Internal waves in channels of variable depth

15 S

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7

ARCHIEF

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bhotheek van d

Onderade!n. .r'--

-epbouwkunde DCUMEN ÍATIE

DAI UM: g OKT. 1973

22

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Internal Waves in Channels of Variable Depth

by

Chia-Shun Yih

Department of Engineering Mechanics

The University of Michigan

Ab st rac t

Internal waves in prismatic channels of variable depth propagating

along the channel axis are studied. It has been shown that for whatever

stratification of the fluid the frequency of the wave motion increases

whereas the wave velocity decreases as the wave number increases. A

general method of solution for an arbitrary channel is then presente in detail, which gives the wave velocity and the fluid motion for a given wave number and a given mode by successive approximations.

long waves are studied in some detail, a few specific examples of long

waves are given, and the Connection of the present theory

with

the

classica. shallow-water theory is shown.

Lakv.

Technische

_Hogeschooß

1. introduction

Known solutions of gravity waves in a prismatic channel of variable depth which have a degree of general applicability are of three

cate-gories. For very long waves (first category) the shallow-water theory

(Lamb 1932, pp. 273-274) gives (gh)2 as the wave velocity, where g

is the gravitational acceleration and h the average depth. For very

short waves (second category) not confined to the edge region the variability of the depth is unimportant, since the motion is confined

to the region near the free surface. The third category is the

category of edge waves, which for short waves have an apnreciahle amplitude only near the shores (or the edges), and are therefore always affected by the geometry, specifically the slopes of the

channel near the shore lines, however short the waves compared to the

maximum or average depth. The solution for edge waves (Stokes 1839,

or Lamb 1932, p. 447) is exact if the region occupied by the water is semi-infinite, hounded only by the free surface and a plane of

constant slope serving as the only solid boundary. If the channel

is finite in both depth and width, Stokes'solution is nevertheless valid for each shore if the wave length (In the longitudinal direction) is very short, since the variability of the depth has then an impor-tant effect only near the shore lines.

Aside from these three categories, and interrelating them, are the exact solutions for water waves in a symmetrically placed trian-guiar channel of vertex angle î/2 (Kelland 1839, or Lamb 1932, pp.

447-449) or of vertex angle 2Tr/3 (MacDonald 1894, or Lamb 1932, _pp.

449-450). These exact solutions are useful because they provide a

In this paper internal waves in channels of variable depth are

studied. The differential system governing the flow of a system of

superposed layers of homogeneous fluids is formulated by first

consid-ering a single layer. Then for the layered systci it is proved by the

use of comparison thorcms that the frequency of the waves increases

but the wave velocity c decreases as the wave number k increases. Then

the differential system governing the flow of a continuously stratified

fluid is derived, and the increase of and decrease of c as k increases

are again proved in general.

After giving a few solutions in closed form (under the restriction of the Boussinesq approximation), a general method of solution for

wave motion in stratified fluids is given. In the form given the method

is for application to continuously stratified fluids, but it can he adopted to deal with homogeneous fluids, and the manner of adoption is briefly indicated in the last paragraph of Section 7.

Finally we study long waves in some detail, and both continuously

stratified fluids and layered systems are considered. A few examples

are given, and the connection of the theory to the classical shallow-water theory for long waves in a single fluid is shown.

2. The diffcrentil ystcm for the case of constant density

If viscous effects are neglected and the motion is supposed

to have started from rest, and if the density of water is constant, the

motion is irrotational and a velocity potential exists, the gradient

of which is the velocity vector. We shall use the Cartesian coordinates

(x, y, z), with z measured longitudinally, y measured vertically,

and x measured across the channel. If the velocity components in

the directions of increasing x, y, and z are denoted by u, y, and w,

respectively, we have

u

v=-2y'

The equation of continuity then becomes

2

d

C +

+ __;_ )

= C)

2 2

which is the equatioa to be solved.

At solid boundaries the normal velocity component vanishes,

so that

where n is measured in a direction normal to the solid boundaries. At the free surface the pressure is constant, so that, with the

square of the velocity neglected and with r denoting the displacement

of the water surface from its equilibrium position, the AScrnoulli

equation is

w = -

(1)

(2)

-5-(7)

+ _{= constant}

₍₄₎

in which

t is the time. Since

v--,

(4) can be written as

t2

+gO

I

(5)

which

is the free-surface condition. We

shall

assume

= f(x,y)expi(kz-t) (6)

in which k is the wave nuinher and o/2rr is the frequency. _{Then (2)}

be comes f

+f -kf=O

xx _yy (3) becomes o

(8)

and (5) becomes

(10)

o

-= gf at the free surface, (9)

with subscripts indicating partial differentiation. Equations (7),

(8), and (9) constitute an eigenvalue problem, of which, k2 being

given, a2 is the eigenvalue to be found.

3. The differential system for superposed layers

If the fluid system consists of superposed layers, each of which is homogeneous, the differential equation for each layer is still

(2), the boundary condition (3) still holds. The interfacial condition

at each surface of discontinuity can be derived in a way similar to

the way in which (5) is derived. The result is

-'

-+ (P2.-P)&, = o

at

in which the subscript 2. indicates the lower fluid and the suhscrint

u the upper fluid and If there is a free surface,

= O there, and (10) reduces to (5). With the form of (for any

layer) given by (6), we can write (10) as

a2[f)2.-(pf)] =

The differential system consists of (7) for each layer) (s), and (1])

4. Variation of a2 or c2 with for constant or step-wise density

We can slow that 2 increases and c2 decreases as k2 increases,

in a very general manner, whether or not the fluid is stratified.

Consider first the case p = constant, and let k2 have two values,

2 2 2

k1 and k2. The corresponding eigenvalues will he denoted by a1

and a22, and the corresponding eigenfunctions by f1 and f2. Then

2 2

V2f1 -

k2

,

V =-- +

2

ax ay

2

-k

f2 =0

and the free surface conditions are

a12f1 = g

2

a f

2 2

'y

Multiplying (12) by f2 and (13) by f, integrating over the domain occupied by water, utilizing (8), (14), and (15) whenever necessary, and applying the well-known Creen Theorem, we obtain

a

=1

g m ml i rnO

2

-7-with fr '

ff dA

mO

12

(

lx2x

f

+f f

ly2y

)dA Ja ₍₁₈₎ a+h «T = f1f2dx ,

)

rn a

where a and a+b are the values of x at the shore lines, so that h

is the width of the free surface. The difference of (17) and (16)

is

1 2 2 2 2

g 1

= (k2 -k1 )I

so that in the limit (as k1 approaches k2)

) _C' O 2 J

>0

dk where a+h = f2dA , «T = f2dx -8-a

One consequence of (20) is that the group velocity cg is always of the same sign as the wave velocity c, since

._J =1

+k2

I cL, 2 g in ml (17)

Since a (la (22) I g dk so that da

cc =>O

:' dk2

Now we wish to see how c2 varies with k2. Rewritinp (19) as

2

C

1 2 2 2

-

_{(c -c )J} (k22-k12) (I 2

g 1 2 1 m riO g m

and going to the limit, we have

2 2

!k2E__JrI

i_j

g

dk2 O g

But from (16) , on making k2 k1, we have

2

= +

g

so that the right-hand side of (25) is negative, and we have

2 dc < dk2 dc Cg = C + k (26)

from (23) and (26) we have

0zcc

g

which means that the magnitude of C is always less than the magnitude

of c, whether or not c i.s positive.

For superposed layers, each of which is homogeneous within

itself, (23) and (26) are obtained in much the same way. All we

need to do is to apply the same procedure to each of the layers, and then apply the interfacial condition (11) at each interface.

4. The differential system for a continuously stratified fluid

If the fluid is continuously stratified, we shall denote the density of the fluid when it is undisturbed by p0, which is a function

of y alone. The density perturbation will he denoted by p, so that

the total density is p0 + p. The mean pressure p0 is related to p0

by the hydrostatic equation

dp0

- gp0 . (28)

Then the linearized equations of motion are

P0 (u,v,w) = - (z-

- , 5)p

+ (0, - gp, 0) (29)

where p is the pressure perturbation. The linearized equation of

incompressibility is

gives

3(p0u)

2ç)

+vp =0

(30)

O

with the accent indicating differentiation with respect to y. The

equation of continuity is

ii

v 2w

+ + = O . (31)

x y z

Cross differentiation of the first and third equation in (29)

(p0w)

)E

O

Since dependent variables will all he assumed to have the time factor

-icyt

e , (32) gives

(p0u) (pw)

=0

from which we have

nw=

(z

being a velocity potential

for u and w

Let F be a function of x, y, z, and t defined by

F

In view of (34) ad the third equation in (2e), differentiation

of (36) gives

-i- tyz

(37)

Assuming for all dependent variables the factor exn i(kz-.cit), we can

write (37) as

F=b

(38)

The subscripts in (37) and (38) indicate partial differentiation. Recalling that the first and third equations in (29) are

- px = - pz

and combining (36) and (38) into

ty

we see that

(36)

+gp

(35)

since the exponential tirie factor is understood.

The equation of continuity is

2

with

or

Substituting (39) into the second equation in (29), we have

-

:-- p0v) + gp = 0 (41)

From (30) and (41) we have

2 (q

-

p0v) + gp y - O (42) 2

V-2 p0+gp0

with y given by (43), (40) becomes (remembering the factor exp i(kz-ct))

2 2 D _y

= O . (44)

. k+p

-O Dy 2

p0gp

The boundary condition at the solid boundary is

flu + = O (45)

where u and y are given by (34) and (43), artd n1, n2 and n3 (= (J) are

the direction cosines of the normal to the solid boundary. The

Bernoulli equation is still valid in the free surface (if there is

one). In fact, the Bernoulli equation obtained by integrating (29)

is simply (39). But we must recall that p is the pressure nerturbation

only. If we require the total pressure p + p0 to he zero on the free

surface, and use (28), we have (with the new definition for ₎

.L

+ = constant (at the free surface) (46)

which gives, upon differentiation with respect to t,

gp

- 2 (47)

p0+gp0

5. Variation of 2 and c2 with k2

Por the purpose of establishing the comparison theorems, we shall write (44) as

-14-2 2

k 2

2

)=0

(J

arid we recognize that this is reoUy the sane as the equation of

continuity (31). We also recognize that (47) can be written in the

riLuch simpler form

CT

-= gv

po

We now consider two wave numbers k12 and k22, with the corres-ponding eigenvalues G2 and a22, and the correscorres-ponding eigenfunctions

and The velocity components (u1, V1, w1) and (u2, y2, w2)

s a tis f: u1 By1 Bw1

-+ -4'- o

Bx By Bz Bw

+ - + - =0.

(50) Bx By Bi Multiplying (49) by

2 and integrating over the fluid domain, and

utilizing the Green's Theorem and the boundary conditions (45) and

(47a) for o

_o,

= q1, V = y1, we have

2i

_{+ci (J}

_-hI)=O

+ 1m 1 2 0m i 1m m in which (48) (47a) (49)

i

ff

.L

q) q) dA ,

l

=

Jf

---lx 2x 1m JJ p0 lm

_{= ff}

q)1q)2 dA 1a+h p = 2 t

,1q2dx

1

O'O

1a

where

a and a#b are the abscissae

of the

shore lines.

Similarly, orì

multiplying (50)

and

integrating, we have

2 2 I + k I o (J - I! ) = O 1m 2 0m 2 2m ni where 2m

-ff

2 0g p dA (53)

The difference between

(52) and (51) is

2 2 2 2 (k2 -k1 Om = 2 - 1 )U1 +K )

mm

whe re K m From (54) we have cr' ' O 'n1v''2v 2 , 2 2 p0+gp0 )(cr1

gp0 )

2 _I -16-(52) (54) dA (55) (56)

where I li, und K are respectively the value of I , U , nd K

Q Orn m m

when k1 = k2, and are obviously positive. Hence

We can write (54) as - - 2 2 (k 2 k 2) [I c +K )] = k 2(c -c )(11 +K ) 2 1 Orn i in in 2 2 1 in in which gives dc2

10-c(}1+K)

dk2

k2(11+K)

It is a simple matter to show that

JK>O

where J is the value of J (or J ) when k = k and a = a

im 2m 1 2 1 2

From (Si) , on making k1 = k2, we have

+ (J-11) < O hence - c2ii - c2K = (I + c211 + c2J) - c2(J+K) < O

12

> dk2

-17-(57)

(58)

and (3S) gives

2

dc

< dk'

As before (57) and (59) imply, respectively,

2

cc

>0

and

0<cc <c

t. t

6. Some solutions for arbitrary wave numbers

If the dcnsity stratification is exponential, and if the inertial effect of density variation is neglected (i.e., if the Boussinesq

approximation is used), it is possihie to obtain solutions in closed form for waves in a symmetric triangular channel, whatever the wave number may he, provided there is a free surface.

Let

Po =

p0(0)e'

(61)

where y is measured from the vertex of the triangular channel, and let the sides of the channel be given by

y = (62)

-18-(59)

With given by (61) , (44) becomes xx +

X2(+)

- k2 = O (63) in which 2 G 2

G -g

If the inertial effect of density variation is neglected, (63) becomes

+x2

-k2=O

xx yy

The boundary conditions at the sides are, in accordance with (45) and

(62) and with u and y given by (34) and (43)

;__q

= o

X n y

for the two sides given by (45) , respectively.

The solutions for symmetric modes have the form

= A cosh cx cosh yy cos k(z-ct+c) , (67)

provided 2

22

2

+XY =k

-19-(64) (68)

The in (67) is an arbitrary constant. Since t!ìe channel is svrulietric

and (67) is symmetric with respect to x, it is necessary to consider only the boundary

y = mx (62 a)

where

=

- y

(66 a)

Substituting (67) into (66a) and using (62a), we obtain

am

- tarih ax coth ymx = i

''X2 which demands Thus

cx=ym

am

=1

2 2 2

X =m

(69) (70)

Since m is real, must he positive. Equations (68), (69), and (70)

2 2

2m' = k

so that

Substitution of (67) and (70) into (72) gives

2 1rn kd

/2 tanh (73)

where d is the maximum depth. Given and m, a2 is known, and (73)

gives k. We can also write (73) as

kX

tanh kd (74)

With given by (64), given and k we can find a from (74). Then

is known and hence n2 is known.

Evidently there is also an antisyrruTietri.c mode, given by

= A sinh OEX sinh yy cos k(z-ct+c)

-21-k k

= (71)

Y and

The dispersion relation is given by the free-surface condition which in this case can be written as

2 -,

(47),

Then (U9) , (70), and (71) still stand, but the dispersion fornula is n ow 2 km kd = coth or 2 g1X kd -coth

-;-It can be easily shown from (63) that if there i.s no free surface

must be negative. The solutions given ahoye all correspond to

positive hence the presence of the free surface is essential

for the existence of these solutions. The waves tLese solutions

represent are therefore largely free-surface waves rather than internal waves, and the densit' strati fication has only a minor effectthat

of affecting the value of the slope n. For this reason these solutions

are not very interesting. We note that if 3 = O we have m2 =

(73) becomes

tanh (77)

which is just the solution of Kellaiid (1839).

We shall now proceed to study truly internal waves.

-22-7. A general method of solution

Given a density stratification p0 and a channel cross section,

our task is to find the relation between a2 and k2 determined by (44)

2

and (45), supplemcnted by (47) if there is a free surface. Since a

appears in the denominator of the third term in (44) and as a

multi-plier of that term, (44) is inconvenient to use as it stands. We shall

transform it into an equation in y. Differentiating (44) with respect

to y, and using (43), we have

gp'0 ¡2

2

(po + _(v

-

k)

+ (p0v')

a Jx

with the accent on p0 and on y indicating differentiation with respect

to y..

The boundary condition (45) has to he written in terms of y alone.

Since u is given by (34) and y by (43) , we can write

gp'0

POULj0

a

1(x)J a

(po+

2)vdy+f

where f1(x) is an arbitrary function of x. It will he shown later

that the boundary condition (45) demands that f1(x) be a constant.

hence 'Y _gp1 +

)vdy

. (80) Jo

p0u=

(p a2

Multiplying (45) by Po and using (80), we have

-23-y

°)vdy+n2p0v=0

gp'

0 2

G

in which the boundary geometry not only determines n1 arid n2, hut will

play a role after y has been differentiated with respect to x in the

first term. We shall show later that the integral form of (81) can

be changed to a differential form.

If there is a free surface the condition there is (47) which again

must be expressed in tenas of y. By virtue of (43), (47) can be written

as

gpv

Applying the operator V2 on this and using (40), we have

V 2 (

2'

2 G (81)

Equations (78) , (81) , and (82) constitute the differential system

defining the eigenvalue problem.

We shall now impose two restrictions on our study: (i) we shall

assume that the channel is symmetric, and (ii) we shall exclude

sloshing modes (with motion in the x-y plane only) from consideration. Asymmetric channels can be similarly treated without any substantial additional difficulties, and sloshing modes need a separate treatment. We shall describe the boundary of the symmetric channel by

2

[f(yfl2 X

and consider only the branch

Then the direction numbers (n1. n, O) of the normal to the boundary are

n1=1,n7=_f'(y),n3=O

(84)

Restriction (ii) enables us to use the following expansions:

y = v00(y) + k2v02(y)

+ kv(y) +

.

22

2 4

+ k x [v20(y) + k

v72(y)

+ k v24(y) + .

44

2 4

+ k x [v40(y) f k f42(y) + k v44(y) . .

f

(85) and = X0 + k2X2

f

k4X4

f

whe re 2

G2

C ₌ (t-) -25-J

x=f(y)

(83)

Substituting (85) and (86) into (78), and extracting the terms of zeroth order in k, we have

2X0gp'0v20 - gp'0A0v00 + (p0v'00) = O (88)

which gives V20 in terms of y00.

If terms of zero order in k in (81) are taken, that equation be-comes, with n1 and n2 given by (84),

2x jX0gp'0v20dy

- f' (y)p0v00 = O

J

With x equal to f(y), this becomes

ry

f,

2 X0gp'0v20dy

- ? p0v00 = 0

jo

This equation is valid for all y. Hence we can differentiate it with

respect to y and obtain

2X0gp'0v20 - (f'C1p0v00)' = o (90)

Eliminating y20 between (88) and (90), we have

(p0v'00)'- (f'fp0v00)' + X0gp'0v00 O (91)

The boundary

condition

at y - O is

-27-(95)

v0(0) = O (92)

If the upper surface is fixed, the boundary condition there is

v00(d) = O (93)

If there is no flat upper surface, and the conduit is full of fluid,

(93) can simply he applied at the highest point. On the other hand,

if the upper surface is free, the boundary condition (82) can be written

as

v'00(d) =

-A0g 2v20(d) - v00(d)} (94)

integrating (90) in the Stieltjes sense over a vanishingly thin layer

at the free surface, we obtain

f' (d)

2X0gv20(d)

f(d) v00(d)

Substituting this into (24), we obtain

f' (d)

v'00(d) =

f(d) v00(d)

which could have been obtained by integrating (91) across the free surface

in the Stieltjes sense.

which agrees with the equation governing wave motion (Yih 1965, p. 29)

in a stratified fluid for k = O. Furthermore, (95) becomes

v'00(d) = A0gv00(d)

which is the free-surface boundary condition for a rectangular channel,

and which agrees with (82) when y i.s independent of x and when k = O

(for which A = A0). The conditions (92) and (93) remain unchanged if

f(y) is constant. Hence the differential system consisting of (91),

(92) and (93), or (91), (92) and (95), agrees, as it should, with that for a rectangular channel when f(y) is taken to he a constant.

From the system (91), (92) and (93) or (91), (92) and (95) we can

determine A0 and v00(y). Then v20(y) is known from (90).

We shall describe the next stage of approximation. The procedure

of successive approximation will then he clear. Taking terms of order

k2 and of zeroth order in x in (78), we have

The equation corresponding to (89) is now

y 2 J (X0gp'0v22 + X1gp

'V

+ 0 20 p0v20)dy + I O -

Ç-

_{P0 [v02(y) + f2v20(y)] = O} (97)

-28-where

Differentiation of (97) gives

2(À0gp'0v22 + p0v20 + X1gp'0v20)

(ftPo

02) + (ff'p0v20)' - I' (98)

Elimination of y22 between (96) and (98) gives

Li

O

L y02 = - f [(ff'p0v20) + p0v00 - X1gp'0v20 +

T,'

in which L is the operator on y00 in (91). We note first that when

(85) is substituted into (78) and terms of order x2 are collected0

y40 can be expressed in terms of V20, and is therefore known. Thus I

in (98) is known.

The boundary conditions are, if there is no free surface,

v02(0) = O = v2(d) (100)

1f we now multiply (91) by v2 and (99) by y00, and integrate

the

resulting equations, by parts if necessary, and using (92), (93), and (100) whenever possible, we obtain two equations the left-hand sides

of which are identical. Taking the difference of these two equations,

we have -29-(99) 'r I = 4f X0gp'0v40dy o

-30-J:

[(ff'p0v20)' + poyo0 + I' - A1gp'0v20]dy = O (101)

which determines À, since y20 and y00 are known. Then (99) can he

integrated by the method of the variation of parameters to give y02.

Then y22 is known. Further approximations follow the same pattern.

If the upper surface is free, the free-surface boundary condition can be found by integrating (99) in the Stieltjes sense, and an equation

similar to (101) can be found. In fact, to obtain it one need only

add the terms

- (f2f' - À1fg)p0(d)v20(d) - 4f2X0gp0(d)v40(d)

to the left-hand side of (101).

It remains to show that the f1(x) in (79) can be taken to he a

constant. The argument is as follows. We have obtained successive

approximations to the eigenvalue and the eigenfunction, at each stage

satisfying all the boundary conditions. If f1(x) is not a constant, it

is an additional term for the potential in (34), which gives rise

to an additional velocity whose y component is zero. That

velocity therefore cannot possibly satisfy (45) unless its x-component

is zero or unless f1(x) is a constant.

In the next section we shall study long waves. But before we leave

this section we shall make two comments: (i) The method of expansion can

be slightly modified to deal with unsymmetric channels. (ii) If the

fluid is not stratified, p is constant, and from (90) one deduces that

for the second approximation, in which terms of 0(k2) are considered,

it is found that y02 is governed by the equation

-1

V"

-02 (f'f y02) = O

which is what (99) becomes. The boundary condition on V02 is

identical with (95)

f' (d)

v'02(d) [X0g

f(d) v02(d).

Equation (102) is easily integrated, giving the result

V'

-f'f

02

1O2

=

and

Substitution of (104) and (105) into (103) gives

2 JO

=gh

c0 g

f(d)

where h is the average depth. Equation (106) agrees with the result of

the classical shallow-water theory for long waves. Thus every

compari-son we have made indicates the correctness of our procedure.

-31-(105)

8. Long waves

We shall give some specific examples of the speeds of long internal

waves, and shall consider two special classes of density stratification.

(i) Exponential density distribution. If the density distribution

is given by (61), we can work directly with (44), which in this case

becomes (63), with À2 defined by (64). If we expand in a power series

2

in k , we have

q00(y) + k202(y) + k4q04(y) +

+ k2x2

[() +

k24)22(y) 4. . . 1 (107)

44

+ k x [q40(y) + k242(y) f . _. I

Since we expect to be negative for internal waves (i.e., waves that

do not owe their existence primarily to the free surface), and since X2 contains the factor a2, which contains the factor k2, we shall write

a2k2 + a4d4 + . . (lOS)

in which

a2 = (109)

We shall endeavor to determine y for a given channel cross-section

and a given in (61).

Substituting (107) and (108) into (63) and taking terms of order

2

k , we obtain

22O -

-2 _{+ - 00 =}

The terms containing k2 in (111) are

2mx20(y) +

y2'00(y)

O (112)

or

2y 20(y) +

y2'00(y)

= o (113)

after substitution of y for mx (it being sufficient to consider one half of the symmetric boundary).

Combination of (110) and (113) gives

Let the channel boundary be given by (62). Then the condition

there is, from (34), (43), (45), and the definition of X2 given by (64),

-33-1 y 00

2o

:p"00+ (+-)

4:'

f

o . (110) (114)

If the inertial effect of the density variation is neglected (Boussinesq approximation), this equation becomes

00 y 00

2

(115)

the solution of which satisfying the boundary condition at y O is

where J0 is the Bessel function of the zeroth order. If the

upper surface is fixed, the eigenvalues for y are the roots of the

equation

J1(yd) = O (116)

since y is proportional to 4'00(y) at this stage of approximation, and

since

J'0(x) =J1(x)

If the upper surface is free, then (47) gives the condition

- _{0(d) = '00(d)} ,

so that (115) is replaced by

J(yd) yJ1(yd) (117)

which gives y. Once y is known, the long-wave speed c0 is calculated

from

-34-_, -,

kc

' -,

O 2 _V

-g

, or c0

lt is important to note that the roots of (116) or of (117) are

for internal waves only. The speed of waves due predominantly to the

presence of the free surface is found in the following way. First of

all, differentiation of (46) with respect to t gives directly

We see that there are no terms iree of k in (63) and (119). hence

400(y) = constant = 1 (say)

which satisfies condition (45) at the channel boundary, is an acceptable

solution. Proceeding to the second approximation, whether or not the

ßoussinesq approximation is used, we reach a nonhomogeneous differential

equation in p02(y) the solution of which together with the boundary

conditions then determine ' or c0. The c0 so determined is not

pro-portional to , but is much larger, and the corresponding waves are

predominantly surface waves, the density stratification merely causing

a minor correction if is small.

Because of the convenience afforded by the exponential density

stratification, we have used the differential equation in instead

of (78). Remembering (61), (86),

f= x=my

-35-(118)

and

z

_f

₊

_k2f

₊

_k4f

mO m4

for the mth layer, counting from the bottom up. Furtherrnore, we shall

write

2 2

22

42

=a0 k

+kc4 +

Substituting (120) and (121) into (7) and taking only terms free from k, we see that the solution is

f L

mO in

2

-y--co

one can readily show that (114) is equivalent to (91). In facts

'00(y) is proportional to p0v00(y).

(ii)

Superposed homogeneous layers.

If the fluid is composed of

superposed layers, in each layer the governing equation is (2) or (7). The boundary conditions are (8) for the rigid channel boundary, (11)

for the interfaces, and (9) for the free surface. 0f course, (9)

is a special case of (11). Note that the f in (7) is not the f in

(83).

The solution is now not restricted to symmetric channels. Suppose

there are n layers. We shall use the expansion

a -36-a0 O

(120)

(121)

(122)

which satisfy all the boundary conditions if only terms free from k

are taken.

For the second approximation we have to solve the equations

2 2

+

)f

-c form=l,2,

2 2 m2 in

together with the boundary conditions. Now (123) is just the equation

for potential flow with uniformly distributed sources of strength Cm If (8) is satisfied (with f now identified with

rb1 I ' (y)dy = A1C1 j 12 o -37-n

c2bpC

gp

E AC

O

nnr.

n

mm

m= i (123) b2

f'22(y)dy = A1C1 + A2C2 (124)

rb

n n

E AC

o m=1

by virtue of continuity. In (124) bm is the width of the mth interface,

and Am the cross-sectional area of the mth layer. (See Figure 1.)

Integrating (11) across h, layer by layer, we have

2

C0 b1(p1C1 - p2C2)

g(1

There arc n unknowns C not all of which are zero. Hence we obtain

In

a determinant which must vanísh. Its vanishing gives n values for

c02. The last of the equations in (125) corresponds to a free surface.

If the upper surface is rigid, it is to he replaced by [since f'2(d) must vanish, d being the height of the upper surface measured from the

lowest point of the channe].i

n

Z AC =0

_mm

m= i

Indeed, the theory given here is a natural generalization of the classical shallow-water theory for a single fluid of constant density.

1f there is only one single fluid, and if it has a free surface, the last equation in (125) gives, with n = 1, b = b1, and A = A1,

2

c02b = gA , or c0 = gh

h being the average depth. This is a classical result.

Acknowledgment

This work has been sponsored by The Office of Naval Research, under

Grant NR-062-448 to the University of Michigan.

-38-Ribliography

Kelland, P., 1839, on iaves, Trans. R. S. Edin., Vol. 14.

Lamb, Fi., _{1932, Hydrodynamics, Cambridge Univ. Press and}

The MacMillan

Co., Sixth Edition.

MacDonald, ILM., 1894, Waves in canals, Proc. _{Lond. Math. Soc., Vol 25,}

p. 101.

Stokes, G.G., 1839, Report on recent researches in _{hydrodynamics, Brit.}

Assoc. Rep., 1846.

Yih, C.-S., 1965, Dynamics of Nonhomogeneous _{Fluids, The MacMillan Co.,}

New York.

-39-Figure Caption

Figure 1. Definition sketch.