Internal and surface wave production in a stratified fluid

WAVE MOTION 3 (1981) 215-229 NORTH-HOLLAND PUBLISHING COMPANY

INTERNAL FLUID

AND

SURFACE

WAVE

PRODUCTION

IN A STRATIFIED

J o s e p h B. K E L L E R Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA 94305, U.S.A.

D o r o t h y M. L E V Y Graduate School of Business Administration, New York University, New York, N Y 10006, U.S.A.

D a l j i t S. A H L U W A L I A Courant Institute of Mathematical Sciences, New York University, New York, N Y 10012, U.S.A.

Received 2 February 1981

Various mechanisms of production of internal and surface gravity waves in a stratified fluid are analyzed. Formulas are obtained for the surface elevation and the pressure, density and velocity of the fluid for both two- and three-dimensional motions in each case. The mechanisms are application of pressure to the free surface, movement of the bottom, injection of fluid below the surface, alteration of the fluid density, movement of a pressure distribution over the free surface, movement of a distribution of bottom velocity along the bottom, movement through the fluid of a source of fluid and movement through the fluid of an alterer of density. The first mechanism may represent a storm or explosion above the surface, the second a submarine earthquake or landslide, the third an underwater explosion, the fourth the subsidence of fluid displaced from its equilibrium height by mixing, the fifth a storm moving along the surface, the sixth a propagating earthquake or sequence of earthquakes or landslides, the seventh an object moving through the fluid and displacing it and the eighth the collapse of the mixed wake of a moving object. The density profile of the undisturbed fluid is permitted to be arbitrary and may be discontinuous, the depth is arbitrary but constant and the linear theory of inviscid fluid motion is employed.

1. Introduction W e c o n s i d e r v a r i o u s m e c h a n i s m s t h a t p r o d u c e i n t e r n a l a n d s u r f a c e w a v e s in a h o r i z o n t a l l y stratified inviseid i n c o m p r e s s i b l e fluid. T h e m e c h a n i s m s a r e a p p l i c a t i o n of p r e s s u r e to t h e f r e e surface, m o v e m e n t of t h e b o t t o m , i n j e c t i o n of fluid b e l o w t h e surface, a l t e r a t i o n of t h e fluid d e n s i t y , m o v e m e n t of a p r e s s u r e d i s t r i b u t i o n o v e r t h e f r e e surface, m o v e m e n t of a d i s t r i b u t i o n of b o t t o m v e l o c i t y a l o n g t h e b o t t o m , m o v e m e n t t h r o u g h t h e fluid of a s o u r c e of fluid a n d m o v e m e n t t h r o u g h t h e fluid of a n a l t e r e r of d e n s i t y . T h e p h y s i c a l i n t e r p r e t a t i o n o f t h e s e m e c h a n i s m s is given in t h e a b s t r a c t . T h e d e n s i t y profile is a r b i t r a r y a n d m a y be discontinuous. O n t h e basis of l i n e a r t h e o r y , w e o b t a i n a m u l t i p l e i n t e g r a l e x p r e s s i o n for t h e w a v e m o t i o n d u e to t h e a c t i o n of all t h e m e c h a n i s m s . T h e n w e e v a l u a t e it a s y m p t o t i c a l l y at p o i n t s far f r o m t h e s o u r c e a n d l o n g a f t e r it has b e g u n to o p e r a t e . W e find t h a t t h e w a v e m o t i o n is a s u p e r p o s i t i o n of t h e p r o p a g a t i n g i n t e r n a l w a v e s a n d t h e s u r f a c e w a v e . T h e a m p l i t u d e of e a c h m o d e is a s u m of c o n t r i b u t i o n s f r o m t h e v a r i o u s m e c h a n i s m s . I n a d d i t i o n to t h e g e n e r a l case, w e t r e a t t h e special cases of axially s y m m e t r i c a n d of t w o - d i m e n s i o n a l sources. T o i l l u s t r a t e t h e results, w e specialize t h e m to s e v e r a l s i m p l e d e n s i t y profiles. P r e v i o u s l y , H u d i m a c [ 1] c a l c u l a t e d t h e w a v e s g e n e r a t e d b y a b o d y m o v i n g h o r i z o n t a l l y at c o n s t a n t s p e e d in a n o c e a n consisting of two l a y e r s of d i f f e r e n t densities. M e i [2] c a l c u l a t e d t h e w a v e s p r o d u c e d b y t h e 0165-2125/81/0000--0000/$02.50

© North-Holland Publishing Company

J.B. Keller et al. / Internal and surface waves in stratified fluid

216

collapse of a mixed region in a stratified fluid. Miles [3] treated both the waves generated by a horizontally moving dipole, which represents a moving body, and by the collapse of a mixed region, which represents a wake. Kranzer and Keller [4] found the surface waves produced by an initial elevation or by an initial impulse applied to the surface, in water of constant density. Keller and Munk [5] calculated the wave pattern due to a uniformly moving disturbance in a stratified fluid, but not the wave amplitude. Our results agree with these previous ones when we specialize them appropriately. Thus they may be viewed as extensions of the previous results to more general density profiles and to other mechanisms of excitation.

2. Formulation Let po(Z) denote the density of a stratified incompressible inviscid fluid at rest between the free surface z = 0 and the bottom z = - h . The density may be discontinuous at the J interfaces z = -hi, ] = 1 . . . . . J. Let p, p and u = (ul, u2, w) denote respectively the density change, pressure change and velocity of a small amplitude motion of the fluid. Let ~i(x, t), ] = 0 . . . . . 3", be the corresponding changes in elevation of the free surface ] = 0 and of the J interfaces. We denote by x = (xb x2) the horizontal coordinates of a point and by z its vertical coordinate. The above quantities satisfy the following equations of incompressibility, mass, and motion, as well as the following boundary, interface and initial conditions: V . u = Q ( x , z, t),

(2.1)

P, + wp~) = R ( x , z, t),

(2.2)

poUt +Vp + pg = S ( x , z, t),

(2.3)

P - gpo~o = P ( x , t),

z = 0,

(2.4)

[p] = g[po]~/j(x, t),

z = -hi,

w = ~?jt(x, t),

z = 0, -hi,

w = W ( x , t),

z =-h,

u(x, z, O) = u*(x, z),

] = 1 . . . . . J,

] = 1 . . . . . J,

(2.5) (2.6) (2.7)

V . u* = Q(x, z, 0),

p ( x , z, O ) = p*(x, z),

~j(x, 0) = ~/*(x), ] = 0, 1 . . . . . J.

(2.8) (2.9) (2.10)

In (2.3) g is a vector in the positive z direction of magnitude g, where g is the acceleration of gravity. In (2.5) the brackets denote the jump in p or p0 across an interface. The functions Q, R and $ represent the strengths of distributed sources of volume, density and m o m e n t u m respectively. The function P denotes the pressure applied to the free surface and W is the vertical velocity of the bottom. The functions u*, p* and v/* are the initial values of the velocity, density change and elevation of the free surface and the interfaces respectively. In (2.8) u* and Q at t = 0 are required to be consistent.

3. Fourier analysis of the initial-boundary value problem T o solve the initial-boundary value problem (2.1)-(2.10) we shall utilize the Fourier-Laplace transform defined by

ZB. Kelleret al. / Internaland surfacewavesin stratifiedfluid O0

(30

217

fit2

](k' z' t')= I-oo [-~o So f(x' z' t) e-i(t"-~') dt dxl dx2"

(3.1,

H e r e k = (kx, k2) is the horizontal wavenumber vector and u is the complex frequency. We also define/r to be the spatial Fourier transform of f evaluated at t = 0: cO

(3O

T(k,z)=I_

[_ f(x,z,O)e-ik"dx, dx2. (3O

(3.2)

O0

In applying (3.1) we shall utilize the facts that )~t - T - i ~ , ~ and )~j = ikj)~ ] = 1, 2 which follow from the definitions above. We now apply (3.1) to (2.1)-(2.7) and (3.2) to (2.8)-(2.10) to obtain equations for the transformed quantities ~,/~, ~ etc. Before writing these equations we express u in the form u = (v, w) where v is the horizontal velocity vector v = (u, u2). We also write S = (T, S) where T = (S, $2) and S = $3. In addition we denote differentiation with respect to z by a prime. Now the transformed equations can be written as follows

ik. t3+ ~ ' = ~,

(3.3)

-ip~ +t~ + ~p~ =/~,

(3.4)

po(-ivt~ + 6) + i/~k = 2, po(-ivff + ~, ) + P' + ~g = g,

(3.5) (3.6)

p-gao~o=/~, z=O,

(3.7)

[P] = g[Po]~i,

(3.8)

z = - h i, ] = 1, 2 . . . . . J,

~, = -iv~j + ¢lj, z = O, -hj, ] = l, 2 . . . . . J,

(3.9) (3.10)

~ __. ,7",

ik •

~ * + ( f f * ) ' = O,

=~*,

(3.11) (3.12)

~=,~*, /=0, 1,...,y.

(3.13)

We now solve (3.5) for ~ and use the result to eliminate ~ from (3.3). This yields a relation between p and ~'. Then we eliminate ~ from (3.4) by using (3.6) and get a relation between if' and ~. Then we eliminate i~ by using these two relations to obtain the following second order ordinary differential equation for ~,: (pol~') ' - k2po(1 - ~-2N2)1~ = -- t2l~,

(3.14)

• =-(poO)'-u-~k • (7"-po~*)'+ik2u-~(S-poff~*)+gk2~'-2(l~-t~*).

(3.15)

where

Here

k 2 --

k 21 + !.2 -~2 a n d

N2(z) -

N(z) is the Brunt-V~iis~ilii frequency defined by

-gp~(z)/po(Z).

(3.16)

T o obtain a boundary condition on • at z = 0 we first eliminate 4o from (3.9) at z = 0 by using (3.7). From the resulting equation we eliminate/~ by using the relation previously obtained from (3.5)and (3.3). In this

218

J.B. Keller et al. / Internal and surface waves in stratified fluid

way we get ~'-gk2v-2~

= ap',

z =0,

(3.17)

where ~- O + ( p o p ) - l k

^

• (T-po8

,

)-ik2(pov)-lP-gk2v-2~l*o,

z =0.

(3.18)

We next obtain interface conditions on ~ by eliminating ~i from (3.8) by using (3.9) and then eliminating/3 as before. In this way we get

[pol~']-gk2v-2[po]l~ = O/,

z =-h~',

] = 1 . . . . . J,

(3.19)

where ~)/= v - l k " [7"]-gk2v-2[po]rl~' + [ p o 0 ] - v - l k

• [pol~*],

z =-hi.

(3.20)

At the bottom z = - h , ff satisfies the boundary condition (3.10). Equation (3.15) with the boundary conditions (3.10) and (3.18) and the interface conditions (3.20) constitute a boundary value problem for ,3(k, z, v) as a function of z. Once ~ is found, all the other quantities can be obtained from the preceding equations.

4. Solution ot the problem To solve the problem just formulated for ~ we introduce two solutions el(z) and e2(z) of the homogeneous form of the differential equation (3.14). We require them to be continuous and to satisfy the homogeneous form of the interface conditions (3.19). In addition e 1(z) must satisfy the homogeneous form of the boundary condition (3.17) at z = 0 and e2(z) must satisfy the homogeneous form of the condition (3.10) at z = - h . These conditions are

[poel]-gk2u-2[po]ei = 0 ,

i = 1, 2; z = hi, ] = 1 . . . . . J,

(4.1)

e'l (0) - gk 2v-2e l(O) = 0,

(4.2)

e 2 ( - h ) = 0.

(4.3)

The functions e~(z) also depend upon k 2 and v 2, which dependence is not shown explicitly. It follows from the homogeneous form of the differential equation (3.14), satisfied by el and e2, that po(z) multiplied by the Wronskian of e~ and e2 is independent of z within each layer. From (4.1) it also follows that it is the same constant in all layers. Upon evaluating this product at z = 0, making use of (4.2), and at z = - h , using (4.3), we obtain

po(ele~ - e2e~ ) = F ( k , v) = p o ( - h ) e l ( - h )e~ ( - h ).

(4.4)

H e r e F is defined by

F ( k , v) = po(O)el(O)(e~ (0) - gk Ev-2e2(O)).

(4.5)

We now use the two solutions el(z) and e2(z) to construct the Green's function of the inhomogeneous problem for ~(z). In the usual way we obtain G, which we write in the form

G(z, z') =

el(max[z, z'])e2(min[z, z'])

F ( k , v)

(4.6)

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219

Then ~ ( z ) is given by

~(z) =

h

G(z, z')q~(z') dz'+po(O)G(z, O)~F+po(-h)

OZ

From (4.6) we see that each term on the right side of (4.7) has F(k, write (4.7) in the form

(z, -h)l~'4- ~ G(z, -hi)(9~..

(4.7)

/'=1

1,) in the denominator. Therefore we can

~(k, z, 1,) = V(k, z, p)/F(k, 1,)

(4.8)

where V is just F times the right side of (4.7). Next by applying the Fourier and Laplace inversion formulas to • given by (4.8), we obtain w in the form

w(x, z, t)= (2~r)-a f f I V(k' z' V) ei(kx-~t) dkl dk2 dt," F(k, 1,)

(4.9)

The k~ and k2 integration extends over the entire real (k~, k2) plane while the v integration extends along a contour from -00 to +00 in the complex 1, plane, lying above all the singularities of the integrand. We now evaluate the v integral by closing the contour with a large semicircle in the lower half plane, along which the integrand vanishes for t > 0. We assume that V has no singularities below the original contour. Then the integral is just a sum of the residues at the zeroes of F. We pointed out above that ei(z) is a function of v 2 and k 2, and from (4.4) we see that F is also a function of ~2 and k 2. Therefore the zeroes of F occur in pairs which we shall denote z, = +to,(k), n = 1, 2 . . . . . When F = 0 we see from (4.4) that e2 satisfies the boundary condition (4.2) at z = 0 in addition to the condition (4.3) at z = - h . Then e2 is a multiple of el and it is an eigenfunction w, (k, z) of the homogeneous form of the problem satisfied by ~,(z). This eigenvalue problem has infinitely many simple eigenvalues and corresponding eigenfunctions. The residue evaluation of the ~, integral yields

-i f f ~ V[k,z,+to,(k)]e~[k.,:~, (k)t]dk~dk2. F~[k, +to,(k)]

w(x'z'tl=(-~w)2

(4.10)

.=I r~

Here Then

F~ = aF/Ov. Since el(z) is a multiple of e2(z) at the eigenvalue, we shall choose the factor to be unity. e~(z) = e2(z) when i, = +(0,, and V can be written in the following simpler form: 0

V[k, z, +to.(k)] = w.(k, z){;_ w.(k, z')O(k, z', +to.)dz'+po(O)w.(k, O)aF(k, +to.) h

•

}

+po(-h)w" (k, -h)lfV(k, +to,)+ ~. w,(k, -hi)O i .

(4.11)

j=l

We shall now evaluate (4.10) asymptotically for large values of r = ]xl and t by using the method of stationary phase. To do so it is convenient to introduce the polar coordinates r, 0 and k, 0 defined by (x, y) = r(cos 0, sin 0),

(kl, k2) = k(cos 0, sin ~).

(4.12)

Then the exponent in (4.10) becomes i~b~ where &~ is defined by

d~ = kr cos (O-O):Fton(k )t.

(4.13)

J.B. Keller et al. / Internal and surface waves in stratified fluid

220

The conditions that the phase be stationary are O~b~ = 0 and akd,~ = 0 which yield s i n ( O - 0 ) = 0 and r c o s ( O - 0 ) • to" (k)t = O. The coefficient of t is just the group velocity c,"(k) of the nth mode, defined by

c,'(k) = to" (k).

(4.14)

If c,'(k)> 0 then the stationary phase conditions yield 0 = O for ~b+, 0 = 0 + xr for ~b- and r/t = c,'(k).

(4.15)

We now calculate the matrix of four s~cond derivatives of rb ± at the stationary points and find that its determinant is krtto~ (k) and its signature is ~:[1 + sgn to: (k)]. Upon using these results in the stationary phase formula for the asymptotic evaluation of (4.10) we obtain

Ik,'to'l'2{V(k., O, z, to.) F~(k,', to.)

-i

w(r, O, z, t ) - - ~ r ~ ~ I to: I

e i(k.r-al~t-~/ a-(~r/ 4) sgn to '~)

+ V(k,', 0 + ~r, z, -to,') e-i(knr-tont--'n'/4--(~/4)$gn to n) ~

F~(k~, -to.)

J

(4. 16)

H e r e the sum over n includes only those values of n (i.e. those modes) for which (4.15) has one or more real roots k,, the sum over k," is over those real roots, and to," is evaluated at the root k,'. The expression (4.16) for w is our main result. In the next section we shall simplify it and obtain corresponding results for other quantities and also for the two dimensional case.

S. Simplification of the result and the two dimensional case The result (4.16) can be written in a more convenient form by first introducing the quantity B,'(r/t) defined by

B,(r/t) =

1 to, x/2 ~rF~(k,', to,,) to"

(5.1)

This is a function of r/t because the root k," of (4.15) is a function of r/t. Next we write V in the form 6

V(k, O, z, to) = w,'(k,z) T: Ai(k, O, to).

(5.2)

i=l

T o obtain (5.2) we use the definitions of ~, ~ and O r given by (3.15), (3.18) and (3.20) respectively, in the definition (4.11) of V. The functions A 1 A6 are defined to depend respectively on F, W, Q, R, $, and the initial data. Explicitly they are given by . . . . .

Al(k, 0, to) = -ik2to-lw,'(k, O)P(k cos O, k sin O, to),

(5.3)

As(k, 0, to) =po(-h)w'(k, -h)lTC(k cos 0, k sin O, to),

(5.4)

/R0

Aa(k, 0, to) = - |

d_ h

w,'(k, z'){po(z')O(k cos O, k sin 0, z', to)}' dz'

+po(O)w,'(k, o)O(k cos 0, k sin O, O, to) J

+ ~. w,'(k, -hi)[po(-h~)O(k cos O, k sin 0, - h i, to)],

(5.5)

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221

o

A4(k, 0, to) = gk2to -2 I- w.(k, z')l~(k cos O, k sin 0, z', to) dz',

(5.6)

h 0

As(k, 0, to) = kto -1 f_ w.(k, z ' ) { ~ (k cos 0, k sin 0, z', to) cos 0 h

+~(k 1

cos 0, k sin 0, z', to) sin O-ik~(k cos 0, k sin 0, z', to)} dz'

A

A

+ kw.(k, O)to- {Sl(k cos O, k sin O, O, to) cos 0 +S2(k cos O, k sin O, O, to) sin O} J

+kto -1 ~ w.(k, -hj)[~l(k cos 0, k sin 0, -hi, to) cos 0 jffil

+S2(k cos O, k sin O, -hi, to.) sin 0],

(5,7)

0

A6(k, O, to) = kto -! I_ w~(k, z'){[0o(Z')~*(k cos 0, k sin O, z') cos 0 h

+Po(Z')~* (k cos O, k sin 0, z') sin 0]'

-ikpo(z')ff,*(k cos 0, k sin 0, z') -gkto-l~*(k cos 0, k sin O, z')} dz' -koo(O)w.(k, 0)to-l{ ~i*(k cos 0, k sin O, 0) cos 0 + t~*(k cos O, k sin O, O) sin O+gkto-l~*(k cos O, k sin 0)} J

- ~. w.(k, -hi){gk2v-2[po]~*(k cos 0, k sin 0) 1=1

+kto-l[po(-hj)6*(k cos 0, k sin 0, -hi) cos 0 +po(-hi)~* (k cos 0, k sin 0, -hi)sin 0]}.

(5.8)

We now observe that each A~ satisfies the relation Ai(k, 0 + ~r, -to) = ,~i(k, 0, to) where the overbar denotes the complex conjugate. In addition, since F is an even function of v, F~(k, -to) = -Fv(k, to). As a consequence of these two facts the second term in braces in (4.16) is minus the conjugate of the first term, so their sum is 2i times the imaginary part of the first term. Therefore we can write (4.16) as follows, by using (5.1) and (5.2):

w(r,O,z,t)~l~,~.B.(r/t)w.(kn, z)Im{ ei(k"r-'%t-'~/4-('~/4)~':) ~ Ai(k.,O, ton)}. r n k.

iffil

(5.9)

In (5.9) w is expressed as a sum of outward travelling modes w,(k,, z), one representing the surface wave and the others the propagating internal waves. The amplitude of each mode is a sum of six contributions from the various mechanisms of excitation. The amplitudes will be considered further in subsequent sections. We shall now present expressions analogous to (5.9) for the other physical quantities. To obtain these expressions we consider the equations of motion (2.1)-(2.7) and assume that the inhomogeneous terms Q, R, $, P and W vanish for r and t sufficiently large. We first use (5.9) for w in (2.3) and solve the equation for p by integration with respect to z. Then we use the resulting expression for p in the xl and x2 components of (2.3) and solve for ul and u2. Next we use w in (2.2) to get p. Finally we use w in (2.6) to get ~j. The results for all these quantities are the same as (5.9) with an extra factor multiplying the exponential function in each

J.B. Keller et al. / Internal and surface waves in stratifiedfluid

222

case. The factors are given in the following list:

p

ito,po(X)w'(kn, z)/k]wn(k,, z),

Ul

i cos Ow'(kn, z ) / k , w , ( k , , z),

u2

i sin Ow',(k,, z)/k,wn(k,, z),

p

rlj

(5.10)

ip~(z)toS 1,

iw~(k,,-hj)/to,w,(k~, z),

ho=-O.

An important special case of these results is the axially symmetric one in which all the prescribed functions depend only upon r and t, but not upon 0. Then the Ai and the solution w, p, etc. are independent of O, so we can set O = 0 in (5.2)-(5.9). This is so because the Fourier transforms (3.1) and (3.2) depend only upon the magnitude k of the wavenumber vector but not upon its direction. These Fourier transforms can also be expressed as Hankel transforms by utilizing the integral representation of the Bessel function 3"0. They become

f(k"'z'l")=(2"tr)l/2fo

oo fo f(r'z't)J°(k~'r)ei~trdrdt" (5.11)

oo f(kn, z) = (2"rr)1/2 fo f(r, z, O)Jo(k,r)r dr.

Let us now consider the corresponding two dimensional problem in which all quantities are independent of x2, u2 ~ 0 and 82 ~ 0. We set xl = x, k~ = k and define the transforms (3.1) and (3.2) without integration over x2, and with k2 = 0. Then the preceding results apply up to and including the result (4.11) provided the k2 integration is omitted from (4.9) and (4.10). The phase in (4.10) is then given by ~b~ = k,x :v w, (k,)t. The stationary points are determined by (4.15) with r = Ixl. The stationary phase evaluation of (4.10) yields a result similar to (4.16) which we can rewrite in the following form:

1 { 6 } w(x, z, t)~,---ZrT~T. ~, kZ~/ZB.(Ixl/t)w.(k., z) Im e i¢'k"lxl-°J"t-Or/4)sgn°j") ~, A~(k, sgn x, to,) . (5.12) IXl'--

k.

i= 1

H e r e A~ is defined by (5.3)-(5.8) with 0 = 0. The results for the other physical quantities are also of the form (5.12) with the factors given by (5.10) included, and with 0 = 0 in them. In (5.12) B , is given by (5.1).

6. Moving sources The preceding results are based upon the assumption that the function V, which is just F times the right side of (4.7), has no singularities below the u integration contour of the integral in (4.9). We shall now consider a source moving with the constant velocity U, for which this assumption is not valid. Therefore we must reevaluate the integral (4.9). Let us suppose that the functions u*, p* and rt*, which occur in the initial conditions (2.8)-(2.10), all vanish. The compatibility condition (2.8) then shows that Q = 0 at t = 0. We assume that the inhomogeneous or source terms Q, R, S, P and W in (2.1)-(2.4) and (2.7) are of the form R e [ f ( x l - Ut, x2, z) e-i"']. Thus they represent sources moving in the positive xl direction with velocity U and oscillating with angular --1 A frequency Ix. Their transforms are then of the form -½i[(u - kl U - Ix)-x + (v - k~ U + Ix) ]f(k~, k2, z). As a

J.B. Kelleret al. / Internaland surface wavesin stratifiedfluid

223

consequence, V is of the form

V(k, z, v ) = - ~/[. ( v - k , U - l z ) - l + ( v - k l U + i z ) - ' ] V l ( k ,

z).

(6.1)

When (6.1) is used in (4.9), the v integration can again be done by residues. There are two additional poles at v = k~ U ±/z and thus instead of (4.10) we obtain w (x, z, t) = - - 2(2"rr)2

±ton(k)-klU-I.~+±ton(k)-klU+lz

n=l ±

Vl(k, z) 2(lr)2ff~F(~,,~l~)~

F~[k,±ton(k)]

e i[k'x:g~n(k)t]

dkl dk2

i[k.x_(klU±la.)t ]

dkl dk2.

(6.2)

The k~ integral of the nth term in the first sum can be evaluated in terms of the residues at the four poles k~ = [±ton (kn) ± ~ ] U -1. The kl integrals in the second sum can be evaluated at the poles of 1/F. The results for both sums are the same and combine to yield i

oo

W(X, Z, t)= --'~ n~=le *i~U-'x' f Vl[(±ton(kn); l't ) u - l ' k2, z] ei[±u_t~.(k.)(x _ut)+k2X:]dk2

(6.3)

F~Ekn, ±~o, (kn)]

::l:

H e r e the sum over ± includes four terms. Two correspond to ±/.~ with +ton(kn) and two to ±/.~ and -con(k). For each choice of signs, kn is determined by the equation k 2 = U-2[ton(kn) :~/z]2+ k~.

(6.4)

To evaluate the k2 integral we introduce r and 0 defined by

x, - Ut = r cos 0,

x2 = r sin 0.

(6.5)

For large r we shall use the method of stationary phase. The phases of the two exponentials which occur in (6.3) are r ~ where =~

cos 0 + k2 sin 0.

The stationary phase condition Oc~n/Ok2 = from (6.4), we can write this condition as k2to~' ( 1 -

knU

tonto'. knU 2]

(6.6)

0 yields U-~to" (kn)Ok,/Ok2 = ;:tan 0. U p o n calculating Okn/Ok2

= ~tan 0.

(6.7)

Equations (6.4) and (6.7) determine the values of k2 and k, at the stationary point. If k2 is a solution of (6.7) with the upper sign, then - k 2 is a root of (6.7) with the lower sign. Only solutions of (6.4) and (6.7) with k2 real yield stationary phase contributions to the integral over k2 in (6.3). We shall denote these solutions kn and kn2. We now evaluate the k2 integral in (6.3) by the method of stationary phase. Then we combine the + and terms by using the facts that V1(-k, z) = ~'~(k, z) and Fv(k,, -ton) = -F~(k,, to,). Next we write V~ in the form

Vl([ta,(k,)- i~]U -1, kn2, z) = w,(k,, z) i~=~Ai( t°n(k-~-- IZ, kn2).

(6.8)

J.B. Keller et al. / Internal and surface waves in stratified fluid

224

The quantity A6 is 0 because the initial data were assumed to vanish. The A~ are given by (5.3)-(5.7) with kn cos 0 replaced by u-l[ton(kn) -/x], k, sin/9 replaced by kn2. The last argument ton is absent because the factor e - ~ ' in P, IV, Q, R and S has been transformed explicitly above. The asymptotic form of (6.3) can then be written

w(r,O,z,t)~ 2nrnk. i_____~,F~[k,, OAn(kn)]l&n+,k2k2l wn(kn, z) k~ 1/2 Im{ exp(mi/zt + i r [ ( t ° n ( ~ : / x ) + ~in -sgn

cos 0 + kn2 sin 0]

:v/~ ,k,2)}. ,n+k2k:)i=t ~.Ai (ton( k_~__

(6.9)

In (6.9) the sum over n includes only those values of n for which (6.4) and (6.7) with the upper sign have a solution kn, kn2 with kn2 real. The sum over kn is over these solutions for each n. The sum over plus and minus is over the two signs of tz. The second d e r i v a t i v e 02~2/02k2, obtained from (6.6) and (6.4), is given by +

¢,,,k:k: = 1

. 2 ton (k.)to"p (kn)] - 3 [to~k~2

knU2

j L k2U

12 , 2 I kaU 1 - ' ~ton + knU\ ton (1 tonk,2

r

2

knUE] J cos O.

(6.10)

The results for the quantities p, gl, u2,/9 and rb are the same as (6.9) with a corresponding factor (5.10) multiplying the exponential function. Let us now consider the two-dimensional case in which all quantities are independent of x2, u2 - 0 and S2-- 0. Then we set xl = x and kl = k and define the transforms (3.1) and (3.2) without integration over x2 and with k2 = 0. Then the above results apply up to and including (6.3) provided the k2 integration is omitted from (6.3) and the preceding equations. The + and - terms in (6.3) can be combined as before and the result can be written in the form

1 w(x, z, t ) ~ - ~

wn(kn, z)

Im{e ;i~''+iv-'"°-;~'''x-U° ~ A [t°n:V/z~/

i=1 i \ " - ~ } j .

E,"Ek. F ~trk -, eo.(k.)]

(6.11)

Here k, denotes a root of (6.4) with k2 = 0. The A, are determined as before and are independent of k2. We can find all the other quantities by multiplying the exponential function in (6.11) by the corresponding factors in (5.10). The result (6.11) remains asymptotically correct for x - Ut large if the exponentially decaying terms are omitted. Then the sum may be restricted to those n and kn for which ton(kn) is real. In the important special case of a uniformly moving steady source we have /x = 0. Then the three dimensional result (6.9) simplifies to

w(r,O,z)~r~rEEFr,"

wn(k,,z)

+ 1/2 ~on(kn)]lCn.~2~l , [ r~on(kn) 8+kn2sin o] +-~-sgn i'rr Cn,k2k2) 5A ×Im/exptirL----~--+ n k.

vL~n,

cos

ita,,(k,,) ' t - - b - - ' kn2)}.

(6.12)

Similarly for/z = 0 the two dimensional result (6.11) becomes

n n knFv[k,,ta,(k,)] Im

y" A, -~ .

eiU-l~°"(x-Ut) i=l

(6.13)

J.B. Keller et al. / Internal and surface waves in stratified fluid

225

7. Specialization to three simple density profiles

We shall now specialize the preceding results for the three cases of a single layer of constant density and finite depth, two layers each of constant density, and a fluid of exponentially varying density. A. When po(z) = p = constant, (3.16) yields N = 0 and then (3.14) becomes w " - k2w = 0. The solution of this equation which satisfies (4.2) and (4.3) is w(k, z) = sinh k(z + h) and the corresponding value of v is u = to(k) = [gk tanh kh] 1/2. Thus there is only one propagating mode, the surface wave, and from (4.14) its group velocity is c = (gh ) 1/2qb(kh ) where 4~(tr) = ~(tr 1 -1 tanh tr) 1/2 + 2 cosh 1 2 o" (o"-1 tanho') -1/2.

(7.1)

Then (4.15) becomes $(tr) = r/(gh)l/2t, which has one positive solution tr if r < (gh)l/2t and no solution if

r > (gh)l/2t. We now determine F from (4.5), calculate F~ = (pk/to) sinh 2kh and use this in (5.1) to get B. Then from (5.9), noting that ~b,, < 0 and sgn to"= - 1 , we obtain

w(r, O, z, t ) -

1 ( 'a'prsinh2tr

gtanhcr~b(°')~ '/2

~

]

sinhcrh-l(z+h) Im

{

6

ei(°'r/h-c°t) i=l ~

Ai(tr/h,O, to)

}

for r < (gh)l/2, t (7.2)

w(r, O, z, t)~0

f o r r > (gh) 1/2.

t

In the two dimensional case, (5.12) yields

w(x,z,t)

1 ( ~rpsinh2~r

g tanh cr~b(cr).~1/2 / sinh°'h-l(z+h)

×im e_[_i(Ixl,,/h-o,t+,,/4) l i~1 ~ A'(hsgnx'w')}

for Ixl< (gh)l/2, t (7.3)

w(x, z, t)-O

forlXl>(gh) 1/2. t

In (7.2) and (7.3), ~r is the solution of ~b(o') = r/(gh)l/2t or ~b(cr) = Ix I/(gh)l/2t respectively, so w ~ 0 for r or Ixl > (gh)l/2t, the Ai are given by (5.3)-(5.8) and in (7.3) we must set 0 = 0 in the equations for Ai. For the steadily moving source, (6.12) and (6,13) become

l(2gh~ 1/2 [ tanho- ~l/Zsinh~rh-l(z+h)

) } \~l~k=k~l/ to

sinh 2or +-~-sgn ~bk2k2) ~ A I ( V , k2)},

(7.4)

i=l

w(x-Ut, z)-

~

ItanhtCxl/2sinhtrh-l(z+h) r i U _ l t o ( x Vt) 5 to Im t e cr -) s~i-n-h 2 ~ ,=~. a , ( ~ ) } .

(7.5,

J.B. Keller et al. / Internal and surface waves in stratifiedfluid

226

Here to = (gh-tor tanh or)l/2 and the sum is over the roots or of the following equation obtained from (6.4) and (6.7): (~_~ cosh or sinh or +or U2 or 1)1/2 (7.6) tan O = tanh or 2(U2/gh)or c o s h 2 Or -- cosh or sinh or - or" In (7.4)

t#;2k 2 O c c u r s ,

gh

+

~)k2k2 =

and it is given by (6.10), which can be simplified by using (6.4) for -3

k2 to

gh

Uto

or c o s h 2 or

gh or2(sinh or cosh or+ or)I x{(or--~2tanhor)(sinh2orcoshEor+orecosh2or)

U2

sinhorcosh2o "

jcos0.

(7.7)

B. When po(Z) = p l for - h < z < 0 and po(Z) = p for -oo < z < - h , then N = 0 in each layer. Then (3.14) becomes w " - k 2 w = 0 and the solution of this equation satisfying (4.1)-(4.3) is (P - p l ) ( v 2 - g k ) e -k(z+3h) (p+pl)v 2-gk(p-pl) ' O>~z >~-h,

Iek(z-h)

=

w(z)

"e k(z-h~ (p_pl)(vE_gk)ek(Z_h ~ (P +Pl)V2-gk(p -Pl) '

(7.8)

- h >!z.

Now (4.5) becomes

-2kh[ kh V2--g k . , -kh v2--g k ] [e v 2 + g k * t p - p l ) e (p+O1)v2_gk(p_01)

F(k,v)=2ple

(7.9)

The two positive roots of F = 0 are

to1 = (gk) 1/2,

0)2 =

P

g k ( p - p l ) ]1/2 c - ~ k-h +-plJ

(7.10)

In this case (6.12) and (6.13) reduce to

., f2- V

w.(k., z) ÷ ~rr . F.[k., ~.(k.)]l,/,..~,=,,=l

w(r,O,z)-Xl-~,., .

f

[.

I'to.(k,,)

1/2

xlm~exp~lr[----~cosO+k.2sin

i,rr

,+

"]'~

/to,,(k,,)

O+--~sgnm..k=k=J] i~=1Ai~----~-, k.2) }, (7.11)

w(x_Ut, z) 1~. w.(k.,z) im{e_iO,u_,(x_u,)~A[to.]~ ~r. F.[k.,to.(k.)]

,~=1 '\-U/J"

(7.12)

Here the sum is over the roots of (6.4) and (6.7), while F~ is given by

F~[k, to(k)] = 8togkpl e

-2khr ekh 101(/9 - - P l ) e -kh ] ](toE+ gk)2 + [(p +pl)to 2 - gk(p -pl)]2J"

(7.13)

The equations (6.4) and (6.7), corresponding to the second equation of (7.10), simplify to o r + a sinh or (oral2- sinh or~1/2 tan O = a:2/2ora2_sin h or_or\ s-~nh~- ] '

(7.14)

J.B. Keller et al. / Internal and surface waves in stratified fluid

227

where Or = kh,

f2 =

U2p (P - P 1) gh'

a = cosh o. + ~p sinh Or.

(7.15 )

Equation (7.14) is exactly the same as equation (40) of Hudimac [1]. In (7.11), ~¢)n+,k2k2 is given by (6.10) where to: is given by the second equation of (7.10) and U o. + a sinh or , f x/a 3o- sinh Or

(7.16)

k2 _ k2 oral 2 - s i n h or 2o.af2

(7.17)

to' =

Furthermore, in this case (7.13) simplifies to a2p F~(k, to) = - g2k 2(p - p l ) "

(7.18)

Corresponding to the first equation of (7.10), the equations (6.4) and (6.7) simplify to If

g

"

g

• ~1/2

tan 0 = ~ : ~ [ ~ - - ~ ( 1 - ~ U 2 ) J

g

(1-2--~)

-1

.

(7.19)

In addition,

---12 F~(k, to)=-~kgk[Pl e -kh + Pl P--(p --Pl) e-3kh].

(7.21)

C. Let us now consider a fluid with an exponential density variation po(z) = p e -"z.

(7.22)

In this case, which has been studied by Lamb [6], the homogeneous form of the differential equation for w becomes w " - otw' - k2(1 - otgv-2)w = 0.

(7.23)

The solution of this equation with v = to, which satisfies (4.3), is given by w ( k , z) = e "z/2 sinh/!l(z + h),

(7.24)

provided to2 # a g and k 2 = ~2_a2/4 1 -agto

-2.

(7.25)

This solution satisfies the free surface condition (4.2) if/!~ is a root of the equation t a n h / / h = k 2 _/~ 2 _ a 2/4.

(7.26)

228

J.B. Keller et al. / Internal and surface waves in stratified fluid

F r o m (4.5) we can compute Fv(k, 0)) and we find 2

(7.27) By using this equation in (5.1), we can reduce (5.9) to 2p e "z/2 w(r, O, z, t)

0).

0 ) ' ( k . ) 1/2

( k . - / 3 . + a /4) smh /3.h

{

x s i n h / 3 . ( z + h ) I m e i(k"r-o'"t-'n/4-(w/4)sgn'")

6

}

~., A i ( k . , O, to.) .

(7.28)

i=l

T h e sum is over those n for which kn is real. If 0 ) 2 = ag, only the trivial solution of (7.24) satisfies (4.2) and (4.3). W h e n otg0)-2 < 1, (7.26) has three real r o o t s / 3 = 0, +/30 and infinitely m a n y pairs of imaginary roots /3 = +i/3,, n = 1, 2 . . . . . The r o o t / 3 = 0 yields the trivial solution w = 0. All the imaginary roots lead, by (7.25), to negative values of k 2 and therefore to nonpropagating waves. The roots +/3o yield one positive value of k 2 and therefore one m o d e of propagating waves. W h e n 1 < ag0)-2 < 2, (7.26) has the trivial real root/3 = 0 and another pair of real roots/3 = +/30 provided that a h / 4 > (ago) - 2 - 1 ) / ( 2 - a g o ) - 2 ) . This pair of roots yields k 2 > 0 and therefore corresponds to propagating waves. T h e r e are also infinitely m a n y pairs of pure imaginary roots/3 = +i/3,, n = 1, 2 . . . . . all of which yield k 2 > 0. Thus in this case there are infinitely m a n y modes of propagation. For large values of n,/3, = n Trh-l+ (~ -g-10)2)/n.ff+ O(n-2). T h e r e f o r e from (7.25) the corresponding value of k 2 is 2+

2

2

2 (n~r/h) a / 4 20) - . -1, k,= agto_2_ l +--g-~+O(n ).

(7.29)

W h e n ago)-2 > 2, (7.26) has only the trivial real root/3 = 0 and infinitely m a n y pairs of pure imaginary roots/3 = +i/3,, n = 1, 2 . . . . for each of which k 2 > 0. For large values of n,/3, and k, are given by the expressions above. When ago) -2 -- 2, (7.26) has no real solutions and infinitely m a n y pairs of pure imaginary solutions for each of which k 2 > 0. For large n the formulas above apply to them.

8. Wave generation by an explosion or a moving body Finally, we shall simplify our results to yield the motion due solely to injection of fluid at the volume rate Q ( x l , x2, z, t) per unit volume. Thus R = S = P = ~j = W = u* = 0. F r o m (5.3)-(5.8), we find that Ai = 0, for i # 3, and P o

A 3 ( k , , 0, 0),) = 1

p o ( Z ' ) w ' ( k , , z ' ) O ( k , cos 0, k, sin 0, z', to,) dz'.

(8.1)

d_ h

This last expression is obtained from (5.5) after integrating by parts once. T o represent an explosion we suppose that Q is an impulsive point source followed by a sink, of " s t r e n g t h " Qo, so that Q = OoS(Xx)8(x2)8(z - Zo)8'(t).

(8.2)

ZB. Keller et al. / Internal and surface waves in stratified fluid

229

Then

0 = -itoQoS(Z -Zo).

(8.3)

Thus (8.1) becomes

Aa(kn, O, ton) = -itonQopo(zo)w" (kn, Zo).

(8.4)

By using this in (5.9) we obtain

w(,, o,z,t)

l"

n k,,

z)w:(kn, zo) n

(8.5)

Here Bn(r/t) is given by (5.1). A moving body can be represented by a point dipole of strength Qo which is moving uniformly,

O = OoS'(xl + Ut)8(x2)8(z - Zo).

(8.6)

In this case w(r, O, z, t) is given by (6.12) with Ai = 0, i # 3 and A3

wn ( ton, U kn) =iOooo(Zo)w" (k., zo)-u"

(8.7)

Acknowledgment Research was supported by the Office of Naval Research, the Air Force Office of Scientific Research, the Army Research Office, and the National Science Foundation. D.S. Ahluwalia was also supported by the Office of Naval Research at Lamont-Doherty Geological Observatory.

References [1] A.A. Hudimac, "Ship waves in a stratified ocean", J. Fluid Mech. 11, 229-243 (1961). [2] C.C. Mei, "Collapse of a homogeneous fluid mass in a stratified fluid", in: Proc. 12th Intern. Congr. Appl. Mech., Springer, Berlin (1969) 321-330. [3] J.W. Miles, "Internal waves generated by a horizontally moving source", Geophys. Fluid Dyn. 2, 63-87 (1971). [4] H.C. Kranzer and J.B. Keller, "Water waves produced by explosions", Z Appl. Phys. 30, 398-407 (1959). [5] J.B. Keller and W.H. Munk, "Internal wave wakes of a body moving in a stratified fluid, Phys. Fluids 13, 1425-1431 (1970). [6] H. Lamb, Hydrodynamics, Cambridge University Press (1932).