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WC-

AN EXTENSION OFTH4E THEORY OF WATER, HAMMER

by

Richard

AFSWP No. 922Office of Naval Research

Contract No. Nonr-266(o8)Technical Report No. 15

CU-15-55.ONR-266(OS) -CE

March 195 5

' umba inivrt

in the (Ittg of Ntem Vork

DEPARTMENT OF CIVIL ENGINEERINGAND ENGINEERING MECHANICS

C0

AN EXTENSION OFTHE THEORY OF WATER HAMMER

by

Richard Skalak

AFSWP No. 922

Office of Naval Research

Contract No. Nonr-266(08)

Technical Report No. 15

CU- 15-55-ONR-266(08) -CE

March 1955

WY The American Society of Mechanical Etr29 WEST 39TH STREET, NEW YORK 18, NEW YORK

AN EXTENSION OF THE THEORY OF WATER HAMMER

R. SkalaxAsst. Prof. of Civil Engrg.

Dept. of Civil Engrg. & Engrg. Mechanics

Columbia University

School of Engineering

New York 27, N. Y.

Contributed by the Hydraulic Division for presentation at the ASME Diamond

Jubilee Spring Meeting, Baltimore, Md. - April 18-21, 1955. (Manuscript

received at ASME Headquarters May 19, 1954.)

Written discussion on this paper will be accepted up to May 20, 1955.

(Copies will be available until Feb. 3, 1956)

ADVANCE COPY: Released for general publication upon presentation.

Decision on publication of this paper in our ASME journal had not been takenwhen this pamphlet was prepared. Discussion is printed only if the paper ispublished in an ASME journal.

Printed in U.S A Price: 50 cents per copy

(25 cents to ASME members)

All 19,T" "SION OF T1E TIEOEY OF -'AT3h HAXIP2

Ly h. Skalak

NOl 7ENCLATURE

The following nomenclature is used in the paper:

c Velocity of sound in fluid

Ce Joukowsky water hari.er velocity (See eq. 1)

K Bulk modulus of fluid

a Radius of tube

h Thickness of tube wall

E Young's modulus of tube wall material

p Pressure in fluid

v xial velocity of fluid

z Distance axially along the tube

t Tim

u Axial displacement of tube

w Radial deflection of tube wall (+ outward)

Poo Initial density of fluid

r Radial distance

4Velocity potential

m Mass of tube per unit surface area

7 Poisson's ratio for tube wall

C0 Velocity of sound in tube wallCoW o ( -V-

q Force per unit area on tube wall

t, Xk Transform variables

n, s Integers

N See eq. (23)

IL See eq. (24)

J o,Jl Bessel functions

oIF

D See eq. (30)

F See eq. (36)

G See eq. (37)

B See eq. (38)

- 2 -

CI; Arbitrary constant velocities Is H Integrals. See eq. (53)

01102 Phase velocities L Length of wave front2

Z: See eq. (48) R - -0

Variable of integration c1zSee eq. (50 A =

1. Introduction

The usual theory of water hammer which is used to compute the magnitude and

velocity of pressure waves in a pipe filled with an elastic fluid was given by

Joukowsky [8] in 1898. This theory predicts that pressure waves travel without

change of shape at a velocity oe:

CCe= C 2 (1)

In the derivation of this formula it is assumed that the pressure is uniform

across any section of the pipe and that the deflection of the pipe is equal to the

static deflection due to the instantaneous pressure in the fluid. The first

assumption neglects, in effect, the inertial forces associated with radial motion

of the fluid. The second neglects the mass of the pipe wall and the longitudinal

and bending stresses in the pipe wall.

For the case of an infinite train of sinusoidal pressure waves in a tube filled

with fluid, much work has been done without resorting to the above assumptions. In

1878 Korteweg [9] showed that the phase velocity of sinusoidal waves varies with

the wave-length when the radial inertia of the fluid and the mass of the pipe wall

are taken into account. Korteweg also gave the expression for ce, eq. (1) as an

approximation of the phase velocity for long wave-lengths. The work of Korteweg

was extended in 1898 by Lamb [10] who considered the pipe as an elastic membrane

thus including the effect of longitudinal stresses in the pipe wall. Lamb showed

there are two finite phase velocities as the wave length approaches infinity.

One of these is very close to Korteweg's result ce and the other is very nearly the

velocity of sound in the material of the tube wall. Recent papers by Jacobi [6]

and Thomson [1143 include the flexural stiffness of the pipe wall. The inclusion

of the bending stresses affects the phase velocities appreciably only for very shortwave-lengths.

r ' the r- ,-r .

In the present paper water hammer waves are considered without some of the

simplifications made in Lhe Joukowsky theory. The equations of motion for a

thin cylindrical tube are used, cf. FlUgge (11],

with one additional term for the effect of rotatory inertia. The fluid is assumed

to be elastic and inviscid but no assumptions are made regarding its motion except

that it is small.

Only the case where the velocity of sound in the material of the pipe wall in

greater than in the fluid will be considered. Much of the analysis would be valid

in the opposite case, but the relative speeds of the various waves would differ.

2. A Formal Solution by Integral Transforms

obiainecL in the form of a. do~te i+egral.

In this section a formal solution of a typical water hammer problem isA

The solution is reduced to terms of single real integrals whose evaluation is

discussed in the next section.

The problem considered is to find the moticn of a thin cylindrical tube filled

with fluid starting from given initial conditions. The conditions, shown in Fig. 1,

are that for t - 0:

p-O, v a 0 for z > 0

P Pop0 v- vo for z e- 0 (2)

u 0 0, w a 0, for all z

The velocity v0 is assumed to be related to the pressure po0 by:

This is the velocity that would exist behind a plane shock wave of pressure p0 in

an infinite body of fluid. Thus the initial conditions for the fluid correspond toa step shock wave moving in the positive direction. An external pressure equal top0 is assumed acting on the outside of the tube for z < 0 to prevent any motion of

the tube at z - -oo , This external pressure is assumed to remain on the tube for

z < 0 for all t.

These initial conditions are intended to simulate the situation arising when

a pipe leading from a re-servoir is filled with fluid at rest below reservoir

pressure and a valve at the rpservoir end is suddenly opened. The part of the tube

left of the origin in Fig. 1 is a convenient device for supplying fluid and energyin place of a reservoir. It is expected that the particular means of generating awater hammer wave will not essentially affect its characteristics when the wave hasprogressed far enough down the pipe.

The motion of the fluid will be described by a velocity potential which must

satisfy the equation:

V 2 + "' (4)

where the subscripts r and z denote derivatives.

The motion of the tube due to a radial force per unit area, q (z, t), isgoverned by two equations:

___w~z y+ 1120o 12C - + 2!a UUz- -j-2 + 1 f z z U(5 )

LL U w + 1- w o (6)ae+ailed.

A derivation of these equations is given in [12], They agree with the equationsgiven by Flhgge [11] except thaot the rotatory inertia term Wzz has been added above.

The fact that the fluid and tube are expected to remain in contact is e np'esedby:

w €] (7)I =r J.o.

The pressure exerted on the tube wall by the fluid in:

r [ o)a (8)

This is the value of q to be used for z ;, 0 in eq. (5). For z < O, the external

pressure - p0 must be added.

It is convenient to consider the solution of the problem as the sum of two parts:

The first part is the shock wave that would be produced by the initial

conditions if the tube were rigid. The tube displacements w1 , ul, for this part

are zero. The velocity and pressure vl, pl, for the fluid are:

V,--Vo' I for Z > = 0Vp I, o~ 2 >C V% (9)

PI POj P I = 0

The second part is the response of the tube and fluid to a progressive load

pressure ps equal to the pressure produced by the shock wave of the first part. This

load pressure acts on the tube wall radially outward. Its magnitude is Po for

0 < z < ct and zero elsewhere.

Inasmuch as wl, ul, vl, etc. are known, the problem reduces to finding the

the response w2. u2 , V2, etc. of the tube and fluid to the load pressure ps" For

this second part of the solution the system is initially at rest and the governing

equations are (4), (5), (6), (7) with:

'2 = [a] _ 2Ps (10)

The solution is begun by transforming the governing equations and eliminatingthe velocity potential 42" The remaining transformed equations are solved simultane-

ously and the inversions of the transformations then yield the solution.

With respect to z a Fourier transform, cf. (15] p. 42p is used. Defining thetransform of a function g (z, t) with respect to z by:

CO

-00then the inversion is:

00S(z't t) ) L I (12)

-00

With respect to time a Laplace transform is usedp but the tranform variable

adopted is-t- times the usual transform variable. This results in a certain

synetry with the rirst transform and convenience in later analysis. Defining the

transform of a function (, t) with respect to t by:

00

then the inversion is: 0

-oo- L o.VW = =a~.tCX (114)

-CO- C f

The double transform of derivatives of g (s, t) may be expressed in terms of the

double transform of g (z, t) by the usual partial integration procedure, (13] p. 27.

Using the fact that initial velocities and displacements are zero, the transform of

the derivatives involved are:

Using q2 as given by eq, (10), the doubw transformation of eqs. (o1), (5), (6)and (7) yields: =

rr r$2r 2C (16)

s !2 2 'W 2 +I 2 SI W (4- 2. + + L 2.

+ P5 (17)

- + t Lk- - W2 W2 = 3h (18)0

r] &

The double transform are functions of k and SL only except for

2 hc s 2 I ) Freemeq. (16) it isfounad that the rdepenenc of 2involves a Bessel function of the first kind. The function pa in known. Solving

eqs. (16), (17), (18), (19 for ,2 , 121 2 and inrtig yields:

- 0 -

00 L c C 2 mo

cO oo1 2. / - h. Lfl.tZ + r1.. a (21)

___ LJ0(- -- )eL t-oo O- (22

+~ +* c'o h2']

a 0 aLa J~iCt i e2 (214)

Sa (a ,- I ,

P, 0- (22)

The evaluation of the radial deflection wifl be considered first. It istypical of the quantities of interest. The inner or fl integral of eq. (20) may beevaluated by Cauohy,'s residue theorem applied in the complex £i plane. Durin thisprocess is regarded as a constant real parameter since the outer integral involvesonly real values of

The factor in the denominator of the integrand in eq. (20) produces a simple

because (c + ) A is finite at this point. There are poles at the points for which

the remaining bracket is zero, i.e., where:

1 (25)

L2~ Nj

For any real this equation has an infinite number of roots rL n* These roots

have an interesting physical significance. They are the circular frequencies of the

modes of free vibration of the tube and fluid system having a wave-length X -

Alternatively the ratios - are the phase velocities cn of progressive sinusoidal

waves of wave-length 2f. These facts may be developed by considering the governing

equations (4), (5), (6) and (7) with q in eq. (5) equal to the fluid pressure only.

A solution is assumed of the form:

w - w e 1 + (26)

where the prime quantities are constants and f(r) is an unknown function of r.

Substituting eqs. (26) in eqs. (4), (5), (6) and (7) the constants and f(r) may be

eliminated, leading to exactly the transcendentaI eq. (25) relating I and fl .

For real values of f all the roots L n of eq. (25) are also real. If any of

the JL n were complex or imaginary the corresponding free vibration would grow or decay

indefinitely in time. Since the system has no means of dissipating or radiating energy,

this is not possible physically.

Fig. 2 shows the S vs t values for a typical case in which the velocity of sound

in the tube wall is greater than in the fluid, i.e., c0 > t, It is found that for any

value of there is one root 1 , for which "A - f ) is negative. This root corre-

sponds to a phase velocity less than c, the velocity of sound in the fluid. There are

also an infinite number of roots for which (c 2 " 2) is positive. These correspond

to phase velocities greater than c. Fig. 3 shows the same data as in Fig. 2 plotted as

phase velocity vs wave-length.

The fact that two sets ofStn curves pass through the origin in Fig. 2 shows that

there are two modes which have finite phase velocities as the wave length approaches

infinity and the frequency approaches zero. This was also shown by Lamb (10), p. 7.

It is mentioned here because in a recent paper [i1), p. 933, it is erroneously statshd

that there is a cutoff frequency below which only one mode may be propagated.

The above discussion shows that the poles of the integrand of eq. (20) are at

SL - 0 and 11 - /l n, all on the r.al SL axis. The path of integration is below

this axis and it will be closed in the usual manner by adding a semicircle of

infinite radius in the upper half of the SI plane. The integral over the semicircle

approaches zero as the radius approaches infinity. The contribution of the pole at

- 0 to w2 is:(O

co

a ---2- -, L (27)I [ K Y2+i Zy~ 11 -- t C

The contributions of the nY poles to w. may be expressed as:

W . O eL 21RL~ z e _ (28)

The sum indicated is over the -L n roots for each value of * The values of the

SL as functions of t are given by Fig, 2. Eq. (28) may be written in the form

L~~j + 1

29 r s al f D (29)

where P~ 0 2I Cl~ 2(~ 2 ~ J2 +(0

+C~_____ +(- + +

The integral in eq. (29) covers all four quadrants of the siC. curves in Fig, 2,9but it may be expressed as an integral over the first quadrant values only because

D is even in a and even in and the -Ln curves are symmetric about both ams.

Thus:

Spn ( i -CLt) S t( + t)}~ L

.L C

where the summation is over the .L. n curves in only the first quadrant of Fig. 2. This

concludes the frzral solution for w. The evaluation of the integrals remaining in

60q4 (27)and (31) will be considered in the next section.

instead of evaluatLi, u2 and 2' certain quantities of interest which are

derivatives of u2 and 2 will be computed directly. These are:

Pressure in the fluid:602 (32)

Axial velocity of the fluid: b t - =(

Longitudinal strain in the tube; :b L

Expressions for P20 V2 and uz2 are derived from the double integrals for u2 and 4 2'

eqs. (21) and (22), through application of eq. (15). The inner integral in S may

be evaluated in each case by using Cauchy's residue theorem as in the case of w2.

The results are:

P2 = P O O C Ie(' [0 n n +SI) (33)

-00 0

(O S in ( 1-At) sin(+: (34)S2I -L . + r I L.( +)

where:=

(37

+(38)

and f(l ) stands for the denoi~naor of the interad in eq. (27).

- Ii -

The solution of the problem originally posed at the beginning of this section

is given formally by:

w UW2a " '2b Eqs. (27) and (31)

P Pl * P2 Eqs. (9) and (33) (39)

V l * v2 Eqs. (9) and (34)

uz a uz2 Eq. (35)

3. Evaluation of the Solution for Large Values of Iz and t.

In the formal solution indicated by eqs. (39) the integrations with respect towould be difficult to carry out in general because the IL n values are known asfunctions of t only through the transcendental equation (25). However, some informa-tion may be obtained by considering approximations of the integrals for large values

of I Ind t.

Consider the part of w given by eq. (31) and, in particular, values of z and t

such that:

z -c' t (40)

where c' is an arbitrarily selected velocity. Substituting in eq. (31):

00

1 -andIt is found by inspection of D, that the functions ( - D D

c) care bounded for all of the P_ n curves for all [ except for the lowest two curves, A,and ft 20 at the point - 0. The distinction between the lowest two and the

remaining sL n curves arises because only the IL 1 and -a 2 curves pass through the

origin. A physical interpretation of this fact is that only these two lowest modes

have finite phase velocities as the wave-length increases indefinitely.

For the higher - n branches, n > 2, the integral (4i) may be approximated by

the method of stationary phase (7J p. 505, for large values of t. This method shows

- 12 -

these integrals to be of the order of

The integrals over the SL 1 and SL 2 curves will be broken into two parts; one

part from I =O to f = 6 and a second part from E to co where f- is aSMaI

positive number. By the stationary phase theorem the portions from f to oo are again

0 ) Collecting these results, eq. (41) becomes:

Po ii_(_c_-) sn( C'j+n)t1 dt +O(L) (42)

JAL (-9.-fC Jo

Since E may be chosen indefinitely small, the functions in the integrand may

be approximated by an appropriate number of terms of their expansions about 0 0.

Consider the first term of the integral with A' - XL which may be written:6

The ~i deomnao W1 s~(-fL ) I A (143)

The denominator of the integrand approaches a constant time approaceszero. The constant is:

where: c- -= Ly

=Ur Dos, 4PO C t + 2.C [ir C 1 h2A 2e 1 22j

The constant c is the phase velocity of the slowst waves for wavelengthsapproaching infinity. It is very close to the Joukowsk velocity e . An ejxmssion

for c1 is derived in the appendix.

In the numerator of eq. (43) S11 is replaced by the first two terms of its

expansion about O 0:

,)I= (46)

This expansion is developed in the appendix. It is necessary to use two terms

of the expansion because the first term alone would result in a vanishing numerator

for the case c' n c 1 .

Using eqs. (44) and (46) in (43) yields%pO0w = PO - a ' + l 3 at (47)

0

where z' is defined by:

zI= (C' - cI) t (48)

The new coordinate 21 is the location of any point relative to an origin moving atvelocity c1 . In eq. (47) the upper limit has been changed from E to cv , The

integral thus added is 0(-) by the stationary phase theorem, and is negligible

in the present approximation.

For z' - O, the integral in eq. (47) has a constant value of provided d1 ispositive which is the case here. For points where 1 1 is Lare, the termcontaining 3 may be neglected. Then the integral in eq. (47) becomes ! i for ! s'*For points near z' - 0 it is convenient to change the variable of integration from

to n IS '. Then:

W:= PC +I (49)

0where: a,'t (EQ)

From eq. (49) it may be seen that for w1 constant must be constant, say o0

Then:

This shows that points for which wI is constant continually move away from the pointz' 0 0. To suvnarize, wT is a wave having constant deflection at the point 2' W 0

i.e., moving with velocity c I . At points far ahead and far behind this wave frontpoint the deflection is also constant, but the wave front continually spreads out

- i/I -

around the point a' 0. Fig. 4 shows the general shape of the wave,

If the same reasoning that has been used to evaluate wI is applied to the

remaining terms of eq. (42) similar results are obtained. Thus for large t:

2 b I-H, * 12 - H (52)

where: 00

IrM(n- -I Dn(53)

The constants e2, D2 , d2 wh.ich appear in H2 and 12 are defined in the same

way as Cl, D1, dl in eqs. (45) and (46) by changing all subscripts from 1 to 2 in

these equations. The velocity c2 is very nearly the velocity of sound in the tube

wall, co.

The term T is the deflection w, discussed above. The H1 term is a wave of

the same velocity and shape as wT but of smaller Alitude and traveling in the

negative direction.

The H2 and 12 terms are also waves of the same shape as wTO but traveling at

the velocity c2 . These precursor waves are found to be very smll compared to the

main water haumer waves 13 HI which travel at velccity Cl.

The complete deflection w is the sum of wbO, eq. (52) and w2., eq. (27)9 Thedeflection w2a is independent of time and its value for large values of Izi =a beapproximated by the saw kind of reasoning as used for w~b for large values of t,

The result is:

h2.20 2 E [a + Pa 4~L l for Zi (54.)

me term 2h will be neglected compared to unity since on4 thin-12a (I - )

walled tubes are considered, Thus:

2

\A + o for (55)C2o- 2 Eh

The procedures used to evaluate w may also be used to obtain approximations of

p, v and u for large I j and t starting with the general expressions, eq#. (39),The results are: w + p O - H +I H or ± :

- 2Eh _

op-Zec~ ((I+ -N2_)v- r _1, ,

LL C1

2. -(I, ,))

These equations lead to four wave fronts in each case as discussed for v.

Numerical results for a typical case of water in a thin steel tube are shown inFig. 5. In the present approximation for large t, the pressure and fluid velocity

are again independent of r, i.e., they are constant across any section.

In order to estimate the length over which a given wave front extends, theslope wz will be computed at the points which move with constant velocity CI or 02.The difference in w before and after a wave front divided by the slope at the wave

front point will be indicative of the length of a wave front.

The double integral for wz is derived from w, eq. (20). The first integral may

be evaluated and the result folded by the aw proced.. es used for v. For a pointmoving so that C - cI t:

(57

o

PU Vc~ ac ' +L +I-, z-H o

2 E iC

- 16 -

The function .( in bounded for all and the first integral in eq. (57)

is 0 (1) by the Iiemann-Lebeague lemmas [16] p. 172.

In the seconi integral the integrand is also bounded, This integral may be

approximated by the method of stationary phase. The dominant part of the integral

will come from the vicinity of stationary phase points which are defined by:

C±n)= 0 L~e. C,(58)

In general such points give a contribution which is 0 -- ), but if:

Sc,-±n = o

then the contribution to the integral from the vicinity of this point is 0

[5] p. 40. Eq. (58) and (59) are both satisfied on the IL 1 curve atv 0 "O t

Hence this contribution will predominate over all others for large values of to

To evaluate the dominant term, the expression for -a 1 eq. (k6), is substituited In

eq. (57). Hence for large t: OD

Wo PO c0s (S t)

w = Po -L 4. (60)

0Changing the variable of integration to t)

00

0

This integral (61) is known, (5] p. 41. Hence:

W 3 (-") (62)

Now the change of w from a location far behind the point z - c t to a locationof + he point

far aheadA Z CI t is by eq. (49):

- 17 -

&w - (63)

Defining the "nominal length" h of the wave front to be aw divided by wz:

'' 3 J-

Computation of the "nominal length", L2 , of the wave fronts moving at velocity o2

results in the same formula with dl replaced by d2.

Some numerical values of L, and L2 are given in Table 1. It may be noted that

although the length of the wave front is long compared to the diameter of the pipe

it is short compared to the distance from the origin that the wave has traveled.

The above computations of the lengths of wave fronts hold also for p. v and

u because they are described by the same integralsk 1 1, Hls 120 H 2 .

4. An Extension of Joukowsky's Method

Some of the results derived by use of integral transforms m also be derived

by a much simpler procedure which is an extension of Joukowsky's method. The

extension consists of taking into account the longitudinal stresses and longitudinal

inertia of the pipe wall. This makes possible the precursor type wav which is

essentially a tension wave in the pipe wall.

The following derivation follows the general plan of Joukowsky'a paper (8].

The pressure and axial velocity of the fluid are assumed to be uniform over amy

cross-section of the pipe. The fluid is assumed compressiblej its bulk modulus

being K. The equation of continuity my then be written in the form t

V I _ P 2 bw (65)- I K T 6t a 6

- 18 -

The equations of motion of the fluid reduce to:

-v bp (66)

So far, these are the equations usually used. The new element is the use of the

equations of motion of the tube considered as an elastic membrame instead of a ring

of no mass. These equations are derived from eqs. (5) and (6) by dropping all terms

arising from bending stiffness and rotatory inertia and substituting p for q:

- - - - (67)a*2 MC 2.

"U w 0 (68)

In eq. (67) the term -i has been omitted because the effect of radial inertia of the

tube and of the fluid is neglected as in the Joukovsky theory.

No attempt will be made to solve the equations (65), (66), (67) and (68) ingeneral. It is sufficient for the purpose at hand to show that the system pernits

solutions which are waves of arbitrary shape moving at either of the velocities aI or

c2 without dispersion. A solution is assumed of the form:

w W, wf f(Z + Zt)

p = p f (f Z t) (69)

v1 = vL,' f (I + -Et)

Ut =LIti(+-Z)where the primed quantities are constants and ; is an unknown constant velocity. The

constants u.L, u' and 3 are related by the requirement:

& t 1 11 (70)

Substituting the assumed form (69) into eqs. (65), (66), (67), (68) and (70), there

result five linear homogeneous equations involving the five primed constants. In order

that these equations have a solution different from zero it is necessary for the

determinant of the coefficients to be zero. This condition yields an equation

- !12 -

involving ; which can be satisfied only if takes on the values - c1 or c2. Thus

the velocities at which waves may travel without dispersion in the present considera-

tion are the same as the pr:'ncipal velocities derived in the integral transform

method,

A further point of interest is that the ratio of the change in w to the change

in p across a wave front is the same in the extended Joukowsky theory and the

transform solution. The same is true of any two of the quantities w, p, v, Uz, etc.

This is shown by solving for the ratios of interest in each theory with c or c2

as the case may be.

These results are of interest because they indicate the significance of the

assumptions made in the uoukowsky theory. It shows that c differs from c becauseIonJittitdl 1 .Odf the. pipe. e

the Joukowsky theory neglects longitudinal stresses and inertia. It also showsA A

that the dispersion predicted by the transform method is due to the effects of the

bending stiffness and of the radial inertia of the pipe and fluid. The effect of

radial inertia is much more important. After the wave disperses somewhat, bending

effects are negligible.

5. Other Initial Conditions

The integral transform method may be applied to certain other initial conditions

beside those assumed in Sections 2 and 3. The problem and results shown in Fig. 6

illustrate a second example. The initial conditions are:

v v for z <0 v=-v for s >00 0

p u - w = ut - wt -'0 for all z.

Under these conditions each half of the pipe simulates the water hamer problem in

which an established flow is suddenly cut off by an instantaneous valve closure.

Since by symmetry no fluid crosses the plane z - 0, each half of the pipe acts as

a closed valve as far as the other half is concerned.

The magnitudes and velocities of the waves may also be established on the basis

of the simolified procedure of Section 4. In general, this will be the simpler

method. It does not predict dispersion, but it may be assumed that the dispersion

characteristics are the same in any case.

6. Conclusions

The theory developed herein shows the following results which are not contained

in the usual theory:

1. If the initial conditions specify a sharp wave front of pressure, this "ront

will gradually disperse over a finite length as it proceeds. Thisdispersion continues Indefinitely so that no steady wave shape is ever

reached.

2. Two waves will be generated in general; one which is similar to the

Joukowsky result and one which is essentially a tension wave in the pipe

wall. The pressure change due to this precursor is very small.

0~.3. AfterAsufficiently long time has elapsed since the generation of a wave

there will be one point in each wave front which has a constant deflection

and fluid pressure and moves at a constant velocity. This velocity is

very close to the Joukowsky velocity for the main pressure wave. For the

precursor this velocity is slightly less than the ve.ocity of sound in

the pipe wall.

4. The bending stresses in the pipe wall have very little effect on the

characteristic velocities and the rates of dispersion after a sufficient

time has elapsed. Such stresses can only be significant in the early

stages of the motion in regions where the wave front is sharp so that the

variation of pressure and deflection is very rapid along the length of

the pipe.

5. The relations between fluid velocity, pressure and hoop stress given by theJoukowsky theory are substantially correct, but there are also longitudinal

stresses in the pipe not predicted by the usual theory.

6. Available experimental datais in very close agreement with the predictions

of the Joukowsky theory as to the pressure rise and velocity of the main

pressure wave. Inasmuch as the analysis presented in this paper gives

essentially the same results as the Joukowsky theory for these items, it isconfirmed experimentally. However, there is no experimental data available

on the precursor wave or the rate of dispersion predicted herein.

ACKNOWLEDGEMENTS

The author is grateful to Professor H. H. Bleich for his

advice and encouragement throughout the preparation of this

paper and the thesis on which it is based. This research was

partially supported by the Office of Naval Research under

Contract Nonr 266(08).

L 115L I 0 GRA PT

(1) Allievi, L., Allgemenie theorie uber die veranderliche Bewegung desWassers in Leitungen, translated from Italian into German by R. Dubsand V. Bataillard, Julius Springer, Berlin, 1909.

(2) American Society of Mechanical Engineers, Symposium on Water Hamer, 1933.

(3) Biot, M. A., "Propagation of Elastic Waves in a Cylindrical Bore ContainingFluid," Journal of Applied Physics, Vol. 23, 1952, pp. 997-1055.

(4) Fay R. D., "Waves in liquid-filled Cylinders," Journal of the AcousticalSociety of America, Vol. 24, No. 5, Sept. 1952, pp. 459-462.

(5) Havelock, T. H., The Propagation of Disturbances in Dispersive Media,Cambridge University Press, 1914.

(6) Jacobi, W. J., "Propagation of Sound Waves along Liquid Cylinders,"Journal of the Acoustical Society of America, Vol. 21, No. 2, 1949,pp. 120-127.

(7) Jeffreys, H. and Jeffreys, B. S., Methods of Mathematical Physics,Cambridge University Press, 2nd Ed., 1950.

(8) Joukowsky, N., "Uber den ydraulischen Stoss in Wasserleitungsrohren,"Yimoires de l'Academie Imperiale des Sciences de St. Petersbourg, 8 serie,

Vol. 9, fNo. 5, 1900.

(9) Korteweg, D. J., "Ueber die Fortpflanzungsgeschwindigkeit des Schallesin elastisches Rohren," Anrlalen der Physik und Chemie, 9 Floge, Band 5,1878, pp. -1h2.

- 22 -

(10) Lamb, H., "On the Velocity of Sound in a Tube, as Affected by the Elasticityof the Walls "'Manchester Literary and Philosophical Society, Memoirs andProc., Vol. 42, No. 9, 1898.

(11) Flhgge, W., Statik und Dynamik der Schalen, Julius Springer, Berlin, 1934,Reprinted by Edwards Brothers, Ann Arbor, Mich., 1943.

(12) Skalak, R., An Extension of the Theory of Water Hammer, Doctoral thesis,Columbia University, 1954.

(13) Sneddon, I. N., Fourier Transforms, McGraw-Hill, 1951.

(14) Thomson, W. T., "Transmission of Pressure Waves in Liquid File 4 -.Proc. of the First U. S. National Congress of Applied Mechanics, ASME,1951, pp. 927-933.

(15) Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals,Oxford University Press, 1937.

(16) Whittaker, E. T. and Watson, G. N., A Course in Modern Analysis,Cambridge University Press, 4th Ed., 1952.

APPenDIX

Expansion of -a 1 and n 2 About " 0

The values -f 1' .I 2 are defined as the lowest two roots of eq. (25). From

the numerical results represented by Fig. 2, it is expected that both - 1 and 912approach zero as approaches zero. Hence, Al 1 and -n- 2 havean expansion of

the form:

= .(71)

where only odd powers occur because the -- are odd functions of ( . The values

of ; and d are determined by substituting this expansion in eq. (25) and equating

the coefficient of each power of I appearing to zero.

1

The terms - and L in the transcendental equation (25) are given in full byN

eqs. (23) and (24). Expanding the Bessel functions, t

may be written in the form;

- \[)- 5, C 4 +-) J (72)++ ....]-

8~~ 9

In order to be able to clear eq. (25) of fractions, the quotient of the two

infinite series in eq. (72) is written as a single series:

S, 0& a(7.3)

Using this value the transcendental eq. (25) may be put in the form:

M 0, 0- I C*'L

+

-6- - ,--o

Now the assumed expansion given by eq. (71) is substituted above for -i. • The

result is a series of the form:

it+±K '+ b3 f + = (75)

Each coefficient b of this series must be zero since the value 1I is a rootn hof the transcendental eq. (25). Equating b I to zero and neglecting - compared

to unity yields an equation involving ; but not d. This is interpreted as an

equation on the yet unknown ;. The roots are:

C = CT 2AR , + R + Rt (\ -- V') A AR + , + ( - -/ 1)(2A+ "1 (76)L2 (2 Ai-?s)

where _ (07)IC7 MoO (77)

Equating b2 to zero yields an equation involving both ; and d. This equation

is considered to determine two values of d, one for c - cI and one for c - 2.

The solution of the equation b2 0 for d yields:

C L T- (78J

_ 4V

-v I

t r74

-777---

-4 a

- 25-

52The ro dire an IMfInafmber of/

addit!"l| branches in i his &red.

4

3

2/

I =

o 2 4- G 8 10 12 14-FIG. 3 C. --

__ - I..

/\ - -,oo---.,, v,,,,Jou ,,1

Qorr

----------- .6

_____LI -20 -16 -12 -8 -0 o 4.

Fig. 4. Form of Tjplcol Wave Front

P-0 -- -. P.O

V v-v V. 949 jV. 9V t i 0

Iiiiial ouid1ow.hio~ p~.~v.~'*",Iitial Corikl~ions

No+* P. V. __. P.

5 9 mabt Cat

C t No*.e.- siCC

6404 *.01jp io a l

I~~ I --Tn-o 1

f- la-z -

S40.+ t c3 fppNotto CAa.

Va .(so ,49 p£n foi_ .07 P . I tuh fil. .ith F5q P. .00wes ofwP.pm o selhb i wt

waij m tr V t~.. C , fs~ i t~ ce o l w v~ r ± ~ oeo ntain o lw

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