Multipole expansions in the theory of surface waves

MULTIPOLE EXPANSIONS IN THE THEORY

OF SURFACE WAVES

Bi R. C. THORNE

Communicated by F. UItsELr

Received 11 March 1953

i. Introduction. Problems dealing with the generation of surface waves in water

involve the consideration of singularities of different types in the liquid. In the casse when bodies are present in the liquid , waves maybe either generated by the movement of the body. or reflected from the body. The two cases are essentially equivalent, and

the resulting motion can be described by a series of singularities placed within the

body. The boundary conditions on the surface of the body give equations from which the exact form of the potential can be obtained. Ursell(10) has solved in this manner the problem, earlier discussed by Dean 1, of the generation of surface waves by

a submerged circular cylinder. In this two-dimensional problem he used a series of complex potential functions arising from multipoles at the centre of the cylinder, but the velocity potential of the motion could have been described, without the introduc-tion of the stream funcintroduc-tion, in terms of the velocity potentials of the multipoles.

It is clear t.hat in analogous three-dimensional problems, such as waves froma

sub-merged sphere, where complex theory is not applicable, the velocity potential could also be described in terms of the velocity potentials of multipoles within the body.

In this paper the motion arising from line and point singularities in deep and shallow water is discussed. These singularities are characterized by their giving rise to potentials

which are typical singular solutions of Laplace's equation in the neighbourhood of the singularity. This specification alone does not give a unique motion, as it is possible to add to the resulting velocity potential a harmonic function regular in the liquid and

satisfying the boundary conditions. To ensure uniqueness we impose the condition that

the motion at infinity should be that of a diverging wave.

For three-dimensional waves in a canal of constant width and infinite depth,

reference may be made to UrsellUl).

Several of the results discussed in this paper have been given before in different

form. For references see § 9.

Statement of the problem. The theory is concerned with the irrotational motion of a non-viscous incompressible fluid, moving under gravity, with a free upper surface; the motion is taken to be simple harmonic in time, of small amplitude, and of period

2iï/. The singularity is at depth f, and near this point the potentials

are typical

singular solutions of Laplace's equation, namely, for the line singularity,

(log r, r" cos nO. r_IL sin nO) cos al,

and for the point singularity,

cos m4z

r_1P(cosO){

if

cosai,

708 R. C. TIIORNE

where r,O,z are polar coordinates with the singularity as origin; the line O = O points vertically down and is an azimut hai anglo about. this lino (see § 2, 5).

In § 2-5 the case of deep water is considered and in § 6 afl(l 7 water of finite depth.

2. The two-dimensional problem. Symmetric o.cillations. A Cartesian coordinate

system is defined with its origin in the mean free surface; the x-axis is horizontal, the

y-axis is vertical, and y increases with depth. Let O. O' be the angles defined by the

relations _x -_.-- x

tan O = , tan O' =

y-f

y+f

and let r, r' denote t.he radial distances of the point (x, y) from the points (O,f) and

(O, f) respectively.

It will be shown that there is a wave potential D(x, y, t) satisfying the equation

+ = r'

,,

¿ ₍₁₎

ax2 _¿y2

everywhere in the liquid, and at the mean free surface (y = O) satisfying the relation

K1)+!=O,

(2)

where K = u211g. Further, D is symmetrical about x = O and the expression

cosnO

V(x,y,t) _,

cosoiisregularfory>O,

(3)

and for large values of x,

(1)(x, y, 1) represents a diverging wave. (4)

Set (1)(x,_{y, 1)} CoSflOcos ot + ç1(x, y) cos ai + (D2(x,y, t), (5)

where ç is adjusted with the function r" cos nO to satisfy the surface condition (2); and an even function of x and also satisfying (2). is added to give the wave con-dition at infinity: (1)2 isa standing wave of the form ecosKxsino-t.

The function ç1(x, y) is harmonic, regular anti bounded in the half-plane y O. It is

an even function of x, and decreases with increasing y; this follows since the effect

introduced by the free surface decreases with increasing y. This suggests that ç1(x, y) be taken in the form

ç1(x.y) ₌

A(k)efcoskxdk.

(6)

.10

Under suitable conditions, this expression gives at the free surface the relation

o

=

[(e- +

K)

,'°]

+ (K - k) A (k) e cos kx dL'.

(7)

k'' e

lW+f)coq kxdk. (8)

k'euI)coskxdk.

(9)

and that, for y >f,

co

It is not difficult, by reference to (13), § 12'2, ex. 2. to show that, for

y>

cosnO'

(-1)"

f

r'tz

_{(n- I)!Jo}

MnO I

_¡

(n- 1)!J() (o

f,

and substituting from (10) and (S) into (7), we are led to consider 1(x, y) of the form

K+k

ç 1(x. y) = '

(n

' -- '4

,,

k'_1 e-+J)cos kxdk, L). _JO

where the symbol indicates that the principal value of the integral is meant. The differentiation under the integral sign, used in verifying the boundary condition (2), is seen to be valid by a consideration of the integral around the contour consisting of the sector with centre the origin, of largeradius, and of small angle, the real axis being indented at the point k = K. The effect of taking an integral other than the principal value is to introduce a standing wave at infinity. This solution for ç51(x, y) is such that

r cos nO + ç1(x, y) satisfies the free surface condition.

The function is now determined from (4). According to equation (5) we have

D(x, y, t) [cosno

+

3J

±

k'' e''cos kxdkl cos o-t + I2(x, y, t).

For large positive values of x,

ç1(x, y)

'«

[ir'

eyjsin

Kx + 9UnJ ±

k'

dk].

This is derived from a consideration of the integral around the contour formed by the quadrant R(k) >0, 3(k) >0, with an indentation at the point k = K. Thus for large x,

( l)'-'

(I)(x,y,

(n 1)!

2irK e(u+f)sin

Kx cos ol + ()2(x, y, t),

and the condition (4) is satisfied with

D2(x, = 2iK' e_K(Y+f)cosKxsin at.

This is symmetric about x = 0. Hence

cosnO

(_1)1

K±k

(b(x,y,t)

= [ r" +

(n 1)!

K_kke

coskxdk] cos at

+ 27TK'' e_K(u+f)cos Kx sin o-t.

(n-1)!

This expression may be expanded in a powerseries:

(_kr)scossO

e1cos kx = R exp (

kr e'°)

8=0 8!

cus nO

Thus iD(x, y, t) = coscrt+ [A3cosa-t±B8sin o-t] rcos8O,

80 (_1)n+8_1

rK±k

I k7L+8

e"1dk,

where _{= (n-. 1)!s!}

Kk

(_ 1)2i

and

B -

_{(n 1)!8!}

K1L+8 e1.

This expansion is valid for r 2f' < 2f.

45-2

Multipole expansions in the theory of surface waves

On the free surface y = O we have the relation

Fi a ,.\ cos nOi

ii

a cos 710,1

= (IO)

Hence the required potential is easily found to be [sinnû

(-1)"

F'.

r" (n - 1)! K

kI

ek1)sinLrdLj

coso't (1)(x,y,t)= + -I _{1fl_1}

+'

'

2irKh1e(U+f)sin Kxsin ai.

(n- 1)!

This function is an odd function of x. 'Fue power series expansion is

sin nO

(1'(x,y,t) = cosa't+ [.48cosut+B8slnot]rMsInsO,

r 8 I)

where A, and B, are the same as in § 2. This is valid for r 2f' < 2f.

The two -dirne n8ional problem. Logarithmic singularity. The singularity is now such

that the expression _{(I)(x, y. t) - logr cos o't}

is regular everywhere in y O. 'ro ensure that 1) is bounded for large values of r, set

y, t) = (log- + ç1(x, )) COS ai + 1)2(x,

where r' is defined in §2. On the free surface y = O,

" \ r _cosO

(k+-.)log ,=2(

r

r

'

Thus ç1(x,y) = 2 - coskxdk.

.

It-k

r

and 1(x, y, ') = [log -, + Coskxdkj cos o'! - 2n- e(Y+f)cos Kxsin ai.

T he power series expansion is given by

(I)(x,y,1) = log!',cosot+_r

IAcosoi+B;sinoi]r8cossü,

(_1)M

kae-2kf

where A 8! 3f0 (IA' (__ 1a-1

B; = 2ir'

'-

e_2KfK 8! As before, this is VfllJ(l for r 2f' < 2f.

and sinnO' r'"

-(_I)1 "

I 0 k? leL1+f)sin _kxdk.

(n- l)

710

R. C. TII0RNE

The two-dirnen.sional problem. Anti-syinmetrieal o8cillaEion. In §2 the potential function discussed was an even function of' X. The case of an 0(1(1 function of x giving rise to anti-symmetrical 05('ilhLtiOflS is flow considered. The singularity at the point (O.f) is now sUch that the expression

sin nu

(1)(x, y. 1) - - CUS ai

r" is regular everywhere in !/ ? O.

Multipole expaníion8 in the ineary of .surface waves

5. The fhree-dinien.ioníil problem. With the origin in the mean free surface, the

x- and y-axes horizontal and the z-axis vertical, z increasing with depth, define the

angles O, O', a by the relations

R

tau O = - tan O'

-where R = /(x2 + y2); let r.r' denote the ra(lial distances of the point (x. y, z) from the

points (0. 0.f) and (0. 0, -f) respectively.

It. will be shown that there is a velocity potential (D(x, y, z, t) satisfying

everywhere in the liquid, and at the mean free surface (z = 0)

K4)+?=0,

(2)

where K = o2/g Further.*

P7(cosO)

cosmcoso-t

is regular in z> 0, and for large valuesof R,

(1)(x, y, z, t) represents a diverging wave. (3)

As in §2, set

P cosO)

4)(x,y,z,t) = _{- 'mfl+I} cosinczcosot+ç51(x,y,z)cosot+ 4)2(x,y,z,t), (4)

where ç and 2 are added in order to satisfy the conditions on 4), and are found in

a similar way to the functions ç and (1)2 in § 2; ç is harmonic, regular and bounded in

the half-plane z 0. Thus we take

ç1(x, y, z) = casrncLJ A(IC) e+f)J(kR) dk. Now for z >f, O) _{= (n}

_m)!Jo k (J(kR)

dk ((2),p. 100). Also

P'(cosO)

(_I)m+nPI(cosO),

and on the free surface z = 0,

1K ¿'] P7(cosO)

-

[K

] P(cosO')

I z]

r'

zJ

r"1

From (2), (4), (5) an 1(6) we deduce that

(- iy'+"-'

K+k

1(x, y, z)

= (n - m)! CO5

rn3J

K - k

k' ef)J(kR) dk,

and this expression for ç can now be substituted into (4). For the behaviour of 4) for large values ofR, note that

.J,,,(kR) = .')I1(kR) where H(kR) = J,n(kR)+1Ym( ((lo), p. 73). H(kR) is bounded for (k)> O and R real.

dM P(cos O)

* ¡)m(COK_{O) is (Iefine(1 tA) ho} MIT)"O

d(eo O

see (13), §155. This differs by a fctA)r nf

-

I)" from the definition ofHobson; seo (4), p. 90.

tanz =

(1) ¿2(1)

-+

Px2 ?24) ¿'24)

=0

-+

z2 R

z+f'

and for large x,

Km(X)()

ir

_{CL+_lI8X}

4in2-12

+o(-

i ((12), pp. 78, 202). Hence for large values of R,

¶35

± k"

e 1)J(kR) dk

irK"' e(fY(KR)

A diverging wave is represented for large R by

e [Y,,,(KR) cos ai IJ,(KR) sin ai] cos ini.

Thus

Pm(cosO)

(-1)"'"-'

_K+k

(1)(x,y,z,t)

= [

r"1

+

(ntn)!

cosm2cosai

2irK" eK(z+f)j (KR) cos m.isin ci.

(nm)!

This potential is an even function of y, and is an even or odd function of x according

as m is even or odd. A potential which is an odd function of y can be formed by the

8ubstitution of sin ma for cos mi in the expression above.

To give (I)(x, y, z, t) as a power series, use the expansion

e')Jm(kR) = e_kTCOJ(krsjnO)

1- 1 m u'

( I)Nsinm+ O COSum O

= '

₂

u=m

-

kr) 28! (in + 8)! (u - m - 28)!

where u' = (uin) or

(u m 1), whichever is an integer. Thus, from Hobson(4),

p. 93 (note the factor ( I)"), we derive

P(cosO)

e'Jm(kR) = (_1)m

(Lr)t'

(u+in)! u m and P'(cos O)

(D(x, y, z, t) = cos ma COS ci +

_'-'Ls

cos ai + D sin ai] rP"(cos O) cos mjx, a m

1_1n+a_1

rK+k

where

C =

_a

'

/ _{ß I} ,.

k"e1dk,

(nm)!(8+m)!

j° A k

-and 1)_g

2nKi+8+1 e'1.

(nm)!(8+m)!

This expansion is valid for r 2f' < 2f.

6. The two-dimen.aional problem with finite depth of liquid, h. The statement of the

problem is similar to that in §2; the same axes are taken. The conditions (1), (2) and

(3) of § 2 on the wave potential aro relevant in this problem. Take these conditions as (1), (2) and (3) for this section.

(7)

712 R. C. TILORNE

By use of the contour of § 2, it follows that

i'f

" K + ik

=

2niK'' e-fJfl(KR) + ilj

K - i/c

/cn e1-1)JI(ikR)dk.

Multipole expanion. in the tàeory of 1surface waves

It is now shown that there exists a wave potential (D(x, y, t) satisfying these conditions

together with the further condition imposed by the liquid having a finite depth.

namely,aty=h,

!!.

(4)

cosnO

Set 1?(x,y,t)=

cosot+1(x,y)cosot+2(x,y,1).

(5)

The function ç is adjusted to fit both the surface condition (2) and the boundary

condition (4). çf is harmonic, and since y is bounded in the liquid, take

1

¡

ç1(x,y) = , I

{A(k)sinhky+B(k)coshk(h-y)]k'coskxdk,

(n- l).j

where A (k) and B(k) are such that the integral converges in the region

-cc<x<co, Oyh.

The boundary conditions (2) and (4) are now used to give integral expressions,

similar to (7) of §2. Note that for the condition (4) the relation (9) of §2 is used. By

equating the integrands of these integrals to zero, the functions A(k) and B(k) are

determined.

The potential function is now of the form

cosnû

cost

'I(x,y,t)

= r'

cosOi+(l)!

" e-"'3'4)(K sinh ky - k cosh ky) + (- 1)' (K + k)ef cosh k(h - y)

x'3I

_{Kcoshkh-ksinhkh}

k''coskxdk

+ (D2(x, y, t). (6)

It can easily be shown that the integral is convergent

everywhere in the liquid. To determine consider the form of t' for large values of x, and as in § 2 take the

quadrant 1(k) > O, 3(k) > O as the contour, but in this case indented at the point

k = k0 (k0> O), k0 being the only positive root of the equation

K cosh k0h - k0 sinh k0h = 0. (7)

The function V2 is found in a manner similar to that used in §2.

Hence the potential function which gives rise to symmetric oscillations is

cosnû cosot

«(x,y,t)

= r"

COS(Tt+(1)

¡' e"1 (K sinh ky

- k

cosh ky) + ( - i )" (K + k)e-"f cosh k(h - y)

k"'coskxdk

Kcoshkh-ksinhkh

2irk" cosh k0(h

-

y)

(n - 1)! [2k0 h + sinh 2k0 h] k0(h,_fl + ( i )' e"o(DJ C08 k0 x sin crt.

Similarly, the potential function which gives rise to anti-symmetric oscillations is

sin nO cos 01

(I)(x,y,t)

= r"

cosot+(1),

e( hf) {K sinh ky - k cosh ky}+ ( - i)"(K + k)e-kf cosh k(h_{- Y) kg-1 sin kxdk} X ¶ß

K cosh kh - k sinh kh

.10

2irk cosh k0(h - y)

714 R. C. THORNE

For the logarithmic 8inJu1arity. the equations for A(k) andB(k) are found in a similar

way. It then follows that

coshk(h-f)cohk(h-y)

I

D(x,y,t) = log-

_{kcoahkh(Kcoshkh-ksinhkh)}

e- sinh kf 4ncosh k0(h -f) cosh k0(h - y)_{cos Ic0x sin crt.}

sình ky

coskxdk-kcoshkh J - 0h+sinh2k0h

For all these two-dimensional potentials, expansions can be made in powers of r, by

use of formulae analogous to (11) of § 2. Those expansions are valid in the region

r2f'<2f.

The three-dimensional case uith finite depth.

The problem is solved in a way

essentially similar to that used in the two-dimensional case. The potential is P(cos O)

I(x,y,z,t) =

cosmxcosol

rtl+'

2 k'' cosh k0(h - z) [e_(h_f) + ( - l)""

eks('1_f)] cos m..z cos cl

Jm(ko05 rn.

sin (Ti +

+ u

(n-rn)! [2k0h+sinh2k0h]

(n-m)!

[e_hf)(Ksjnhk - k

cosh k:) + ( - i )m+n1 (K + k) e' coshk(h

-

k'Jm(kR) dk.

xç

Kcoshkh-ksinhkh

Further expre.s8ion.s for the potentials. In the above work the potential functions have been expressed in terms of integraLs with a singularity in the path of integration;

the principal value is taken. By using the quadrant contour indented onthe real axis, these singularities can be avoided. Furt her, the potential of the singularity can be

included in the integral, in each case by use of the expressions, valid for z> O,

eniO i P

r" - (n - i)!

_J

k"' e_

exp -

i[k(y

f)

-r

log-;

-2

k

P(cosO)

2

- 1T(n

-

ni)!_Jo k" cos [k(:

-f) - (n - ni)

Km(kR) dk.

If we write (I)1(x, y, t), (I)2(x, y, t), 4(x. y. t) and (1)4(x, y, z, 1) for the potentials

dis-cussed in §2, 3, 4 and 5 respectively, we can then derive the following expressions,

valid for z> O: e"10 cos cl . 2nK"

(i)1(x,y,t)+i(I)2(x,y,t) = -

--- +

t( - 1)"-'

r"

(n-cos al Joe K

- ik

I

k"-' edk

+(- I)

_{_}

_1)! ,

k+ikPiLk(Y+f)_

2]

2irK"

= i( - 1)"-'

(n-

1pexP{K(Y+f)i(Kx-oI)J

2cosol j'

+

(n-I)!J0

[Ksinky-kcosky]

K2+k2

A(k)dk,

where

A(k)

_{= (-}

l)R+h[iK+k]eikf for n = 2s,

Muitipole expanions in the theory of surface waves

(b3(x, y. 1) = log cos o-t + 2ire-(U*f)sin (Kx- o-t)

ekL

+2coso-tJ [Ksink(y+f)_kcosk(y+f)]2Qdk

= 2ire-K(u+D sin (Xx - o-t)

2c0so-tÍ

-

[K sin ky kcos ky] [K sin kf kcos kf]L[K2+

o

Ptm (cos O)

(I)4(x,y,z,t) = -'

cosrncosot

(-+

_(nrn).

21TK'' e_K(f)cos mx[}Ç,1(KR)cos o-t - J,(KR) Sifl (71]

2cosal

K-ik

+ ( -

1)-'--'

_ir(nm)!

cos rn.x _{Jo K + ik}exp i[k(z +1) - (n - m) 4irJ k"Km(kR)dk

(-

2irK1 e_K(Z+f)cos mcz[Ym(KR) COS (TI - 1m(K1) sino-t]

= (nrn)!

4

+ cos ai cos rnx [K sin kz - k cos kz] k'Km(kR)Bn_m(k) dk,

lr(n-m).jo

K2+k

where Bii_m(k) =

9. Conclusion. The above forms for the potentials are those most generally needed.

It is also possible to express the potentials in terms of the original singularity, the

image singularity in the free surface, and afurther term.

The earliest work on this subject was published by Lamb ((8) and (9)). These papers are concerned with a two-dimensional source on the surface, and thethree-dimensional

point source, both cases being for deep water. Havelock (3) has given several integral theorems which can be used to derive the potentials for line and point singularities of simple type. Holstein (5) has proved the resultfor a logarithmic line source in deep

water (4). The case of the point source for deep water, given in §5 with m = n = 0,

has been discussed by Hsien-Chih Liu (6). John derives the solutions for the logarithmic

line singularity in both deep and shallow water ((7), pp. 99, 100). The expressions are

given in a different form from the above. He also considers the point source mentioned

above (pp. 93, 98), but his result for the deep-water case has a mistake in sign in front

of the integral. The results of § 2 and 3 are equivalent to the expressions given by

Ursell(io). This is best seen by a comparison of the expansions in power series of the functions in question.

I wish to thank Mr F. Ursell for suggesting these problems and guiding the work from its inception.

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