NPS ARCHIVE
1969
BOWMAN, W.

LONG WAVE PROPAGATION IN A
TRIANGULAR CHANNEL

by

Wi 11 iam Toft Bowman

NAUAL,EIKN°XL,BRARV

United States
Naval Postgraduate School

THESIS

LONG WAVE PROPAGATION

IN A TRIANGULAR CHANNEL

by

William Toft Bowman

October 196 9

Tku document ken bten appAjovzd rfoi pub tic ie-
Ivue. and 6alz; lti> dUVUbution lb unUmittd.

\tf>r**9 Pnstrraduate School
u.s. Haval Pos^nla 93940
Monterey, Can

Long Wave Propagation
in a Triangular Channel

by

William Toft Bowman

Lieutenant, United States Navy

B.A. , Knox College, 1962

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

from the

NAVAL POSTGRADUATE SCHOOL
October 196 9

' ' ABSTRACT

The purpose of this paper is to examine long wave propagation
in a shallow channel of triangular cross section. Solutions for
small scale (f = 0) , large scale (f ^ o) , symmetric and asymmetric
channels are obtained. Results are shown to be consistent with
earlier work for the fundamental mode (tA = O) , and with edgewave
solutions over a gently sloping bottom for the higher modes. A second
class of waves (quasigeostrophic waves) is also obtained when the
Coriolis effect is included. The results are compared with those for
a rectangular channel.

Library

Ls;/aval/ostgraduate school

Monterey, California 93940

TABLE OF CONTENTS

EauAL,EIKNOXL,BRARY

I.

ii.

in.

IV.

INTRODUCTION
ANALYSIS

A. EQUATION FOR X > 0

B. EQUATION FOR X < 0

C. SOLUTION
SMALL-SCALE CHANNELS

A. SYMMETRICAL CHANNEL (n = /• 0)

1. Asymptotic Results

2. Numerical Results

3 . Wave Profiles

B . ASYMMETRICAL CHANNEL ( n * /. o")

1. Asymptotic Results

2. Numerical Results
3 . Wave Profiles

LARGE-SCALE CHANNELS

A. SYMMETRICAL CHANNEL (n = /. o)

1. Asymptotic Results

2. Numerical Results

B. ASYMMETRICAL CHANNEL (n * /. o)

1. Asymptotic Results

2. Numerical Results

9

10
11
12
13
16
16
16
18
21
21
21
23
23
28
28
28
29
29
29
32

V. COMPARISON OF RESULTS FOR A RECTANGULAR CHANNEL 35

AND A SYMMETRIC TRIANGULAR CHANNEL

A. SMALL-SCALE (i = O) 35

B. LARGE-SCALE ({ * 0) 37
APPENDLX A - Fortran IV Program to Solve Patching Equation 3 9

for Small-Scale Channels

APPENDIX B - Fortran IV Program to Solve Patching Equation 41

for Large-Scale Channels

APPENDIX C - Dispersion Relation for Dronkers' Rectangular 43

Channel

BIBLIOGRAPHY 45

INITIAL DISTRIBUTION LIST 46

FORM DD 1473 47

LIST OF FIGURES

Figure Page

1 Coordinate System 10

2 Solution Region for x > 0 12

3 Solution Region for * < 0 12

4 Dispersion Relation: Symmetric Channel 19
(W. < < i)

5 Dispersion Relation: Symmetric Channel 20
(f=0, n=l)

6 Wave Profiles: Symmetric Channel (n=l) 22

7 Dispersion Relation: Asymmetric Channel 24
(f=0, n=2.0)

8 Dispersion Relation: Asymmetric Channel 25
(f=0, n=5.0)

9 Wave Profiles: Asymmetric Channel (n=2.0) 26

10 Wave Profiles: Asymmetric Channel (n = 5„0^ 27

11 Dispersion Relation: Symmetric Channel 30
(f^O, n=l)

12 Dispersion Relation: Quasigeostrophic Waves 31
(f¥0, n=l)

13 Dispersion Relation: Asymmetric Channel 33
(f*0, n = 2o0)

14 Dispersion Relation: Asymmetric Channel 34
(f%0, n = 5.0^

15 Triangular vs . Rectangular Channels 35

16 Dispersion Relations: Symmetric Triangular vs . 36
Rectangular (f = 0)

17 Dispersion Relations: Symmetric Triangular vs . 38
Rectangular (f # 0)

5

LIST OF SYMBOLS

he maximum depth of channel

g gravitational acceleration

(",v) velocity components

(U,V) "X -variation of velocity components

j* surface height

^ "X -variation of surface height

S slope

n slope factor (n £ l)

<r frequency

f Coriolis parameter

^ wave number

S transformed lateral distance

Wo non-dimensional wave number

<=* non-dimensional frequency (squared)

fv. Laguerre polynomial with argument a.

C^iB) arbitrary constants

c wave celerity

ur non-dimensional frequency

fA, wave mode

X channel scale parameter

ACKNOWLEDGEMENT

This problem was suggested by Professor Theodore Green.
His advice and assistance were invaluable and made the results
contained in this study possible.

I. INTRODUCTION

Investigations of long wave propagation in channels with
cross sections which don't vary along the channel began with
Kelland [1839], who found solutions for progressive waves in a
uniform channel of triangular cross section with sides inclined at
45° to the horizontal. MacDonald [l894J examined wave propaga-
tion in a triangular channel with sides inclined at 30° to the hori-
zontal. Most investigators have limited their work to small scale
channels (i.e. the Coriolis effect was neglected as in the above
cases) in which velocities normal to the axis of the channel were
zero. An exception is Dronkers {_1964J, who studied waves in a
rotating rectangular channel, allowing cross -channel motion.

Many rivers , estuaries , and shallow seas can be approximately
represented by a shallow triangular channel (at least this is more
realistic than a rectangular channel). The shallow-water solutions
for long waves progressing along such a channel are closely related
to the well known solutions for edgewaves [Ursell, 1952; and Reid,
1958J , Such a solution is carried out in this paper and the results
compared to the work of Dronkers. Most of the results are obtained
numerically. When possible, they are compared to asymptotic work.

9

II. ANALYSIS

Long waves are studied, propagating along a channel with a

uniform triangular cross section. The coordinate system is shown in

Figure 1 .

x = 7 *Vh s

Figure 1

The maximum depth of the channel is ho . The slopes of the channel
sides are s for ~x > o and ns (n;>l>) for y. < o (under the shallow
water approximation s, ns < c 1 ) . The depth then is

h(V) = ho + r\SX

(* < o).

(2)

The equations to be satisfied are the shallow-water equations of motion

^u r ail

IT ~fv +g^

c^V

=r 0

^r 0

(3)
(4)

and the continuity equation

2

*■«•■*♦»•# +■»-

(5)

10

where u = -x - component of velocity

v - y - component of velocity

£> = surface height
and y is along the channel. A wave motion in the y direction may
be prescribed by the substitutions

f = n\ (x) e 3

;. (rro <J + «"t )

i(^4rt) (6)

where >r* is the wave number, and cr the frequency. Then for <rx-\f*

-9 ^ f Jus (7)

from equations (3) and (4). Substitution into the continuity equation
yields

^ + 77^+(^^-W^ = °

(8)

for all y and any h C?) .

Equation (8) will now be dealt with in the regions X >0 and x<0
separately, and the results will be patched to insure wave-height and
velocity continuity at X = 0 .

A. EQUATION FOR X > 0

Let £ =ho/s - * . Then <#Jx = - ^1, ^x* = <**«*/<*?. and
h (V) =h0-S*x = S^ . The solution region is shown in Figure 2.

11

I = NyJr

f=v?

SOLUTION
REGION

Figure 2

Then (8) becomes

ffi^ +'& + fa-'**f>t -

where

^0

_ ***1

f

(9)
(10)

B. EQUATION FOR X < O

Let £' = "X -+■ ho/n5 . Then h^^fio-f n S*=ns£ and the corres
ponding solution region is shown in Figure 3.

SOLUTI OH

REG-ION

Figure 3

Then (8) becomes

?'^+4£ +(F'-'«*S*)n

dr j?

= 0

(11)

where

(12)

12

SOLUTION

Using the substitutions

nCO= e""^' c*>«) <13)

equation (9) and (11) are put into hypergeometric form (Slater [1960J ):

where W= 3l£ m , and w' = Si g'on •

These equations have solutions of the form

G = A F*, (w) + C H (w) (x > o)

G= BfrCwO + DHV) (x < o)

where

(15)

(16)

P-vn

ex —

2 <vf\

a' =

p'~/^

(17)

But H /-/' ( V in Slater's notation) are unbounded as £ and £
become small and must be discarded. Then the solutions of (14) and
(15) become

-on

^5>Be" fv<V') C*< <0

(19)

13

where

F»M = J 4 »W + ^iw' ♦ "(a"\^/aV

i'(a' + t) ,a a'Ca'+OCa'+S") ,,'

The quantities f\ and B are constants; we can set f\ = 1 as the
results will be independent of wave amplitude in this linear theory.
At "X - 0 the patching conditions are:

n(i)= Ml') «t X-0 (W---Wc, W'= W.')

(i.e., the wave-height is continuous), and

(2 0)

(21)

(i.e. , the cross-channel velocity component is continuous). Note
that Wo = a^h./s and W.' - Z'mh./ns - Wo/n are convenient

nondimensional wave numbers.
Using (20), (21) becomes

or

(22)

(23)

Then, using (7), (20), and (23), V is also continuous at x - 0

For F«. (Wo) = 0 / (20) gives: 1) B = ° which means (21)

cannot be satisfied or 2) f^< (wJ) - ° which corresponds to a

node at X = O. For f^ (V,^ \ 0 , equations (20) and (23) give,

eliminating B /

14

F. (w.) F.-fa) - F. (u/.) ^ flft - E, (■*) ^g-Cw.) - o . (24)

Note that the solutions F». (W.) = /v (Wo') = 0 are contained in
(24). Also, when the channel is symmetric (o-l.p), (24) becomes

F*(w)[K(w.)- 2 ^-(W.)] = 0. (25)

15

III. SMALL-SCALE CHANNELS

For relatively short waves in narrow channels, where the
Coriolis effect may be neglected, (10), (12), and (17) become

P = ITT" - n P (26)

and

- I - 3 cl

*"? g -s

Equation (2 7) can be written

(27)

3.

k

04 ■ ^sl " (l a*-'~T (28)

where <>< is a convenient nondimensional frequency. Similarly,

a- *
^9n* = ' " ^* (29)

and -a- = -^x = 0 - a*) -3* . (30)

Equating like terms in (28) and (30) yields the following relationship
between a. and a/ :

0- a«.) = n(i- 3*') • (3D

Then equation (24) relates nondimensional wave numbers Wo and
frequencies c< .

A. SYMMETRICAL CHANNEL (n = /.o)

1 . Asymptotic Results

For channels with sides of equal slope (31) gives a. - a! .
If VJo < < 1 / and a. is bounded, an asymptotic result can be

16

obtained. Dropping terms of 0(W0 ) and higher in (25) gives

(l -3«0 ~ W, [Z« -f «.<>- l)] (w0<<lN) (32)

so that a. ~ '/^ • Thus, using (28), (32) gives

w* — — - ^71 <w- < < ^ 03)

or

8 *< ~ Wo (w. << 1) ,

(34)

Since wave celerity c = <s*/>v^ , this can be written

c1 = % g h. (35)

which is identical to Kelland's result for a symmetric triangular channel.
Since the average depth is ho/'x > then this result also corresponds
to the classical result for long waves over a flat bottom: C = q h •

Other asymptotic results for V/o <<1 and <k unbounded can
be obtained by iteration. An example is given below.

Let Wn be the root obtained by keeping terms through w"
in F*. (w) = 0 . Then:

- a. W, = I

f a. 4 0 ("a + >) . .3

Tin^ w*

aW., = etc.

17

This converges rapidly to

- clWo = I. va 4 0 (-£)

and from (28) ,

^ ~ /. V5 . (36)

Similar techniques could give the other asymptotes suggested by the
numerical results presented below.

For W. >> 1 (short waves), it is expected on a physical basis
that the waves in the two parts of the channel become independent of
one another, and that Ursell's edgewave result

«-> = Ci*-H) (M» 0> l,lv ..) (37)

would hold. Although this has not been confirmed here analytically,
it is borne out by the numerical results.

2 . Numerical Results

A Fortran IV computer program to solve (24) for small scale
channels (f zr c/) is given in Appendix A. The results for the
symmetric channel (n = l>0) are shown in Figures 4 and 5. In Figure 5,
the dispersion relation for the fundamental mode (p\ •=. o) and the
next five modes are plotted.

For Vn/o < < 1 (Figure 4) the calculations agree well with the
asymptotic results (34) and (36).

For W« > > 1 / note that

ex ~ fart + 0-T^ (K* *>'»*i • • 0

or u __ ( j p\ f j ) g ^ s (M« °»'^) "'^

which is the same as (37).

18

Dispersion Relation: Symmetric Channel ( W, << I )

(f=0,n=1 .0)

19

IOO0

ro

to 0

no

Figure 5
Dispersion Relation: Symmetric Channel (f=0,n=1.0)

20

3 . Wave Profiles

Figure 6 is a sketch of the wave profiles for modes
0 through 5 showing the nodal points for the symmetrical channel.
Note that the nodes for all modes are symmetric about the center of
the channel. The profile for the fundamental mode fM = o) has no
nodes along the channel. Wave mode (7*1 = l) is the first asym-
metric profile, corresponding to the solution n(£)= -??(£') fp^(vy.)= oj.
Wave mode (W-2) has a symmetric profile with two nodal points.
The higher modes have symmetric profiles for even mode numbers
and asymmetric profiles for odd (tf[) , with the number of nodal points
equal to (tA)- Tne nodal points (except X - o ) move toward the
channel walls with increasing Wo • This corresponds to the edge-
wave condition for short waves (large y*\ ) .

B. ASYMMETRICAL CHANNEL (n ^ I. d)
1 . Asymptotic Results

For Vv/„ < < 1 and a. bounded, (24) can be written

I - ("* + «') = wc[««W) + al^O + °L(£D] + o(w;) . (38)

Since VJ0 < < 1 , then a. + a. ~ 1 . Using (3 1) , this gives

a. + £7j" (a ol + n- /) ~ I

so that

a. » /i ; *' ~ 'A •

Let a. = ^ - € and a'« & — €*. Then (38) gives

W^ ~7 O + O (39)

21

Figure 6
Wave Profiles: Symmetric Channel (n=1.0)

22

which with (28) and (30) gives

This is the same as (34), and is independent of n.

Other results for V/> << 1 similar to (36) are expected for
asymmetric channels but are not examined here.

2 . Numerical Results

The results for n = 3.0 and r\ - 5.0 are shown in
Figures 7 and 8. In each case, the asymptotic result (34) is confirmed
In fact, the entire dispersion relation for the fundamental mode (M = o)
appears to be independent of n . The higher modes have all been
shifted to higher frequencies in comparison with the symmetric case.

3. Wave Profiles

Profiles for n= 1.0 and n =i 5"0 are shown in Figures 9
and 10. Again the fundamental mode (M»0) is unchanged, but the
higher modes have been altered significantly from the symmetric case.

23

loao

Dispersion Relation: Asymmetrical Channel (f=0,n=2.0)

24

loao

Dispersion Relation: Asymmetrical Channel (f=0,n=5.0)

25

Figure 9
Wave Profiles: Asymmetric Channel (n=2.0)

26

Figure 10
Wave Profiles: Asymmetric Channel (n=5.0)

27

IV. LARGE-SCALE CHANNELS

For large channels where the Coriolis effect must be considered,
(10), (12), and (17) give

//=/-a«.»-4 (Va-0 - Z7

and (40)

a,

where <*^- = «"/f .

Equating like terms in (40) ,

c^ =-*ktl

and

" = *<£&)[ feS&fe)1- '1

(41)

(42)

where V = At. .y.?. .

Now W* must satisfy both (24) and (42).

A. SYMMETRICAL CHANNEL (n = /. o)

1 . Asymptotic Results

From (41), a. and a.' are related by ck =■ a.' ■+• /or. But
for Wo << 1 and ur bounded, (42) implies that Jcu-ha'J is large,
so that a. -si a.' • Then (40) gives

This can be written

-** = ~^(^1 0 - 7ZF. (44)

28

From (36) and (44) ,

l^4 = -&-Cur»-0 - -b Cu.«A

U, w.

Dropping the small term gives

us*= ±^ + | . (45)

2 . Numerical Results

A Fortran IV computer program to solve (24) and (42) is given
in Appendix B. A value of )( = 0. 1 is chosen, which corresponds approxi-
mately to the dimensions of the Red Sea. The results for the symmetric
channel are shown in Figure 11. Note that for Wo < < 1 / the asymp-
totic result (45) is confirmed. For LJ~ < < 1 a new class of waves
appears. These are analogous to the quasigeostrophic edgewaves
obtained by Reid. Figure 12 is an expanded plot of these low frequency
waves .

For UT, V/» >> 1 (short high-frequency waves) , the Coriolis

effect should be negligible. Then, for a fixed W., the frequencies <r

for f=0 and f* 0 should be the same. Thus,
*t o-ah,

X

T = X

urx ~ 9 s / f

This relation is observed to hold for Cor, W.>> 1 .

(46)

B. ASYMMETRICAL CHANNEL (h*/. o)
1 . Asymptotic Results

For Wo < < 1 and <x unbounded, asymptotic results similar
to (45) are expected, but are not derived here.

29

Figure 11
Dispersion Relation: Symmetric Channel (f*0,n=1.0)

30

ll

C

o

II

<M

*- '

w

V

>

(0

s

o

•H

£

ft

O

CM

u

T-l

■p

tfl

0)

0

u

0

3

bO

bO

•H

•H

tn

b.

(0

c
o

•H

•p

CB
H
<1>
«

O
•H

«

-

31

2 . Numerical Results

The results for n = a.o and n ■ S. o are shown in Figures
13 and 14. Note that all modes have been shifted to higher frequencies
in comparison with the symmetric case. Also for uT, \JD > > 1 the
f = o and f * o results give good agreement via (46) .

32

ur

Figure 13
Dispersion Relation: Assymetric Channel ( f *£> ,n=2.0)

33

Figure lk
Dispersion Relation: Asymmetric Channel ( f xO ,n=5 • 0 )

34

V. COMPARISON OF RESULTS FOR A RECTANGULAR CHANNEL
AND A SYMMETRIC TRIANGULAR CHANNEL

The dispersion relation due to Dronkers for a channel of uniform
depth (h) and width (w) is (Appendix C)

CM = o,t»a> . ..),

(4 7)

It is reasonable to compare channels of equal average depth, (Figure 15'

Figure 15
Then h = ho/a. and W= 2h*/s , and (47) becomes

A. SMALL-SCALE (f = 6)

For small-scale channels (f = 0) , (48) becomes

(48)

(49)

This agrees with (34) for the fundamental mode (tA-0). This dispersion
relation (4 9) is compared with that for a symmetric triangular channel
in Figure 16 .

35

IOOlO

5ao

zoo

5.0

l.o

Rectangular Channel
Triangular Channel

r.o

/«.o

Figure 16

Dispersion Relations : Symmetric Triangular
vs. Rectangular (f=0)

too

36

B. LARGE-SCALE (f * 0)

For large-scale channels (f^o), (48) becomes

WoJ = HX(<±x-i) -ttV (m« o,ina ) (so)

This dispersion relation (50) is compared with that of a symmetric
triangular channel in Figure 17.

37

Dispersion Relations : Symmetric Triangular
vs. Rectangular (fxO)

38

APPENDIX A

FORTRAN IV PROGRAM TO SOLVE PATCHING EQUATION
FOR SMALL-SCALE CHANNELS

IMPLICIT REAL*8(A-H,0-Z)

DIMENSION C(36)/C(36)/CP(36),DP(36)/CCP(36)/CDP(36)/DCP(36)/WORK
* (36) , B(36) , ROOTR(36) ,ROOTI(36)

1 READ(5,5)A

5 FORMAT (F 10. 5)
IF(A.EQ.-9.50)GOTO99
S=5.0

AP=(S+2 . 0*A-1 . 0)/(2 . 0*S)
WRITE(6,10)A,AP,S
10 FORMAT (35X, 'A = ' , F10 . 5 , 5X, 'AP= ' , F10 . 5 , 5X; 'N = ' , F5 . 2 ,/)
C(l) = 1.0
C(2)=A

DO 2 N =3,12
C(N)=C(N-l)*(A+(N-2))/((N-l)*(N-l))

2 CONTINUE
D(1)=A

D(2^=A*(a+l)/2.0
DO 3 N=3,12
D(N)=D(N-1)*(A+(N-1))/((N-1)*N)

3 CONTINUE
CP(1) = 1.0
CP(2)=AP/S
DO 4 N=3,12
CP(N)=CP(N-l)*(AP+(N-2))/(N-l)*(N-l)*S)

4 CONTINUE
DP(1)=AP

DP(2)=AP* (AP+l)/(2 . 0*S)

DO 6 N=3, 12

DP(N)=DP(N-1)*(AP+(N-1))/((N-1)*N*S)

6 CONTINUE
DO 7 K=l,12
CCP(K) = 0.0
DO 7 1=1,12
DO 7 1=1,12
IF(I+J.NE.K)GOT07
CCP(K-1)=CCP(K-1)+C(I)*CP(J)

7 CONTINUE
DO 8 K=l,12
CDP(K) = 0.0
DO 8 1=1,12
DO 8 1=1,12
IF(I+J.NE.K)GOT08
CDP(K-1)=CDP(K-1)+C(I)*DP(J)

39

8 CONTINUE
DO 9 K=l,12
DCP(K)=0.0
DO 9 1=1,12
DO 9 1=1,12
IF(I+J.NE.K)GOT09
DCP(K-1)=DCP(K-1)+D(I)*CP(J)

9 CONTINUE
DO 11 N=l,ll
B(N)=CCP(N)=CDP(N)-DCP(N)

11 CONTINUE

CALL DPOLRT(B/WORK/ 10 ,ROOTR,ROOTI, IER)

WRITE (6 , 25) (ROOTR(I) , ROOTI(I) ,1=1, 10)
25 FORMAT (//,4X,2D25. 16,/)

WRITE (6, 3 0) IER
30 FORMAT (5 OX, 12)

GOTOl
99 STOP

END

40

APPENDIX B

FORTRAN IV PROGRAM TO SOLVE PATCHING EQUATION
FOR LARGE-SCALE CHANNELS

IMPLICIT REAL*8(A-H,0-Z)

DIMENSION C(36) ,D(36) ,CP(36) , DP(36),CCP(36), CDP(36), DCP(36), WORK
* (36) ,B(36) ,ROOTR(36) , ROOTI(36)
CF=1.0D-04
H=1000.0
G=9.8
SLP=0.1
S=5.0
CAPA=.10

1 READ(5,5)W

5 FORMAT (F 10. 5)
IF(W.GT. 14.5)GOT099
WRITE (6, 10) W

10 FORMAT(50X/,W='/F10.5,/)

OMEGA=C5 0
12 OMEGA=OMEGA-.001

IF (OMEGA. DE.0.0)GOTOl

A=. 5* (1 . 0/OMEGA+l . 0-(CAPA* (OMEGA* OMEGA- 1 . 0))/W)

AP=. 5* (1.0-1. 0/OMEGA-(CAPA* (OMEGA*OMEGA-l . 0)/(W*S)))

C(l)=1.0

C(2)=A

DO 2 N=3,12

C(N)=C(N-l)*(A+(N-2))/((N-l)*(N-l))

2 CONTINUE
D(1)=A

D(2)=A*(A+l)/2.0
DO 3 N=3,12
D(N)=D(N-1)*(A+(N-1))/((N-1)*N)

3 CONTINUE
CP(1)=1.0
CP(2)=AP/S
DO 4 N=3,12
CP(N)=CP(N-l)*(AP+(N-2))/((N-l)*(N-l)*S)

4 CONTINUE
DP(1)=AP

DP(2)=AP* (AP+l)/(2 . 0*S)

DO 6 N=3,12

DP(N)=DP(N-1)*(AP+(N-1))/((N-1)*N*S)

6 CONTINUE
DO 7 K=l,12
CCP(K)=0.0
DO 7 1=1,12
DO 7 J=l, 12
IF(I+J.NE.K)GOT07 41

CCP(K-1)=CCP(K-1)+C(I)*CP(J)

7 CONTINUE
DO 8 K=l,12
CDP(K)=0.0
DO 8 1=1,12
DO 8 J=l, 12
IF(I+J.NE.K)GOT08
CDP(K-1)=CDP(K-1)+C(I)*DP(J)

8 CONTINUE
DO 9 K=l,12
DCP(K)=0.0
DO 9 1=1,12
DO 9 1=1,12
IF(I+J).NE.K)GOT09
DCP(K-1)=DCP(K-1)+D(I)*CP(J)

9 CONTINUE
DO 11 N=l,ll
B(N)=CCP(N)-CDP(N)-DCP(N)

11 CONTINUE

FCN=B(1)+B(2)*W+B(3)*W*W+B(4)*W**3+B(5)*W**4+B(6)*W**5+B(7)*W**(
* + B(8)*W**7+B(9)*W**8+B(10)*W**9+B(ll)*W**10

WRITE(6,15)A,AP,OMEGA,W,FCN
15 FORMAT(2X,4F15.7,D25.16)

GOT012
99 STOP

END

42

APPENDIX C
DISPERSION RELATION FOR DRONKERS' RECTANGULAR CHANNEL

A. FORMULATION

The equations to be satisfied are:

Substitution of

into (1) and (2) gives

kU- fV + f#- = 0

(2)

(3)

(4)

Then , for <r x *? f * ,

and the ordinary differential equation for "r\ is
where a ^ a r*- \

fl = (-^l. _ ^») (6,

with solution of the form

43

B. BOUNDARY CONDITIONS

The coordinate system is shown in Figure CI. The flux normal
to the channel walls must vanish at the walls.

X = W

////// ///////////// S ./;/ /

A*

x= o

\\ \ \ \ \ \ \<\ \ \

Figure CI

WWW \ \ \ \

Then (4) gives

(x* oy) .

(8)

Substitution of (7) into (8) yields

f

23 + A W

AW m

Jtosn f\W s 0

(9)

so that,

Then (6) gives the dispersion relation

(10)

7*1* =

a r *

gh

(11)

44

BIBLIOGRAPHY

1. Dronkers , J. J., Tidal Computations, p. 249-261,

North-Holland, 1964 .

2. Kelland, P., "On Waves," Trans. Roy. Soc. Edin.,

v. 14, 1839.

3. MacDonald, H. M. , "Waves in Canals," Proc. Lond.

Math. Soc. , v. 25, 1894.

4. Reid, R. O. , "Effect of Coriolis Force on Edge Waves

(I) Investigations of the Normal Modes," J. Mar.
Res. , v. 16, p. 109-144, 1958.

5. Slater, L. J., Confluent Hypergeometric Functions,

p. 1-8, Cmabridge University Press , 1960.

6. Ursell, F. , "Edge Waves on a Sloping Beach," Proc.

Roy. Soc. Lond. (A) v. 214, p. 79-97, 1952.

45

INITIAL DISTRIBUTION LIST

1. Defense Documentation Center 20
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2 . Library 2
Naval Postgraduate School

Monterey, California 93940

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Alexandria, Virginia 22314

4. Naval Oceanographic Office 1
Attn. Library

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5. National Oceanographic Data Center 1
Washington, D. C. 20390

6. Professor Theodore Green 6
Department of Civil Engineering

University of Wisconsin
Madison, Wisconsin 53706

7. Lt. William T. Bowman 2
USS Monticello (LSD-35)

FPO San Francisco, California

8. Department of Oceanography 3
Naval Postgraduate School

Monterey, California 93 940

46

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(Security claetltlc ation ol till*, body ol abetract and indexing anno ta (Ion munt be entered when the overell report le claeellled

ORIGINATING ACTIVITY (Corporate author)

Naval Postgraduate School
Monterey, California 93940

2a. »IPO»T IICU»ITY CL ASIIFIC A TlOt

2b. amour

3 REPORT TITLE

Long Wave Propagation in a Triangular Channel

4 descriptive NOTES (Type of report and,lnclusive datee)

Master's Thesis; October 1969

5 AUTHOR(S) (Flret name, middle Initial, laat name)

William Toft Bowman

6 REPORT DATE

October 1969

7a. TOTAL NO. OF PACE!

47

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•a. CONTRACT OR GRANT NO.

b. PROJEC T NO.

9a, ORIGINATOR'* REPORT NUMBER!*)

9b. OTHER REPORT NOI1I (Any other numbert thai may be aaelfxed
thle report)

10. DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distribution
is unlimited .

It. SUPPLEMENTARY NOTES

12. SPONSORING MILI T ARY ACTIVITY

Naval Postgraduate School
Monterey, California 93940

13. ABSTRACT

The purpose of this paper is to examine long wave propagation in a shallow
channel of triangular cross section. Solutions for small scale (f=0) , large scale (f^O) ,
symmetric and asymmetric channels are obtained. Results are shown to be consistent
with earlier work for the fundamental mode (M=0) , and with edgewave solutions over
a gently sloping bottom for the higher modes. A second class of waves (quasi-
geostrophic waves) is also obtained when the Coriolis effect is included. The results
are compared with those for a rectangular channel.

DD

""» 1473

I NOV 65 I "• / W

S/N 0101 -807-681 1

(PAGE 1)

47

Security Classification

»-ll40i

,/

Security Classification

KIY WONDI

Triangular Channel
Edgewaves

Hypergeometric Equation
Laguerre Function
Quasigeostrophic Waves

DD ,'.r..1473 back)

KOL I

S/N 0101 -807-6821

48

Security Classification

'•!U0

Thesis

115979

B782

Bowman

c.l

Long wave prop-

agation in a

tr iangu lar channe 1 .

Thesis

B782

c.l

9

Bowman

Long wave prop-
agation in a
triangular channel .

Thesis

B782

c.j

Bowman
tr»angu)ar channel.

557'

theiB782

Long wave propagation i

( h

ill

;

1

3 2768 002 07388 4

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