&&&&&
&&&
ψψψψψψφφφ
⋅⋅⋅+⋅⋅⋅+⋅⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+⋅⋅+=−
112
112
112
666666
646464
6262626
222
222
222
222
cybyay
cba
cba
ycybyaX
TTT
Th
&&&
&&&
&&&
&&&
In here, Ty is half the distance between the centre planes.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
310
13.4 Exciting Wave Forces and Moments
The first order wave potential for an arbitrary water depth h is defined in the new co-ordinatesystem by:
( )[ ][ ] ( )µµωζ
ωsincossin
coshcosh
⋅⋅−⋅⋅−⋅⋅⋅⋅+⋅
⋅−
=Φ bbeab
w ykxkthk
zhkg
This holds that for the port side ( )ps and starboard ( )sb hulls the equivalent components ofthe orbital accelerations and velocities in the surge, sway, heave and roll directions are equalto:
( ) ( )( ) ( )
( ) ( )
( ) ( )µµωζω
µζ
µµωζω
µζ
µµωζµζ
µµωζµζ
sincoscoscos
sincoscoscos
sincossincos
sincossincos
*1
*1
*1
*1
*1
*1
*1
*1
⋅⋅+⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅−⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅+⋅⋅−⋅⋅⋅⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅⋅−=
Tbeaw
Tbeaw
Tbeaw
Tbeaw
ykxktgk
sb
ykxktgk
ps
ykxktgksb
ykxktgkps
&
&
&&
&&
( ) ( )( ) ( )
( ) ( )
( ) ( )µµωζω
µζ
µµωζω
µζ
µµωζµζ
µµωζµζ
sincoscossin
sincoscossin
sincossinsin
sincossinsin
*2
*2
*2
*2
*2
*2
*2
*2
⋅⋅+⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅−⋅⋅−⋅⋅⋅⋅⋅+
=
⋅⋅+⋅⋅−⋅⋅⋅⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅⋅−=
Tbeaw
Tbeaw
Tbeaw
Tbeaw
ykxktgk
sb
ykxktgk
ps
ykxktgksb
ykxktgkps
&
&
&&
&&
( ) ( )( ) ( )
( ) ( )
( ) ( )µµωζω
ζ
µµωζω
ζ
µµωζζ
µµωζζ
sincossin
sincossin
sincoscos
sincoscos
*3
*3
*3
*3
*3
*3
*3
*3
⋅⋅+⋅⋅−⋅⋅⋅⋅+
=
⋅⋅−⋅⋅−⋅⋅⋅⋅+
=
⋅⋅+⋅⋅−⋅⋅⋅⋅−=
⋅⋅−⋅⋅−⋅⋅⋅⋅−=
Tbeaw
Tbeaw
Tbeaw
Tbeaw
ykxktgk
sb
ykxktgk
ps
ykxktgksb
ykxktgkps
&
&
&&
&&
From this follows the total wave loads for the degrees of freedom. In these loads on thefollowing pages, the ''Modified Strip Theory'' includes the outlined terms. When ignoringthese outlined terms the ''Ordinary Strip Theory'' is presented.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
311
The exciting wave forces for surge are:
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ∫
∫
∫
∫
⋅++
⋅+⋅
⋅−⋅+
⋅+⋅⋅⋅
+
⋅+⋅+=
LbFKFK
bww
L be
L
bwwbe
LbwwTw
dxsbXpsX
dxsbpsdx
dMVN
dxsbpsdx
dNV
dxsbpsMX
'1
'1
*1
*1
'11'
11
*1
*1
'11
*1
*1
'111
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
The exciting wave forces for sway are:
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ∫
∫
∫
∫
⋅++
⋅+⋅
⋅−⋅+
⋅+⋅⋅⋅
+
⋅+⋅+=
LbFKFK
bww
L be
L
bwwbe
LbwwTw
dxsbXpsX
dxsbpsdx
dMVN
dxsbpsdx
dNV
dxsbpsMX
'2
'2
*2
*2
'22'
22
*2
*2
'22
*2
*2
'222
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
The exciting wave forces for heave are:
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ∫
∫
∫
∫
⋅++
⋅+⋅
⋅−⋅+
⋅+⋅⋅⋅
+
⋅+⋅+=
LbFKFK
bww
L be
L
bwwbe
LbwwTw
dxsbXpsX
dxsbpsdx
dMVN
dxsbpsdx
dNV
dxsbpsMX
'3
'3
*3
*3
'33'
33
*3
*3
'33
*3
*3
'333
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
312
The exciting wave moments for roll are:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
32
'4
'4
*2
*2
'42'
42
*2
*2
'42
*2
*2
'424
TwTTw
L
bFKFK
bwwL be
L
bwwbe
L
bwwTw
XyXOG
dxsbXpsX
dxsbpsdx
dMVN
dxsbpsdx
dNV
dxsbpsMX
⋅+⋅+
⋅++
⋅+⋅
⋅−⋅+
⋅+⋅⋅⋅
+
⋅+⋅+=
∫
∫
∫
∫
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
The exciting wave moments for pitch are:
( ) ( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( ) ∫
∫
∫
∫
∫
∫
∫
∫
⋅⋅+−
⋅+⋅⋅
⋅−⋅−
⋅+⋅⋅⋅⋅
−+
⋅+⋅⋅−
⋅⋅+−
⋅+⋅⋅
⋅−⋅−
⋅+⋅⋅⋅⋅
−+
⋅+⋅⋅−=
LbbFKFK
bwwbL be
Lbwwb
be
Lbwwb
LbFKFK
bwwL be
Lbww
be
Lbwww
dxxsbXpsX
dxsbpsxdx
dMVN
dxsbpsxdx
dNV
dxsbpsxM
dxbGsbXpsX
dxsbpsbGdx
dMVN
dxsbpsbGdx
dNV
dxsbpsbGMX
'3
'3
*3
*3
'33'
33
*3
*3
'33
*3
*3
'33
'1
'1
*1
*1
'11'
11
*1
*1
'11
*1
*1
'115
ζζωω
ζζωω
ζζ
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
&&
&&&&
&&&&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
313
The exciting wave moments for yaw are:
( ) ( )
( ) ( )
( ) ( )
( ) ( ) 1
'2
'2
*2
*2
'22'
22
*2
*2
'22
*2
*2
'226
TwT
LbbFKFK
bwwb
L be
L
bwwbbe
L
bwwbTw
Xy
dxxsbXpsX
dxsbpsxdx
dMVN
dxsbpsxdx
dNV
dxsbpsxMX
⋅+
⋅⋅++
⋅+⋅⋅
⋅−⋅+
⋅+⋅⋅⋅⋅
+
⋅+⋅⋅+=
∫
∫
∫
∫
ζζωω
ζζωω
ζζ
&&
&&&&
&&&&
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
314
13.5 Added Resistance due to Waves
The added resistance can be found easily from the definitions of the mono-hull ship by usingthe wave elevation at each individual centre line and replacing the heave motion z by:
( ) φ⋅+= Tyzpsz and ( ) φ⋅−= Tyzsbz
13.5.1 Radiated Energy Method
The transfer function of the mean added resistance of twin-hull ships according to the methodof Gerritsma and Beukelman [1972] becomes:
( ) ( )∫ ⋅
+
⋅
⋅−⋅
⋅⋅−=
L
ba
za
a
za
bea
aw dxsbVpsVdx
dMVNkR2*2*'
33'332 2
cosζζω
µζ
with:( ) ( ) ( )( ) ( ) ( )( ) ( )
( ) ( )µµωζω
ζ
µµωζω
ζ
φθθζ
φθθζ
sincossin
sincossin
*3
*3
*3
*3
*3
*
*3
*
⋅⋅+⋅⋅−⋅⋅⋅⋅+
=
⋅⋅−⋅⋅−⋅⋅⋅⋅+
=
⋅+⋅+⋅−−=
⋅−⋅+⋅−−=
Tbeaw
Tbeaw
Tbwz
Tbwz
ykxktgk
sb
ykxktgk
ps
yVxzsbsbV
yVxzpspsV
&
&
&&&&
&&&&
13.5.2 Integrated Pressure Method
The transfer function of the mean added resistance of twin-hull ships according to the methodof Boese [1970] becomes:
( )[ ]
( )[ ]
( )
( )sbzaae
pszaae
bb
w
sbL a
sba
a
xa
bb
w
psL a
sba
a
xa
a
aw
z
z
dxdxdy
hk
xkszg
dxdxdy
hk
xkszg
R
θζζ
θζζ
ζ
ζ
εεθωρ
εεθωρ
ζεµ
ζρ
ζεµ
ζρ
ζ
−⋅⋅⋅⋅∇⋅⋅+
−⋅⋅⋅⋅∇⋅⋅+
⋅
−⋅
⋅⋅
−⋅⋅−⋅⋅−−−⋅⋅⋅
⋅
−⋅
⋅⋅−⋅⋅−⋅⋅
−−−⋅⋅⋅=
∫
∫
cos21
cos21
tanh
coscos21
21
tanh
coscos21
21
2
2
2
2
2
2
2
with:( ) ( )( ) ( )( )( )( ) ( )( ) ( ) φθζ
φθζφθφθ
µµωζζµµωζζ
⋅+⋅+−=⋅−⋅+−=
⋅−⋅−=⋅+⋅−=
⋅⋅+⋅⋅−⋅⋅=
⋅⋅−⋅⋅−⋅⋅=
Tb
Tb
Tbx
Tbx
Tbea
Tbea
yxzsbsbs
yxzpspss
yxzsbz
yxzpsz
ykxktsb
ykxktps
sincoscos
sincoscos
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
315
13.6 Bending and Torsion Moments
According to Newton's second law of dynamics, the harmonic lateral, vertical and torsiondynamic loads per unit length on the unfastened disk of a twin-hull ship are given by:
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )
( )bT
bmTxxbT
bTwbThbT
bbT
bTwbThbT
mbbT
sbTwbThbT
bT
bTwbThbT
xqz
gxyzkxm
xXxXzxq
xzxm
xXxXxq
gzxyxm
AgxXxXxq
bGxxm
xXxXxq
21
'2''
'4
'414
'
'3
'33
''
'2
'22
'
'1
'11
,
2
⋅+⋅+⋅+⋅−⋅⋅−
++=
⋅−⋅−
++=
⋅+⋅−⋅+⋅−
⋅⋅⋅⋅+++=
⋅−⋅−
++=
φψφ
θ
φφψ
φρ
θ
&&&&&&
&&&&
&&&&&&
&&&&
In here:'
Tm mass per unit length of the twin-hull ship'
Txxk local sold mass radius of inertia for roll
sA sectional area of one hull
The calculation procedure of the forces and moments is similar to the procedure given beforefor mono-hull ships.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
316
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
317
14 Numerical Recipes
Some typical numerical recipes, as used in the strip theory program SEAWAY, are describedin more detail here.
14.1 Polynomials
Discrete points can be connected by a first degree or a second degree polynomial, see Figure14.1–1-a,b.
Figure 14.1–1: First and Second Order Polynomials Through Discrete Points
14.1.1 First Degree Polynomial
A first degree - or linear - polynomial, as given in Figure 14.1–1-a, is defined by:( ) bxaxf +⋅=
with in the interval 0xxxm << the following coefficients:( ) ( )
( ) 00
0
0
xaxfb
xxxfxf
am
m
⋅−=
−−
=
and in the interval pxxx <<0 the following coefficients:
( ) ( )
( ) 00
0
0
xaxfb
xx
xfxfa
p
p
⋅−=
−−
=
Notice that only one interval is required for obtaining the coefficients in that interval.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
318
14.1.2 Second Degree Polynomial
A second-degree polynomial, as given in Figure 14.1–1-b, is defined by:( ) cxbxaxf +⋅+⋅= 2
with in the interval pm xxx << the following coefficients:
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) 02
00
00
0
0
0
0
0
xbxaxfc
xxaxx
xfxfb
xx
xxxfxf
xx
xfxf
a
pp
p
mp
m
m
p
p
⋅−⋅−=
+⋅−−−
=
−−−
−−−
=
Notice that two intervals are required for obtaining these coefficients, valid in both intervals.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
319
14.2 Integrations
Either the trapezoid rule or Simpson’s general rule can carry out numerical integration.SEAWAY uses Simpson's general rule as a standard. Then, the integration has to be carriedout over a number of sets of two intervals, see Figure 14.1–1-b. Numerical inaccuracies canbe expected when 00 xxxx pm −<<− or mp xxxx −<<− 00 . In those cases the trapezoid rulehas to be preferred, see Figure 14.1–1-a.SEAWAY makes the choice between the use of the trapezoid rule and Simpson's ruleautomatically, based on the following requirements:
Trapezoid rile, if: 2.00 <−−
mp
m
xxxx
or 0.50 >−−
mp
m
xxxx
Simpson’s rule, if: 0.52.0 0 <−−
<mp
m
xxxx
14.2.1 First Degree Integration
First-degree integration - carried out by the trapezoid rule, see Figure 14.1–1-a - means theuse of a linear function:
( ) bxaxf +⋅=
The integral over the interval 0xxp − becomes:
( ) ( )
p
pp
x
x
x
x
x
x
xbxa
dxbxadxxf
0
00
2
21
⋅+⋅⋅=
⋅+⋅=⋅ ∫∫
with:( ) ( )
( ) 00
0
0
xaxfb
xx
xfxfa
p
p
⋅−=
−−
=
Integration over two intervals results into:
( ) ( ) ( ) ( ) ( ) ( ) ( )2
000 ppmpmmx
x
xfxxxfxxxfxxdxxf
p
m
⋅−+⋅−+⋅−=⋅∫
14.2.2 Second Degree Integration
Second-degree integration - carried out by Simpson's rule, see Figure 14.1–1-b – has to becarried out over a set of two intervals. At each of the two intervals, the integrand is describedby a second-degree polynomial:
( ) cxbxaxf +⋅+⋅= 2
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
320
Then the integral becomes:
( ) ( )p
m
p
m
p
m
x
x
x
x
x
x
xcxbxa
dxcxbxadxxf
⋅+⋅⋅+⋅⋅=
⋅+⋅+⋅=⋅ ∫∫
23
2
21
31
with:( ) ( ) ( ) ( )
( ) ( ) ( )
( ) 02
00
00
0
0
0
0
0
xbxaxfc
xxaxx
xfxfb
xx
xxxfxf
xx
xfxf
a
pp
p
mp
m
m
p
p
⋅−⋅−=
+⋅−−−
=
−−−
−−−
=
Some algebra leads for the integration over these two intervals to:
( )
( )
( )( ) ( ) ( )
( )
3
2
2
2
0
00
000
2
0
00
mp
pp
mp
pm
mp
mm
pm
x
x
xx
xfxx
xxxx
xfxxxx
xx
xfxx
xxxx
dxxfp
m
−⋅
⋅−
−−−
+⋅−⋅−⋅
−
+⋅−
−−−
=⋅∫
14.2.3 Integration of Wave Loads
The total wave loads can be written as:( )
( ) ( )tFtF
tFF
eFwaeFwa
Fewaw
ww
w
⋅⋅⋅−+⋅⋅⋅=
+⋅⋅=
ωεωε
εω
ζζ
ζ
sinsincoscos
cos
The in-phase and out-phase parts of the total wave loads have to be obtained fromlongitudinal integration of sectional values. Direct numerical integration of bFwa dxF
w⋅⋅
ζε 'cos' and bFwa dxF
w⋅⋅−
ζε 'sin' over the ship length, L , require integration
intervals, bx∆ , which are much smaller than the smallest wave length, 10minλ≤∆ bx . Thismeans that a large number of cross sections are required.
This can be avoided by writing the integrands in terms of ( ) dxxxf ⋅⋅ cos1 and ( ) dxxxf ⋅⋅ sin2 ,
in which the integrands ( )xf 2,1 vary much slower over short wave lengths as the harmonicsitself.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
321
The functions ( )xf 2,1 can be approximated by a second degree polynomial:
( ) cxbxaxf +⋅+⋅= 2
When making use of the general integral rules:
( ) xxxxdxxx
xxxdxxx
xdxx
sin2cos2cos
sincoscos
sincos
22 ⋅−+⋅⋅+=⋅⋅
⋅+=⋅⋅
+=⋅
∫∫
∫
and:
( ) xxxxdxxx
xxxdxxx
xdxx
cos2sin2sin
cossinsin
cossin
22 ⋅−−⋅⋅+=⋅⋅
⋅−=⋅⋅
−=⋅
∫∫
∫
the following expressions can be obtained for the in-phase and out-phase parts of the waveloads, integrated from mx through px , so over the two intervals mxx −0 and 0xxp − :
( ) ( )
( )
( )( ) ( )[ ]
( ) ( )
( )
( )( ) ( )[ ] p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
p
m
xx
x
x
x
x
x
x
x
x
x
x
x
x
xx
x
x
x
x
x
x
x
x
x
x
x
x
xbxaxaxf
dxxcdxxxbdxxxa
dxxcxbxa
dxxxfdxxF
xbxaxaxf
dxxcdxxxbdxxxa
dxxcxbxa
dxxxfdxxF
sin2cos2
sinsinsin
sin
sin
cos2sin2
coscoscos
cos
cos
2
2
2
2
⋅+⋅⋅+⋅−−=
⋅⋅+⋅⋅⋅+⋅⋅⋅=
⋅⋅+⋅+⋅=
⋅⋅=⋅
⋅+⋅⋅+⋅−+=
⋅⋅+⋅⋅⋅+⋅⋅⋅=
⋅⋅+⋅+⋅=
⋅⋅=⋅
∫∫∫
∫
∫∫
∫∫∫
∫
∫∫
with coefficients obtained by:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
322
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) 02
00
00
0
0
0
0
0
xbxaxfc
xxaxx
xfxfb
xx
xxxfxf
xx
xfxf
a
pp
p
mp
m
m
p
p
⋅−⋅−=
+⋅−−−
=
−−−
−−−
=
With this approach, the wave loads on a barge, for instance, can be calculated by using twosection intervals only for any length of the barge.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
323
14.3 Derivatives
First and second degree functions, of which the derivatives have to be determined, have beengiven in Figure 14.3–1-a,b.
Figure 14.3–1: Determination of Longitudinal Derivatives
14.3.1 First Degree Derivative
The two polynomials - each valid over two intervals below and above 0xx = - are given by:
for 0xx < : ( ) mm bxaxf +⋅=for 0xx > : ( ) pp bxaxf +⋅=
The derivative is given by:
for 0xx < :( )
madx
xdf=
for 0xx > :( )
padx
xdf=
It is obvious that, generally, the derivative at the left-hand side of 0x - with index m (minus) -
and the derivative at the right-hand side of 0x - with index p (plus) - will differ:
zero ofright or (plus zero ofleft or (minus 00 pxxmxx dxdf
dxdf
≠
==
A mean derivative dxdf at 0xx = can be obtained by:
( ) ( )
mp
pxxp
mxxm
xx xx
dxdfxx
dxdfxx
dxdf
−
⋅−+
⋅−
=
==
=
00
0
00
14.3.2 Second Degree Derivative
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
324
The two polynomials - each valid over two intervals below and above 0xx = - are given by:
for 0xx < : ( ) mmm cxbxaxf +⋅+⋅= 2
for 0xx > : ( ) ppp cxbxaxf +⋅+⋅= 2
A derivative of a second degree function:( ) cxbxaxf +⋅+⋅= 2
is given by:( )
bxadx
xdf+⋅⋅= 2
This leads for 0xx < to:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )201021
212
10
102
21
101021
201021
212
10
102
21
201021
212
10
102
21
211021
2
2
0
1
2
mmmm
mmm
mmm
mmmm
xx
mmmm
mmm
mmm
xx
mmmm
mmm
mmm
mmmmm
xx
xxxxxx
xfxfxx
xfxfxx
xfxfxxxx
dxdf
xxxxxxxfxfxx
xfxfxx
dxdf
xxxxxxxfxfxx
xfxfxx
xfxfxxxx
dxdf
m
m
−⋅−⋅−
−⋅−−
−⋅−+
−⋅−⋅−⋅+
=
−⋅−⋅−
−⋅−+
−⋅−+
=
−⋅−⋅−
−⋅−+
−⋅−−
−⋅−⋅−⋅−
=
=
=
=
and for 0xx > to:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
325
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )021201
012
12
122
01
121201
021201
012
12
122
01
021201
012
12
122
01
011221
2
2
0
1
2
xxxxxx
xfxfxx
xfxfxx
xfxfxxxx
dxdf
xxxxxx
xfxfxx
xfxfxx
dxdf
xxxxxx
xfxfxx
xfxfxx
xfxfxxxx
dxdf
pppp
ppp
ppp
ppppp
xx
pppp
ppp
ppp
xx
pppp
ppp
ppp
pppmp
xx
m
m
−⋅−⋅−
−⋅−−
−⋅−+
−⋅−⋅−⋅+
=
−⋅−⋅−
−⋅−+
−⋅−+
=
−⋅−⋅−
−⋅−+
−⋅−−
−⋅−⋅−⋅−
=
=
=
=
Generally, the derivative at the left-hand side of 0x - with index m (minus) - and the
derivative at the right-hand side of 0x - with index p (plus) - will differ:
zero ofright or (plus zero ofleft or (minus 00 pxxmxx dxdf
dxdf
≠
==
A mean derivative dxdf at 0xx = can be obtained by:
pm
pxxp
mxxm
xx dd
dxdfd
dxdfd
dxdf
+
⋅+
⋅
=
==
=
00
0
with:
( )
( )
( )( )01
0212
01
10
2021
10
32
32
xx
xxxx
xxd
xx
xxxx
xxd
p
ppp
p
p
m
mmm
m
m
−⋅
−⋅
−−−
=
−⋅
−⋅
−
−−=
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
326
14.4 Curve Lengths
Discrete points, connected by a first degree or a second degree polynomial, are given inFigure 14.4–1-a,b.
Figure 14.4–1: First and Second Order Curves
The curve length follows from:
( ) ( )∫
∫
+=
=
p
m
p
m
x
x
x
xmp
dydx
dss
22
14.4.1 First Degree Curve
The curve length of a first degree curve, see Figure 14.4–1-a, in the two intervals in the region
pm xxx << is:
( ) ( ) ( ) ( )20
20
20
20 yyxxyyxxs ppmmmp −+−+−+−=
14.4.2 Second Degree Curve
The curve length of a second degree curve, see Figure 14.4–1-b, in the two intervals in theregion pm xxx << is:
++−+⋅−
++++⋅+
⋅=2
112
11
200
200
2
1ln1
1ln1
pppp
pppppsmp
with:
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
327
( ) ( )
( ) ( )
( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )( ) ( )
( ) ( )4
sincos
sincossincos
sincossincos
sincos
sincos2
sin
cos
00
00
00
00
001
000
22
22
αα
αααααααα
αααα
π
α
α
⋅−+⋅−=
⋅−+⋅−⋅−+⋅−
−
⋅−+⋅−⋅−−⋅−
=
⋅−+⋅−⋅−+⋅−
⋅+=
−+−
−=
−+−
−=
yyxxp
yyxxyyxx
yyxxxxyy
p
yyxx
yyxxp
yyxx
yy
yyxx
xx
pp
pp
mm
mm
mm
pp
mpmp
mpmp
mp
mpmp
mp
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
328
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
329
15 References
Abramowitz and Stegun [1964]M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions with Formulas and Mathematical Tables,National Bureau of Standards, U.S. Department of Commerce, Applied MathematicsSeries 55, June 1964.
Adegeest [1994]L.J.M. Adegeest,Non-linear Hull Girder Loads in Ships,PhD Thesis, 1994, Delft University of Technology, The Netherlands.
Athanassoulis and Loukakis [1985]G.A. Athanassoulis and T.A. Loukakis,An Extended-Lewis Form Family of Ship Sections and its Applications to SeakeepingCalculations,International Shipbuilding Progress, 1985, Vol. 32, No. 366.
Bales [1983]S.L. Bales,Wind and Wave Data for Seakeeping Performance Assessment,Technical Report, 1983, Prepared for the 17th ITTC Seakeeping Committee, Athens,Greece.
Boese [1970]P. Boese,Eine Einfache Methode zur Berechnung der Wiederstandserhöhung eines Schiffes inSeegang,Technical Report 258, 1970, Institüt für Schiffbau der Universität Hamburg, Germany.
van den Bosch and Vugts [1966]J.J. van den Bosch and J.H. Vugts,Roll Damping by Free Surface Tanks,Technical Report 83-S, 1966, Netherlands Ship Research Centre TNO, ShipbuildingDepartment, Delft, The Netherlands.
Conolly [1974]J.E. Conolly,Standards of Good Seakeeping for Destroyers and Frigates in Head Seas,International Symposium on the Dynamics of Marine Vehicles and Structures inWaves, 1974, No. 8, London, U.K.
Dimitrieva [1994]I. Dimitrieva,DELFRAC, 3-D Potential Theory Including Wave Diffraction and Drift Forces Actingon the Structures,(Description of the 3-D Computer Code DELFRAC, written by J.A. Pinkster),Technical Report 1017, 1994, Delft University of Technology, Ship HydromechanicsLaboratory, The Netherlands.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
330
Faltinsen and Svensen [1990]O.M. Faltinsen and T. Svensen,Incorporation of Seakeeping Theories in CAD,International Symposium on CFD and CAD in Ship Design, 1990, MARIN,Wageningen, The Netherlands.
Frank [1967]W. Frank,Oscillation of Cylinders in or below the Free Surface of Deep Fluids,Technical Report 2375, 1967, Naval Ship Research and Development Centre,Washington DC, U.S.A.
Fukuda [1962]J. Fukuda,Coupled Motions and Midship Bending Moments of a Ship in Regular Waves,Journal of the Society of Naval Architects of Japan, 1962.
Gerritsma and Beukelman [1972]J. Gerritsma and W. Beukelman,Analysis of the Resistance Increase in Waves of a Fast Cargo-ship,International Shipbuilding Progress, 1972, Vol. 18, No. 217.
Grim [1953]O. Grim,Berechnung der durch Schwingungen eines Schiffskörpers Erzeugten Hydro-dynamischen Kräfte,Jahrbuch der Schiffsbautechnischen Gesellschaft, 1953, No. 47, pages 277-299.
Grim [1955]O. Grim,Die Hydrodynamischen Kräfte beim Rollversuch,Schiffstechnik, 1955, No. 3.
Grim [1956]O. Grim,Die Schwingungen von schwimmenden zweidimensionalen Körper,Technical Report, 1956, HSVA-Bericht No. 1090, HSVA, Hamburg, Germany.
Grim [1957]O. Grim,Die Schwingungen von schwimmenden zweidimensionalen Körper,Technical Report, 1957, HSVA-Bericht No. 1171, HSVA, Hamburg, Germany.
Haskind [1957]M. Haskind,The Exciting Forces and Wetting of Ships in Waves (in Russian),Izvestia Akademii Nauk SSSR, Otdelenie Tskhnicheshikh Nauk, 1957, No. 7, pages65-79.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
331
Huijsmans [1996]R.H.M. Huijsmans,Motions and Drift Forces on Moored Vessels in Current,PhD Thesis, 1996, Delft University of Technology, The Netherlands.
Ikeda, Himeno and Tanaka [1978]Y. Ikeda, Y. Himeno and N. Tanaka,A Prediction Method for Ship Rolling,Technical Report 00405, 1978, Department of Naval Architecture, University of OsakaPrefecture, Japan.
John [1950]F. John,On the Motions of Floating Bodies, Part II,Comm. on Pure and Applied Mathematics, 1950, Vol. 3, Interscience Publishers, Inc,New York, pages 45-100.
de Jong [1973]B. de Jong,Computation of Hydrodynamic Coefficients of Oscillating Cylinders,Technical Report 145-S, 1973, Netherlands Ship Research Centre TNO, ShipbuildingDepartment, Delft, The Netherlands.
Journée [1991]J.M.J. Journée,Motions of Rectangular Barges,Proceedings 10th International Conference on Offshore Mechanics and ArcticEngineering, 1991, Stavanger, Norway.
Journée [1992]J.M.J. Journée,Strip Theory Algorithms, Revised Report 1992,Technical Report 912, 1992, Delft University of Technology, Ship HydromechanicsLaboratory, The Netherlands.
Journée [1997]J.M.J. Journée,Liquid Cargo and Its Effect on Ship Motions,STAB'97 Conference, 1997, Varna, Bulgaria, Internet: www.shipmotions.nl.
Journée [2001]J.M.J. Journée,Verification and Validation of Ship Motions Program SEAWAY,Technical Report 1213a, 2001, Delft University of Technology, Ship HydromechanicsLaboratory, The Netherlands, Internet: www.shipmotions.nl.
Journée and van ‘t Veer [1995]J.M.J. Journée and A.P. van ‘t Veer,First Order Wave Loads in Beam Waves,
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
332
Proceedings 5th International Offshore and Polar Engineering Conference, 1995, TheHague, the Netherlands, Internet: www.shipmotions.nl.
Journée and Massie [2001]J.M.J. Journée and W.W. Massie,Offshore Hydromechanics, Lecture Notes,First Edition, January 2001, Delft University of Technology, Ship HydromechanicsLaboratory, The Netherlands, Internet: www.shipmotions.nl.
Keil [1974]H. Keil,Die Hydrodynamische Kräfte bei der periodischen Bewegung zweidimensionalerKörper an der Oberfläche flacher Gewasser,Bericht Nr. 305, 1974, Institut für Schiffbau der Universität Hamburg, Deutschland.
von Kerczek and Tuck [1969]C. von Kerczek and E.O. Tuck,The Representation of Ship Hulls by Conformal Mapping Functions,Journal of Ship Research, 1969, Vol. 13, No. 4.
Kirsch [1969]M. Kirsch,Zur Tauch- und Stampfbewegung eines Schiffes,Schiffstechnik, Band 16, 1969.
Kirsch [1969]M. Kirsch,Die Berechnung der Bewegungsgrößen der gekoppelten Tauch- und Stampf-schwingungen nach der erweiterten Streifenmethode von Grim und die Berechnungder Wahrscheinlichkeit für das Überschreiten bestimmter Schranken durch dieseGrößen,Bericht Nr. 241, 1969, Institut für Schiffbau der Universität Hamburg, Deutschland.
Korkut and Hebert [1970]M. Korkut and E. Hebert,Some Notes on Static Anchor Chain Curve,Offshore Technology Conference, 1970, No. OTC 1160, Dallas, Texas, U.S.A.
Korvin-Kroukovsky [1955]B.V. Korvin-Kroukovsky,Investigations of Ship Motions in Regular in Regular Waves,Transactions SNAME, 1955, No. 63, pages 386-435.
Korvin-Kroukovsky and Jacobs [1957]B.V. Korvin-Kroukovsky and W.R. Jacobs,Pitching and Heaving Motions of a Ship in Regular Waves,Transactions SNAME, 1957, No. 65, pages 590-632.
Kumai [1959]T. Kumai,
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
333
Added Mass Moment of Inertia Induced by Torsional Vibration of Ships,Reports of the Research Institute for Applied Mechanics, 1959, Vol. VII, No. 28.
Landweber and de Macagno [1957]L. Landweber and M.C. de Macagno,Added Mass of Two-dimensional Forms Oscillating in a Free surface,Journal of Ship Research, Vol. 1, 1957.
Landweber and de Macagno [1959]L. Landweber and M.C. de Macagno,Added Mass of Two-dimensional Forms Oscillating in a Free surface,Journal of Ship Research, Vol. 1, 1957 and Vol. 2, 1959.
Lamb [1932]H. Lamb,Hydrodynamics,Dover Publications, Inc., New York, 1932, 6th Edition.
Lewis [1929]F.M. Lewis,The Inertia of Water Surrounding a Vibrating Ship,Transactions SNAME, Vol. 37, 1929.
Lloyd [1989]A.R.J.M. Lloyd,Seakeeping, Ship Behaviour in Rough Weather,ISBN 0-7458-0230-3. 1989, Ellis Horwood Limited, Market Cross House, CooperStreet, Chichester, West Sussex, P019 1EB England.
Miller [1974]E.R. Miller,Unknown Title of a Report on Roll Damping,Technical Report 6136-74-280, 1974, NAVSPEC.
Newman [1962]J.N. Newman,The Exciting Forces on Fixed Bodies in Waves,Journal of Ship Research, 1962, Vol. 6, No. 4, pages 10-17.
Ochi [1964]M.K. Ochi,Prediction of Occurrence and Severity of Ship Slamming at Sea,Proceedings of 5th O.N.R. Symposium, 1964, Bergen, Norway.
Ogilvie [1963]T.F. Ogilvie,First and Second Order Forces on a Cylinder Submerged under a Free Surface,Journal of Fluid Mechanics, 1963, pages 451-472.
Porter [1960]
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
334
W.R. Porter,Pressure Distribution, Added Mass and Damping Coefficients for CylindersOscillating in a Free Surface,Technical Report 82, 1960, University of California, Institute of EngineeringResearch, Berkeley, California, U.S.A.
Reed and Nowacki [1974]M.A. Reed and H. Nowacki,Interactive Creation of Fair Ship Lines,Journal of Ship Research, 1974, Vol. 18, No. 2.
Salvesen, Tuck and Faltinsen [1970]N. Salvesen, E.O. Tuck and O.M. Faltinsen,
Ship Motions and Sea Loads,SNAME, 1970, Vol. 78, pp 250-287
Sargent and Kaplan [1974]T.P. Sargent and P. Kaplan,Modifications to Lloyds Register of Shipping Strip Theory Computer Program (LR2570,Technical Report 74-103, 1974, Oceanics Inc.
Schmitke [1978]R.T. Schmitke,ROLLRFT, a Fortran Program to Predict Ship Roll, Sway and Yaw Motions in ObliqueWaves, Including the Effect of Rudder, Fin and Tank Roll Stabilizers,Technical Report 78/G, 1978, Defence Research Establishment, Atlantic, DartmouthN.S., Canada.
Stoker [1957]J.J. Stoker,Water Waves,Interscience publishers, Inc, New York, 1957.
Tasai [1959]F. Tasai,On the Damping Force and Added Mass of Ships Heaving and Pitching,Technical Report, Research Institute for Applied Mechanics, Kyushu University,Japan, 1959, Vol. VII, No 26.
Tasai [1960]F. Tasai,Formula for Calculating Hydrodynamic Force on a Cylinder Heaving in the FreeSurface, (N-Parameter Family),Technical Report, Research Institute for Applied Mechanics, Kyushu University,Japan, 1960, Vol. VIII, No 31.
Tasai [1961]F. Tasai,
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
335
Hydrodynamic Force and Moment Produced by Swaying and Rolling Oscillation ofCylinders on the Free Surface,Technical Report, Research Institute for Applied Mechanics, Kyushu University,Japan, 1961, Vol. IX, No 35.
Tasai [1965]F. Tasai,Ship Motions in Beam Waves,Technical Report, Research Institute for Applied Mechanics, Kyushu University,Japan, 1965, Vol. XIII, No 45.
Tasai [1969]F. Tasai,Improvements in the Theory of Ship Motions in Longitudinal Waves,Proceedings 12th I.T.T.C., 1969.
Tasaki [1963]R. Tasaki,Researches on Seakeeping Qualities of Ships in Japan, Model Experiments in Waves,On the Shipment of Water in Head Waves,Journal of the Society of Naval Architects of Japan, 1963, Vol. 8.
Timman and Newman [1962]R. Timman and J.N. Newman,The Coupled Damping Coefficients of a Symmetric Ship,Journal of Ship Research, 1962, Vol. 5, No. 4, pages 1-7.
Troost [1955]L. Troost,A Simplified Method for Preliminary Powering of Single Screw Merchant Ships,New England Section of the Society of Naval Architects and Marine Engineers,October Meeting, 1955.
Ursell [1949]F. Ursell,On the Heaving Motion of a Circular Cylinder on the Surface of a Fluid,Quarterly Journal of Mechanics and Applied Mathematics, Vol. II, 1949.
Vantorre and Journée [2003]M. Vantorre and J.M.J. JournéeValidation of the Strip Theory Code SEAWAY by Model Tests in Very Shallow Water,Colloquium on Numerical Modelling, 23-24 October 2003, Antwerp, Belgium,Internet: www.shipmotions.nl.
.
Verhagen and van Wijngaarden [1965]J. Verhagen and L. van Wijngaarden,Non-linear Oscillations of Fluid in a Container,Journal of Fluid Mechanics, 1965, Vol. 22, No. 4, pages 737-751.
Theoretical Manual of “SEAWAY for Windows” TUD Report No. 1370J.M.J. Journée and L.J.M. Adegeest Revision: 14-12-2003
336
Wehausen and Laitone [1960]J.V. Wehausen and E.V. Laitone,Surface Waves,Handbuch der Physik, edited by S. Fluegge, Vol. 9, Fluid Dynamics 3, SpringerVerlag, Berlin, Germany, pages 446–778, 1960.
de Zwaan [1977]A.P. de Zwaan,A Method for Solving Equations with Constant or Variable Right Hand Terms (inDutch),Technical Report, 1977, No. 5, Delft University of Technology, Department forShipbuilding and Shiphandling, Centrale Werkgroep Wiskunde, The Netherlands.