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AD 7 ..... .. ICHIGAN UNIV ANN ARBOR DEPT OF NAVAL ARCHITECTURE --ETC F/G 20/4HEADSEAS WAVE DIFFRACTION COMPUTER PROGRAM. USER MANUAL, (U)
UNLSIID AUG 79 R F BECK N00014-78-C-0109UuuuuuuuFuuuuN
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USER ANUA ADSEAS WAVE DIFFRAC
Robert F Beck
Ii I iI Dc
This research was carried out under the
Naval Sea Systems Command
General Hydromechanics Research Program
Subproject SR 023 01 01, administered by the
Naval Shio Research and DevploDment CenterContraqv "W'-0I o1-8C rO
Reproduction in whole or in part permittedfor any purpose of the United States Government
Approved for public release; distribution unlimited
LAJ
CI
-Department of Naval Architectureand Marine EngineeringCollege of Engincering
The University of MichiganAnn Arbor, Michigan 48109
80
~. t.
DISCLAIMER NOTICE
THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DDC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOT
REPRODUCE LEGIBLY.
i -
IT
TABLE OF CONTENTS
Introduction 1
Theoretical Analysis 1
Numerical Techniques 7
Required Input 10
Output 12
Limitation of the Program 14
List of Subroutines 16
References 18
Appendix I
Program Listing of Diffracted 19
Forces Program
Appendix II
Input Listing of Ore Carrier 94
S.J. Cort
Appendix III
Output Listing for Ore Carrier 98
S.J. Cort A.,
•I-I
puirp;ase of tic. cc..puter program is, Lo co~apnte thre dynamic preazs..r-
distrLjution and relazed cuantities of interest due to the diffraction of sin-
usoidal head waves. The nethod of computation is based on slender-body theory.
The theoretical analysis -'s based on the assumption that the ship is slender.
In addition', it is assumed that the incide nt waves are of snail amplitude and
thelrr va:eength is short relative to the ship length. In -the next section
we s.-a- -gIve -a brief sumanary of' the thecretical. ana'ysls in order to far-ili-
tate unae-rstanding of the computer program. Letails of the theoretical
ar~i~<.-.ay be found in Beck (1979). In the followiing sections, details of
t~ec~iteirhv'ique and the computer progran will b~e discussed.
7H- AALY S TS
.ecoordinate syste. is shown in Figure 1. The origin is at the bow
with --he x-axis pointing aft. The z-axis is vertical upward and the x-y plane
z
waves
Figure 1. Coordinate System
is coincident with the calm-water plane. In this coordinate system, the
incident wave potential is gziven by
-W -t
where ivxgavz -iV
I W
acceleration gravity
a = wave amplitudev = wave number
W21g
U = forward speed
Wo = absolute wave frequency
Se frequency of encounter
= W + W2U/g
The diffraction potential is written as
D(x,y,z)e i et (2)
To find D we must solve both the near-field and far-field problems. In the
far-field, the solution is represented by a line distribution of pulsating
sources, the strength of which is given by
oxWei (wet-Vx)
The source strength, a(x), is found by solving the Volterra integral equaticr.
xga + =0x) c - a (3)
where
-i/2 T = 0
1 - T>>1/42/F2,'
V -i 7T/4
27T(1+2 T*) e
WeUg
g
-2-
9J
lioX) : near field source strenoth.
The near-field source strength is found by solving the near field problem and
may be assumed known when solving equation (3).
In the near-field, the diffraction potential is written as
, x,y,z)e
B-t!.- first and second order near-field solutions must be letermined. The
-cer sclution is simply the negative of the incident wave. The second-
CJ: Sol"*1.;? '!1st satisfy the Jielid.mcltz equa1,iun sLabect to boundary condl-
4c ,.-" n the frec surface and thu body srfac.m At irfirl itS the near-field
:i:. i zast match the far-field solution.
As d4scussed in Beck (1979), the two-term near-field solution can be
.ritten as
-(x,y,z)-- e + A(x) e +Bo(X)S(Y,Z) + Bn(X)On(Yz)
W0 n=1
(5)
-.c coefficient A(x) is determined ,y ratrhing ,itr' the far-field solution and
w-, find
o (X) (6)
The terms inside the square brackets represent Ursell's (1968a) solution
to the Helmholtz equation subject to the free surface and body boundary condi-
t!ons. S(y,z) is a source-like term and On are wave-free potentials. The coef-
ficients B0 , Bn are determined by satisfying the body boundary condition, which,
ir this case, is obtained by setting the normal derivatives of the terms in
square brackets equal to zero on the body surface.
The determination of the coefficients B0 , Bn for arbitrary body shapes
is often very difficult. Troesch (1976) has avoided this problem by using an
integral-equation technique. Troesch's technique and computer program were
1. 7 '.s type of solution is often called a multi-pole expansion.
-3-
actually developed for the solution of the oblique seas case, but they can be
modified for use in the present problem. We first write Ursell's solution in
the form
+Vz(y,z;x) = e + ¢(y,z;x)
(7)
The potential ¢(y,z,x) must satisfy the Helmholtz equation
ay 2 + v22 = 0+ (8)
subject to boundary conditions on the free surface and the body. At infinity,
the behavior of T must match the far-field solution. The free-surface boundary
condition is
30 - V(D = 0 on z=03z
On the body, the boundary condition is
-- evz + (zY;x)] = 0
or
Di N 3e on h(y,z;x) = 03 N D N(9 )
where
h(y,z,x) = equation of body surface in y-x plane
N = two-dimensional unit normal to body surface in the y-z plane
= NN 3)
The positive sense of N is into the body.
The problem for the potential 4 is now equivalent to the boundary value
problem solved by Troesch. He irites the solution as a distribution over the
body surface of two-dimensional sources. Thus, we have
(D(y,z;X) = dl y(n,C)G(y,z;n,C)
c(x) (0)
-4-
where the line integral is ~Qnalong the bod'. contour. y (q,r ' is the two-
dimen~sional source streng--, which is determino&. by satisfyinE T.he body boundily
condition. G(Y,z;fl,C) is -.he Green function, w.- Lch satisfies the Helmholtz
equation and the free surface boundary condition (for details see Ursell
(1968b)). By matching the IL-iting values of bo-.h the multipole solution and
the Green f'nction solutic: for large values of y, it can be shown that.
B0 "X) -2 f dI y(h. )e"
C (x)
j Equation ill) allows the deterraination of the net- r-field source strength, B0(x),
by using the integral-equation solution technique without developing the multi-
pole expansion.
Mhe pressure acting cn the body is found using the linearized Bernoulli
equation, which may be wrlitten as
- = eiWet
=C + -) (; +- ) ie
O(iwe +U ax I ID (2
Sub:;tituting the express'-:nS for i: and %D into equation (12) and retaininr_
only the lowest-order ter-.s (see Beck (1979)), we find the following expression
for the nondimensional pressure amplitude:
p ga
where
aYWWa*(x) =
ga
(y,z;x) =e + Oiy,z~x)
Likewise, we find the linearized wave amplitude in the near field is given
by tho expression
(X,y) = i (x) (y,O;x)ea TrBO (x)
(14)
The exciting forces and moments are found by integrating the pressure over
the body surface. As shown in Beck (1979), the vertical, sectional exciting
force can be written as
f W - iVX IU Cr (x__ ) dZ (y,z;x)N ] e. Vf 3 (x)e = i B0 B O ( x ) f 3
C(x)
(15)
The total heave exciting force and pitch moment about midship are given
by the following integrals along the ship length:
F 3 = dx f3 (x) e
0 (16)
L
F 5 = f dx (L/2-x) f3 (x)e
-i vx
0 (17)
where
F3 = total heave exciting force
F 5 = total pitch exciting moment
L = ship length
The wave induced bending moment is found by twice integrating the vertical
exciting force up to the desired station. Setting the shear at the bow equal
to zero and integrating by parts once, we arrive at
xr -iv&
BM -J d&(E-x) f 3 ()e
0
In the computer program, the value of x, the station at which the bending
7foment is evaluated, is specified by the user.
-6-
j
Fo- ;," :- c; heuplicate:, ci the prtviouo,- c; La p-rraile. .
body), i,.2 ie,;ults @1 o': oi re'.'ous s~l--tion ar(-, co-~drt; ia i:esolvirs ,r
near- iie Id prolheu :. ricart- ield probie-' is; solved ui>th- -:cthod of
Troen-ch (1??9). Tea- lt-Q-h' etho~d hras been dco suite-j shr (rch
(~y,/ 1 ) ~ 'Lt T-ill not 1-, discusL(-1 hecr. -{C uou-r -
ha2 2Lf- to r~in jiead seas by eliiniinatirng the: J.,,aginar1Y !ar -I l'
:r~r''u:ti-oand zetting k=v. Many of' Troesch's subroutines have e'uec
droe:hers need ud o-nly slight modification. It, slho:Uld -1-, noted t--?
I rt .-. bcr n double, pr ci; ion, so thc'. -c.- iohc
mo .e ihe v .:ri :1.es in the present p jrograii anu h -f-coi n
Tro --'Lne Is nnecessa:ry. This fewitching occurs in UBT' ~~7
and 73IIFO-7U,-T':T~A
i'o .curacy, theo Volterra integral equation !-:. stl[,A sL'ILI.:ci-
along th ~'leng-"h -.. an ,Just the .;nput stations. T1,e lipu'.
located a--' XAxis(I). --ne stcations for the integral kcqua' V on arK bseccc:.
calculaticL-s are louc-td at XI(I). T'he increased numb': of ttat-1,,ns is cp:
in SUR'P2r,.' Jj;C'u subroutine inserts more an 5Z -, n'.Oeb-f To
snacir.- iF- aorjntL cosine diE-tribu-.io.n alon-;: the ent. Th ,ze
stations are neve-r closer than ± EPSIL to the input stati.on. At f-;resent P:
is set ecr.al to ~o-' the ship length. Figure 2 show.. hetca sys Tr
along the lengrth. in -Ie figure, there are 7 input siLtcns, ari 1-3 X-statftor.S.
The array TSNUtj-(I) i zdto store the riruber of the XI--statlon a-t each irz,.t
station. i'or example, in Figure 2 ITNU(4)7. The values ('if the various cruartitics-
in'putstationnumber -1 2 3 15C7
Xf statiof1i2 3 4 5 6 7 8 9 10 11 1213Number
Figure 2. Numbering of the XAXIS(I) and XI(I)axis systems
-7-
which are computed in the two-dimensional problem at thc input station are
determined at the XT-sta±on by cubic-spline interpolation in SUBROUTINE
INTRPL.
The 7olterra integral ecuatlon (equation (3)) is solved at each XI station
by a marching process starti~ng a: the bow. o(x) is assumed to vary linearly
over each segment. The value of o(x) between the j+l and j station is
then i e by
(E- -l-C
A Oj+l ( . j (19)J J
where
0, = value of O(x) a- station number jU
,= value of c(x) at station number j+l
j+1 il
To deveioD the marching process w;e first rewrite the integral equation as
_a + -- k+1
S j+l L1Bo+ 1 k=1Xk ]+l
(20)where equals the value of at station number j-l. The integral in
equation (20) can be evaluated analytically by substituting equation (19) as
followsXk+1
].' f d& ( k)ok+l + ( k+l-)Ok
Xk +
Gk+1 GEGCCAk [G(kP j+l 'k+l k j+l,1 k
(21)
k[G(" - G( . 4)Ak k+1' j+1' k+1 k+lj+ k
where G( =,, Y) (2a- 3 (22)3 3Multiplying both sides cf ecuation (20) by 2rB0 j and substituting equation
(22) we arrive at the fcllowing expression for 0j+l in terms of all the pre-
vious o..
-8-
j41 j'> !+ Ij p1 +i I +
=-2TB 2 - 2-B F,~+, 2 .; :oj ! , -,] ( , . ,-,( )-- 'I . , , )}Oj.; " C :+ A-. "j+ j . j+ 1 j":+1 j+
0 3
2BoA 1 k4 k "1t Y 41 -k
-k{G( k+I '+j .]'k+ I)-G(' ' j 'k)
(2-))
In equaWli (23) the funcici '(eSy) is ,1Iven b.' equation (22),
To s-ar the marchin- -reess, it is assumed that the ner-fie2d and th----
dimensic:i. source stren':z:. cre zero at the bo;., (i.e. ol=BOl= 0). By
expandin .he integral eq-:E- mcn for O(x) around x-O and taking the limit as x-0,
it can be shown that
0 -7T ga B'(x) at x=O0 (15)
where the prime denotes differentiation with respect to x. The linear approxi-
mation used for a(x) in the nunerical scheme leads to exactly this same slope
at x=O.
At the stern, there ___ two possible cases. For cruiser sterns, o(x)
and BO(x) are set equal to zero. For trans{om sterns, the prograr- uses the
values of B0 (L) and a(L) s:- o.p'utcd for the transom stern section. Thus,
a(x) and B0(x) are not eqc.al to zero at the stern section. The validity of
this result should be fur_-er investigated, but it seems to give reasonable
answers.
'-. pressure and near-f-eld wave amplitude are computed by equations (13)
and (i._ In computing t'-e quantities, the quotient u(x)/ BO(x) c.-r not be
computed fcr points at which Bo(x)=O (i.e. at the bow and for cruiser sterns).
The proper limits for the quotient are found by expanding thc integral equaL-;.
for 0(x) around x=O or x=L and taking the proper limits. At the bow, we find
1rn (y W) ga
x-)O Bo (W t 0At the stern, the limit need only be taken for cruiser type sterns, in this
case, the result is
L_ (X)___ o( )
i ga + a | dr
x-L TB° W W0 ]0 L
The exciting forces and midship bending moment are found from the evalua-
tion of equations (15), (16), (17) and (18). These equations all involved
integrals of the form
x
i= d f(U )e - V
0where f(s) is a function which varies smoothly along the length. To evalua-e
these integrals a simple Filon-Trapezoidal rule is used (see Tuck (1967). -his
integration is carried out in SUBROUTINE TRAP.
REQUIRNI I!PUT
The required input is subdivided into two separate parts. The first part
contains all the control information such as number of stations, wave frequencies,
ship speed, etc. The second part is the offsets which describe the ship hull.
The control information is read in on device number 5. The ship offsets are
read in on device 7.
Control Information (Read in on device 5)
Card 1: FORMAT (715)
NSTA - number of ship stations to be read in
ND - number of divisions to be used in developing cosine station
spacing used in setting up the XI(1) axis system.
NVEL - number of velocities at which calculations are to be made(maximum of 8)
NFREQ - number of wave frequencies at which calculations are to be -ade(maximum of 16)
NPRES - number of stations at which pressure information is desired(maximum of NSTA)
-10-
COntrol A i t anL. for wave ar..1 t+..C c '. .'0 u. tavt-, aiplitude calcula io.
=1 compute wave amplitude along ship sioc;
M-iD - the number of the station at which the midship bending moment 2iscomputed.
Card 2: FORMAT (4F10.h)
RAC - water ,!nsI-'.y (slugs/ft 3 )
GPAV accele> I:. ' r of g.'vc Lty (ft/ee2
XLYP ship length (it)
7: i,:u incident wave --,nplitude (ft)
Ca: : (CIF>0,A-)1h*c -, u a'... . -o : ,> " ].ere ho _
be '?: values Of'F;U()
Cai.:=: :ti:T ( F 0 Z,
, Wavelcn;.L+t --to-ship-length ratios at which calculations are to :niade. r77, Values of WJTL(E) shu 1 d be rad in.
Card ,' (1015)
14U. (r 7he numubers of the stations at which the pressure distributionis to be completed. There should be NqPRES values of NTM(I).NOTE if NPRES=NSTA or NPRES=O data card 5 must be omitted.
Ship Cffsets (Read in on device 7)
There is one complete set of data cards for each ship station. At leas-
one offset (up to a maximum of 25) must be given for each station. Only one
offset should be given at the bow; it may be either the point (0,0) or, for a
plum bow, the point (O,T) where T is the fore-foot draft. The ship may have
a cruiser (one offset point) stern or a transom (many offset points) stern.
Figure 3 is a picLure of the means by which the offsets are read in for a
given station. For accuracy in the calculations, more points should be entered
near the waterline and in areas where the shape changes rapidly (i.e. the turn
of the bilge). The pom:ns are always entered starting at the negative wat3r'.ne
and reading counterclockwise. Only half the section should be entered. Thcre
must be data points at (-B/2,O) and (O,T).
For each station, the following data cards are necessary.
Card 1: FORMAT (15,FI0.4)
NT2(ISTA) number of offset points on the half section for the stationnumber ISTA.
I -11-
z
YAXIS(5,1), ISTAZAXIS(5,1) NT2 = 8
YAXIS(5,4), ZAXIS(5,4)IC(5,4)=l.o
YAXIS(5,8), ZAXIS(5,8)
Figure 3. Input offset points around a ship station
XAXTS (TSTA) = x locatic- of the station being read in. x=O is the bow.
Card 2: FORMAT (2FI0.4,-:)
YAXIS(ISTA,I) = y - lozation of the Ith point
ZAXIS(ISTA,I) = z - location of the Ith point
IC(ISTA,I) = if --=1, the Ith point is considered a chine point.Nor£ 1ly IC=0.
There should be NT2 5a-a cards number 2 for each station.
Note for each parallel nidbody station, NT2(ISTA) is set equal to 999
and data card 2 is omittei. This causes the program to copy the results of
the previous station.
Sample input data are shown in Appendix II.
OUTPUT
A sample output listing is shown in Appendix III. The sample is for the
input shown in Appendix
The first set of ou-u; is a listing of the input control constants. In
addition, the computed values for the area of the waterplane (AWL) and the full
beam (BEAMWL) at the NEI- station are listed. These values are used in the
nondimensionalization at a later point in the program. It should be noted
that the table of offsets read in on device number 7 is not reprinted as output.
This was done in order t: shorten the output.
-12-
0_.1 I' " (%r4;shic er. <.h. Since O(,) do c s n-- , Wv Lh -: . ..... ,h va.2',-
only printed once for each freqcncy.
The next set is the three-dimensional sou:cec stren-th ((j(x)) distribution
alo.i the ship length. The magritude of 0(x) is printed in the column labeled
NAG (SIGT NA), The n~agrntude of he norid.1mensi-ona- sgma and it: real and
iI: L:ary, parts are j-ri nted f su!, ".--t !ruc. columns. The rondie, ensicnal
... ,,e.:Inp . as
o ())W 0
9a
nhc ur,_.-usre distribu-iu. o.'sr (tn,, or! :.sJ-', f_, Ld ) the input follows
-u ioirce J,.A str" -but ion listis. II7TiS- no P.r'2sure distcibutions are
pr'n:ecL. The pressurc distributfon cir only 'be obLained at the ship st~to.-
ga:.; in the input. The y,z -ocrdinates and angle up from the keel are prin--?d
in %- firs- three colunuis. The two columns under tho heading "2-D POT.N1i-"
coricspond to the two terms of ecuation (7). PFThL- -qual; e+ '7 and P}R 1_'.
the v.-alues of' '(y,z;x). The magnitude and phas&< amhle of the nondimension-li
Drec. Lre are printed in the last -two columns. Thc pressure is computed by
equatcn (13). The phase angles are all relauivm :c -, wave rode at the bow.
if NWAVE=l, the wave amplitude along the sht.r 1.ungth is computed and
printed out. Ihe nondimensional wave amplitude is co::puted by equation (14)
and includes both the incident plus diffracted wave. As with tho pressure,
the phase angle is relative to a wave node at the bow.
The sectional exciting force distribution is print out under the I-uding
EXCITING FORCE DISTRIBUTIC. 7he sectional exciting force (f3(x)) is aon-
putei using the expression inside the square brackets of equation (15). Note
that the e-ivx is not included in the expression for f3(x). In the print out,
the sectional exciting force is nondimensionalized in the following manner
F (x) - f3(x)
3 ga B(x)/2
where B(x)/2 is the LOCAL half beam. The use of the local half beam facili-
tates comparisons between F3 and purely two-dimensicnal calculations.
Finally, the total heave force, pitch moment -ut midship and the bending
moment at station number ';!ID are printed out. The results are computed uzin.,
-13-
equations (16), (17), and (18). The printed results are nondimensionalized in
the following manner:
F3F* -3 PgaLB
F 5
5 PgaL 2B
- BMBM* PgaL 2B
where
L = ship length
B = full beam at the waterline of the NDIID station.
-M= bending moment at NMID station.
The phase angles are all relative to a wave node at the bow.
LIMITATIONS OF THE PROGRAN
As presently written, the program has several limitations of which the
user should be aware. The limitations of the subprograms which calculate the
two-dimensional solution are discussed by Troesch (1976b). When the hull
section shape is very thin or has areas of high curvature more input points
are needed. As the number of input points is increased, the run time of the
program will be greatly increased. Furthermore, there are eigen frequencies
at which the solution blows up as described by Troesch (1976b). this is a
result of the use of an integral-equation technique to solve the two-dimensional
problem. It only occurs at high frequencies and for normal ship operating
ranges should be no problem. It should be noted that at springing frequencies
the effects of the eigen frequencies may become apparent.
The second major limitation has to do with end effects. As with any
slender-body theory, the results near the ends are of questionable accuracy.
The proper means of handling large bulbous bows and transon sterns is not
obvious. At the bow, the program can not handle bulb sections which protrude
underwater forward of the fore-perpendicular. The program can handle normal
bulb sections which intersect the free surface. Thus, the user must "fair
out" the protruding section of the bow. In the stern region, it is importar.t
-14-
_that T j5, i tr(ff' uata, in ld n- t~he last, Ftat.D )i * i - 7n C~ thut :I t
direcLion. In particular, for cruiser sterns, -. :r'e Cf th-
station are (O.0, O.O),it has been observed that the predicted wave arplitude
sometimes shows a marked change from the previous station. This type of
result at the stern section should be viewed with caution.
The final limitation deals with the forjard speed results. The theoreti-
cal analysis (see equation (3)) was carried out for the two speed raxiges -=0
and r .25. The rangc between these two extre.les has not been formally
analyzed. The program as presently written has a switch in it at =.25.
For t ., the zero speed value of C is used, and for t>.25, C is set equal
to thtu c T value. This ,-witch can lead to a discontinuity !n the results
at
. J-.c]d be pointed out that the theoretical analysis assumes high fte-
cuenc.u (or short wavelengths). The progran appears to give reasonable
rEsU1-z: over the entire frequency range. However, in deriving the expressicn
-or Lhe Dressure (equation (13)) certain forward speed terms were neglected
zecause f the high frequency azsumption. Under certairn combinations of for-ard
speed and wave frequency these terms may become important.
-15-
LIST OF SUBROUTINES
BESINT - initializes tables of I Bessel functions for use in determining
the Green's f-_ction for the Helmholtz problem.
BESK - computes the Bessel function for a given argument and order.
BI - is a single wrecision subroutine that evaluates the integral of
Ko(t) for t r-sgina from zero to X.
BKlOD - computes the M Bessel function minus its 1/X singularity
for a given argument.
DBS - solves a system of simultaneous equations using back substitution.
The L-U deccmosition of the coefficient matrix is computed by
DLUD.
DE!, NATSEI, FCN ,ON, EFTA- are subroutines that evaluate the real
exponential Lsnegral. They are part of a mathematics package
from Argonne -aticnal Laboratory.
DE:CCE - calculates 7he co7i:lex exponential integral.
DLUD - computes the T-U decomposition of the coefficient matrix by
Gaussian elimnation with partial pivoting.
DQL, DQL8, DQL12 - evaluate the integral exp(-X) times some function
of X for X rar.ging from zero to infinity by a four, eight and
twelve poin- Laguerre quadrature formula respectively. These
subroutines are from the IBN-SSP listing.
FNT1, FNT2, FNT3 - are all function subroutines that are called by the
DQL routines.
G - function used in computing the integral fo/xN d
GRFUN - evaluates the complex Green's function that satisfies the
Helmholtz e:uation in the fluid domain and the linear free
surface boundazry conditions.
-16-
Oluek &TXte curva, Urtl' alln 11-n t( !L
J O i YL il he paxamel t ic'teru.4-yed ~frairtir -rcii f-itted Lihrough themep: ~S
eie. £;L~ttion:a!oe ali- htrs L.r.t ; hat a eo:(sifle. station
spacirep resilits.
- irier-joiates valu.-es along shIp lergth using piecewise cjbic
fit.
v - etermanes the normal to a line drawn rbetweer tl.1 po*Pnus.
oonpuu ,he pressure distributiz &:~m ±1( sni S tatciri
>7 -evaluates t!,e sine and cosin~e iteLrals.
- -riaps. stLu Offse-z' fran. de-Iine 7 ;deri - . c,
quantities.
-compute,% t-E: 3-D source strength 1-,r SOiVIIIth oI~r~
-: El cor-.l)ex fun;ction which computeF the irtegra.J d ()/ r
S XY finds a series that helps define the scuc-potc. dl- in CR}Y'1I.
TRAP -. compuites the integral ff (x)e' 'x dx
Tk'WD-T computes tn-e '2--D potential,ex;:LItlhqf forces and T.,jxYI u.: ng
TWO.' - compLtcs 'he solu-tior to thc p-J robleii usi2Jig (uo:ech's
method.
WAVE -finds the wave a:-.plitude along the ship length.
-17-
REFERENCES
I. Beck, R.F., "Wave Diffraction in Head Seas," to be published.
2. Troesch, A.W., "The Diffraction Potential for A Slender Ship MovingThrough Oblique Waves," Report No. 179, Department of Naval Architectureand M'arine Engineering, The University of Michigan, Ann Arbor, Michiga.,1976a.
3. Troesch, A.W., "Computer Progra that Solves a 2-D Diffraction Problemof Incident Waves on a Symmetric Body and Satisfying a Helmholtz Equation,"Report to Naval Sea Systems Command, General Hydromechanics ResearchProgram, Departrsmt of Naval Architecture and 4arine Engineering, TheUniversity of Michigan, Ann Arbor, Michigan, 1976b.
4. Troesch, A.W., "The Diffraction Forces for a Ship Moving in Oblique Seas,"Journal of Ship Research, Vol 23, No. 2, pp. 127-139, 1979.
5. Tuck, E.O., "A Sirple Filon-Trapezoidal Rule," Mathematics of Computaticn,Vol. 21, No. 98, 1967.
6. Ursell, F., "The Expansion of Water-Wave Potentials at Great Distances,"Proceedings Cambridge Philosophical Society, Vol. 64, pp. 811-826.
7. Ursell, F., "On Head Seas Travelling Along a Horizontal Cylinder," Journalof The Institute of Mathematics and its Application, Vol. 4, 1968b, pp. 414-427.
I
-18-
APPENDIX 1
Program Listing of Diffracted Forces Proqram.
V I I
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