On the parabolic equation method for water wave propagation

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On

the parabolle

equatiori

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method

torwater

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wave

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propiigatfon

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by

a. c. radder

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july 1978

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I Report DIV 78863 ,,!.:.":,-.~:.

-~.....'..,,':._':...-r:.

-On the parabolic equation method for water wave propagation.

by A.C. Radder

Ministry of Transport and Public Works, Rijkswaterstaat,

Data Processing Division, Nij_verheid$straat 1, 2288 B~ Rijswijk, The Netherlands, Tel.: (070) 906628 and 907640, telex 32683

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Contents

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Abstract 1 2 4 Introduction

Reduction of the mxd-slope equation to the

Helmholtz-equation

Derivation of the parabolic approximation

Asymptotic analysis for the one-dimensional case

Numerical solutions for the general case Application to circular shoal

Summary and conclusions Appendix References Figures 1 - 7 5 8 13 16 19 21 23 26

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Note. The present report contains the preliminary text,

submitted for publication.

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Abstract

A.parabolic approximation to the reduced wave equation is investigated

for the propagation of periodic surface waves in shoaling water.

The approximation is derived from splitting the wave field into transmitted

and reflected components.

In the case of an area with straight and parallel bottem centourlines,

the asymptotic form of the solutien fer high frequencies is compared

with the geometrical opties appreximation.

Two numerical solution techniques are applied te the prepagatien

of an incident plane wave ever a circular shoal.

----..-_---_. ---'-" .. ---,---_. -_.-

-

-

-

-

-

-

---...

-

-

-

--- .--_-

---

_

.

_

-

---

-

--

-

-

._---_{---_..}

..-2.

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1. Introduction

The propagation of periodie, small amplitude surface gravity waves

over a seabed of mild slope can he described by the solution of the

reduced wave equation

c

V'·(cc V'~) + _[ w2 ~

=

0

g

c

(1)

with appropriate boundary conditions. Here ~ (x,y) is the complex

two-dimensional potential function, V':: ( ~x ' ~y )

the hozä.zontia.L'gradient operator,

w

the angular frequency, and c and

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Cg

are the corresponding local phase and group veloeities of the

wave field. This reduced wave equation accounts for the combined

effects of refraction and diffraction, while the influences of hot tom

friction, current and wind have been neglected.

The wave equation (11 has been derived by several authors, for the

first time by Berkhoff (1972}, and by Schönfeld (1972) in a different

form.

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Svendsen (1967).derived the equation for one horizontal dimension, as

is pointed out hy Jonsson and Brink-Kjaer (1973). Smith and Sprinks (1975)

gave a formal derivation of (11. Booij (1978] has proposed a new wave

equation, which includes the effect of a current, and which reduces to

(1} in the current-free case.

The equation (1} is essentially of elliptic type, and therefore defines

a problem which is in general properly posed only when a boundary

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condition along a closed curve is given. In order to obtain a numerical

solution for short waves over a large area in the horizontal plane, a

great amount of computing time and storage is thus needed. However, in

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propagated without appreciable reflection into a preferred direction,

and it should be natural to consider methods which make use of this

property. In the classification of Lundgren

(1976

1

,

such methods can be

distinguished as R-methods (refraction methodsL and P-methods (propagation

methodsL~ both of which represent an approximation to the mild-slope

equation

(11.

Refraction methods are based on the geometrical opties approximation,

which.fails to give a reliable solution near caustics and crossing

wave rays, where diffract~on effects become important.

Propagation methods should be able to account for such situations.

Methods of this type have been proposed by Biésel (1972), Lundgren

(lq761

and Radder

(1q?7l,

but these are lacking, among other things,

in the possibility of making systematic corrections which are needed

if one wants to recover the complete wave field.

In ths present work, a parabolic approximation to the reduced wave

equation (1} is derived from splitting the wave field into transmitted

and reflected components. The result is a pair of coupled equations

for the transmitted and reflected fields. By assuming that the

reflected field is negligible, a parabolic equation is obtained for the

transmitted field. Thîs procedure has been applied to opties by

Corones (1975), and to acoustics by McDaniel (19.751.The derivation

is based on the Helmholtz-equation; therefore, in § 2, a reduction of

the mild-slope equation (11 to the Helmholtz-equation is given, and

in § 3 a parabolic approximation is derived. An asymptotic form of the

solution for high frequencies is presented in § 4, in the case of an

area with straight and parallel bottom contourlines.

Finally, in § 5 and § 6 numerical solutions to the parabolie equation

are obtained in the form of two finite-difference schemes, with application

to plane wave propagation over a circular shoal with parabolic bottom

4.

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2. Reduetion of the mild-slope eouation to the Helmholtz-equation

Although a parabolie approximation ean he direetly derived from

equation (11, it is useful, to simplify the notation and applieations,

to reduee equation (lL to the Helmholtz-equation, without 10ss of

generality.

A sealing factor is introdueed

(2)

whieh turns (11 into the Helmholtz-equation

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(3)

Here the effeetive wave number kc is defined by

iJ2

lCc

k2

=

k2 _ g

C ICCg

(4)

and the wave number k is the real root of the dispersion relation

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W2

=

g

k tanh (kh)

(5)

with h the loeal water depth and g the gravitational aeeeleration.

W dW

The phase and group veloeities are then given by C

=

k ' Cg

=

dk

In shallow water, the difference k2 -k2 may become appreeiable:

C in this case one has

, C

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Th

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~

4h

(6)

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It follows, that kC may be approximated by k if

(7a)

(7b)

implying a slowly varying depth and a small bottom slope, or high

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frequency wave propagation.

Unless stated otherwise, k

c

assuming that {7al and (7bl are satisfied.

will be approximated by k in this paper,

3. Derivation of the parabolic approximation

The Helmholtz-equation (3) can be written in the form

(8)

where x denotes the preferred direction of propagation, and the

sub-script c of the wavenumber k has been dropped (however, for the

derivation of the parabolic approximation the restrictions (7) are

not necessaryl.

+

The wave field ct> should be split into a transmitted field ct> and a

reflected field ct>

ct>

=

+

ct> + (9)

This can be achieved by the use of a splitting matrix T which defines

the transmitted and reflected components by

6.

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The matrix T is formally arbitrary, but some genera 1 physical criteria

limit the choice of T and lead to the governing parabolic equation in

a natural way.

Firstly, equation (9) is valid for arbitrarily chosen

ep

+and

ep

-

only

if T satisfies

a

+ y

=

1

B

+

0

=

0

(11)

Using equation (81, it follows that

a<p

+ _

[-

k2B

+

ay

+

aa

y aB

_{a2 ]}

<p

+ +

d.X -

_B

_ax

+ ---

_{B ax}

B-ay2

[- k2B

a2

aa

a aB

a2 ]

ep

(12a) + - - +

ax

- î3

ax -

B-B

_ay2

aep

[k2B

y2

+

ay

y ~

+

B~

]

ep

+ +

ax-

= + -

_B

_ax

-

_{B ax}

_ay2

+

[k2B

-

;:;-

ay

+.."..-

ay

+-

a

.."..-+B-

as

a2l

eb (12b) p áX

B

_ex

ay2

J

.

I

_{Further, when k is a constant, solutions of the form}

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n,+

ikx

'I'

=

e

=

e

-ikx

(13)

should result, and the equations (12a) and (12b).should naturally

decouple in this·case. This can be achieved by choosing

+

o

+ (14)

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and th.e resulting splitting matrix is (cf. Corones (1975Il

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while (121 reduces to <p+ +

(.!_

2k (16a)

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t

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Î

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( " 1 ak i (2)_ -lk- 2k dx - 2k dy2 <p (16b)

This pair of coupled equations is equ~valent to equation

(Sl.

By

e

neglecting the reflected field <p-, a parabolic equation for the

transmitted field <p+ is obtained

(17)

In a similar way, a parabolie approximation can be directly derived

~+

from equation (1), which yields ior the transmitted field ~

[ ik - -=2~k-!-e-g + 2keei

a

ay

eeg

ay

al

4>+

g .

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Using (2t and (71, equation (17} is recovered.

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Î

a

2

By adding to the left hand sides of (14)_ me operator

S

(

₎

dy2

th 1"· " d " d T ' 1 -i/A

ano er sp 1tting matrl.XlS er1ve: A

=

2 1 i/A

A

=/k

2 +

1,

and a closer approx~ation to equation (SLmay be

ay2

obtained. Unfortunately, the square root operator A makes the resulting

, where

parabolic equation practically untractable, and a satisfactory approximation

must be found for the operator A, in order to obtain numerical results

(cf. McDaniel (1975)).

In the following, the parabolic equation (17) will be considered, in

8.

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a

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direction of the incident plane wave.

4. Asymptotic analysis for the one-dimensional case

In order to test the validi ty of the parabolie equa tion (171 as an

approximation to equation (SL solutions to both. equations will be

compared in the case of an area with straight and parallel bottom

contourlines. The problem is equivalent to pláne wave propagation in a

plane stratified medium in opties and acoustics, and the asymptotic

analysis of Seckier and Keiler (19591 will he followed here.

Dropping the

+

superscript, equation (171 can be written as

2

_{~ on}

0

k

d

_dX

<j> ₊ (18)

where ko denotes a constant wave number, and n

=

k/ko the index

of refraction. By introducing a new coordinate system

(p,cr)

p

=

x

cosa

+ y

sina

(19)

cr

=

-

x

sina

+ y

cos

a

the bottom is defined through h == h (p),

n

==

ne

p

);

dn

_dy

=

À

_dX

dn À

=

tana

(20)

TI

wnere

a

is the angle of incidence, with

l

al<

2 .

It will be assumed that k(P} tends to the constant value ~ (i.e. n(P) -+ 1}

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*

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t

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ikox

Now suppose a plane wave e

is incident from x

=

_00 • The field <p can then be written in t.."1e

form

<p = A (p) exp [iko (x - pfndp)] ₍₂₁₎

with P =

cosa

sin

2

a

A satisfies

Upon inserting (21) into (18). one finds that

(22)

At P

= -

00, A ( pL is supposed to behave like

·

1

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(23)

where the constant R denotes a complex reflection-coefficient.

At P =+ ·00,A(pl should satisfy aradiation condition, i.e. no

incoming wave from +00 •

The equation (221 is in general not explicitly solvabie, and the

solution must be represented by an approximation, which usually takes

on an asymptotic form for high frequencies, in the limit ko+ 00

A point at which the coefficient of A in (221 vanishes is cal led a turning point, where the character of the solution changes from

oscillatory to exponential. In the geometrical opties approximation

of tne problem, a caustic line is formed at these turning points.

If tnere is no turning point, and n + 2 À2(n

-1

»

0,

the asymptotic form has an oscillatory character with R=O,

and can be found by the WKB-method (Cf. Langer (1937}). Let

-

~

A

=

In2 +

2À

2n(n

-

1

)

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p (24)

10..

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then ~~e WKB-approximation to ~ is given by

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A

ikoF

~p

=

pep (26)

A similar analysis for equation

(al

results in the geometrical opties

approximation. Let (27) (28)

1

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1

then the asymptotic form is given by

A

ikoF

~g

=

g

e

g (29)

In the special case

a.

=À =

0,

both ~ pand ~gagree (if the sealing

factor~giS taken into account) with the classical shoaling

formula for a progressive wave

ct> :::: _1_

exp [ifx

k dx] .

g

IC

Xo

g

(30)

~p and ~gare compared in table I, for some values of À and n.

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It is assumed, that the incident wave is starting in deep water,

ko=

wJg, and a correction factor cA

=

-;===n====~~_

/1 + k0h(n2-1)

should be applied for the wave amplitudes, according te equation (2).

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Table

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Comparison of

w

ave amplitudes

A, w

a

v

e

num

bel's

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V'F

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and

wave directions

e

(

no turning point

)

.

c

_A

* A

_p

c:

_A

*' A

_g

l

V'Fp

l

I

V'F

_g

I

e

e

P

g

0.88

0.91

2.01

2

24.

4

°

24.3°

1.01

1.07

3.03

3

,

31.5°

31.4°

0.70

0.74

2.04

2

37.4°

36.9°

0.79

0.85

3.12

3

46.8°

46.1°

-, cr. À

n

4

5°

1 2.

45°

1

3

63.4°

2

2

-

63.4°

2

3

The

'

agreement is rather close, even for comparatively

l

arge values

of

À •

Now suppose there is just one turning point at

P=

P

o ,

a point where

the coeff

i

cient of A

(

P) in

(221 vanishes.

This

will

occur when n takes on the value n

p

n

=

P

1+2.À

2

(31)

In

case of equation

(8},

the corresponding value is given by

n

g

(

3

2)

An

analysis of the turning

p

oint problem can be

f

ound

i

n

th

e artic

l

e

of Langer

(19.37

1

:

for

P

>Po

A(

p

l

takes on an exponentiall

y

decreasing

f

orm

12.

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1

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and for P < Po an oscillatory form

A(p)

=

2

I

Q

I

-~

cos[

f~

O

Q

dp - ~ ]

where

Q

=

kop

Iln

2 + 2À2

n(n

-1)

I

Near the turninq poi.nt, the asymptotic form of the solution can be

(

33

b)

(

34

)

represented by Airy functions.

Uponinsertinq (331 into (211 one obtai.ns the asymptotic form

4>

p.

Here, only the behaviour of

4>p

at - 00 will oe qiven explicitly

(35) .

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1

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1

·

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wi.th.

I

Rpi =

1, i. e-:a fully reflected plane wave arises, with. wave

number

kp and

wave direction

a. p

qi ven by

k

=

.

ko

11

+ 4·

/À

1+

P

,

(36)

For the qeometrical opties solution, the correspondinq formulas

are qiven by

4>

(_00)

=

exp[ikox]

+

R

exp

[

iko(x(À

2

-1) -

2

Y

À)j

(I+

À

2)]

g g (37)

k

=

k

o

g 2À

tan

a.

_g

= ~À

1-For somevalues of À , a comparison is presented in table

Ir.

--~--- --- - - ---_-_-

---Table 11

Comparison of reflected plane waves at a turning po

i

n

t

.

n p n

g

k

_P

/

ko

kg/ko

a.

p 2 1

0.89

0.67

0.89

0.71

1.1

2.2

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When À. approaches zero, the reflected waves in the solutions

<l>

p

and

<P

g

deviate more and more from each other, as would be expected.

Actually, the parabolic approximation is valid provided À.2

»

1

otherwise the coupling between the transmitted and reflected wave

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fields in equations (161 must be taken into account, if one wants to

recover the complete wave field. For systematic corrections to the

parabolic approximation, see Corones (1975}.

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5. Numerical solutions for the general case

The parabolic equation (171-may be solved by using finite-difference

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techniques. In this section, two alternatives will be dealt with.

Assuming plane wave incidence

ikox

e

₍₃₈₎

I

then equation (181 yields for the complex potential function ~

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(39) where f

=

k~n [2(n _ 1)

+ i

ko

d

~

n(n

)

J

dX (40)

A Crank-Nicholson finite-difference equation is used for the numerical

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solution to equation (39), cf. Richtmyer and Morton (19671:

let a rectangular grid be given with grid spacings ~x and 8y, and

be denoted by ~~ J

J

let the approximation to

~(~~x,

j~y)

~,j=O,1,2, .•• The scheme I is then defined by

~~

+

1

~

+

1

₊ ~ + ~~ +

[-2

+

(~y)2

f

~

+~] .

(~~

+ 1

_{+ ~~} + ~. 1 ~. 1 )+

J+

1

_J-

J+

J-

1

_J

_J

_J

+

4 ik

(

~

y) 2

~

+~

_.(~~+l

_-

_~~

₎

=

0

o ~x

n.

_J

_J

_J

(41)

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.14. where

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i+~

n.

J i+l i

=

(n. + n. )

/2 ,

J J i+l

.

n.

f

in

(+

)/

Ax] o n. J

and with initial condition (42)

~~ =

1,

j

=

0,1,2,•..

J (43)

and appropriate boundary conditions, to be specified later on.

Another solution technique, which may be preferabIe, is based

on the change of variabIe

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(44)

which turns (391 into

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a

2ç

aç.

3ç

21'k

n

3ç

+

f

=

0

3y2

+

3y

3y

+ 0

3x

It

may

be expected that the solution

ç

is a less rapidly

(45)

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varying function than ~, thus providing a more accurate appraximation

on the same grid.

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However, the transformation (44)_ is singular at points where

'1'=0(branch-points, or: àmphidromic points), and a direct

application of a scheme like (41l is not possible. In order to

prevent the non-linear instabilities involved, it appears to

be useful to add to the left hand side of (451 an artificial

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viscosity term of the form

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(46)

where

S

is a dimensionless constant of the order of 1

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_{shocks, where an analogous dissipative term has been introduced}

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to insure stabiIitYi see Richtmyer and Morton (1967), chapter 12}.

Let

(47)

then the scheme 11is defined by

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+

R.+l [_ 2 R.

+

8 i.k SË1]_2 R.+~]

+

2 (R.

+

rR.. )

+

l;;j' gj 1 0 Sx. nj l;;j+l "'J-l (48)

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with initial condition

l;;~

=

0

J j

=

0,1,2, ...

(49)

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and appropriate boundary conditions.

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Both schemes being implicit, a system of simultaneous linear

equations has to be solved. For systems like (41} or (481 very

efficient methods are available.

The rate of convergence wiII be exemplified in the next section,

where numerical solutions are obtained for the case of a circular

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snoal.

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_..

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16.

6. Application to circular shoal

As an example, the propagation of an incident plane wave will

be considered over a circular symmetrie shoal with parabolie

bottom profile. Calculations for this severe test case have been

made by Berkhoff (19761., Bettess and Zienkiewicz (19771,

Flokstra and Berkhoff (19.77)_, and Ito and Tanimoto (19721, who

additionally conducted some laboratory experiments.

The shoal is represented by the ~epth profile

for r<R ,

(50) where

h

=

ho

r

2

=

(x _

X )2 + (y _ m

eo

=

-

(ho - h )/R

2 m and

To be definite, short wave propagation is considered, and the

assumptions (71 should apply, which amount to:

(51)

This imp lies that the curvature of the bottom is much less

than the wave number, regardless of the value of the minimum depth.

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The value of the angular frequency

w

follows from the dispersion

relation (51:

w

2

=

g

ko tanh (koho)

i denoting the corresponding

wavelength by

Lo

=

2IT/ka , the problem is then defined through

the parameters hm

IR ,

ho/R

and Lo/R.

In order to specify the boundary conditions, it is useful to

analyse the asymptotic character of the solution for large distanee x

(see appendixl. The governing equation stands for the

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wave packet. The behaviour of this wave packet for large x is a

well known problem in wave mechanics: the spreading of the packet

increases linearly with the distance x, and the magnitude approaches

zero, as

11

IX .

It follows, that the required boundary conditions for schemes I

and II, in case of a shoal, can be given by the undisturbed initial

values of the solution, ~=1 and ç=O, provided these boundaries

are taken sufficiently far away from the area of interest.

In this way, the artificial reflections which may occur at the

boundaries, can be avoided.

Some calculations with the numerical schemes have been performed,

for two configurations of the shoal:

- configuration I, defined through:

hm/R = 0

.

0625

;ho/R = 0.1875

;Lo/R = 0.5

)

- configuration II, defined througn:

hm/R = 0

.

016

;ho/R = 0.116

jLo/R

=

0

.

288

The parameter eog/w2 takes on the value 0.01 for configuration I,

and the value 0.005 for configuration II, so the inequality (51]

is valid in both cases.

The constant

8

in (461 is chosen to be i, and the grid spacings

have been varied according to

~y/~ = !

J

~x/Lo= 1,!,~

and

1/8 .

Configuration I has been studied by Ito and Tanimoto (1972), who

use a finite difference timestep method, and by Flokstra and

Berkhoff (19.771.,who use a finite element elliptic methode Table III

demonstrates the agreement between the various methods for the

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_,

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--

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18.

rate of convergence of ~~e numerical schemes I and II.

A detailed view of the solution is given in figures 1-5, which show

a comparison of wave amplitudes between the mentioned methods,

contourlines of the amplitude and of. the phase, and energy flux

lines (waveorthogonalsl. A grid is used with 281 ~ 449 grid points;

the centre of the shoal is located at xm=33, Y =113(in grid units),

_m

_,

and the radius of the shoal is R=16 ~x.

-Energy flux lines are defined throuqh the energy streamfunction G:

~

====:====================

====-

-=-=-

=-

-

~=-~

=-

--=

-

-

-

_

--

-

-

_

--=----

-

=

-

_

-.

---(52)

---,.---

----

-

---

-

-

----

-

-

-

-

._---_. ---- - --- .

Table 111

Comparison of maximum wave amplitudes for

a circu1ar shoa1.

~y/t.x

=

1/2

~x/La

1

Scheme

I

1/2

1/4-1/8

1

Scheme

11

1/2

(a=

1)

1/4-1/8

Ito and Tanimoto (1972)

Flokstra and Berkhoff

(1977)

Bettess and Zienkiewicz

(1977)

Configuration

I

Maximum

location

A

x/La

2.17

8.5

2.08

7.0

2.05

6.8

2.05

6.6

2.10

7.0

2.03

6.5

2.04

6 •

6

.

2.05

6.6

2.1

6.3

2

.

04

6.4

Configuration

II

Maximum

location

A

x/La

2.57

9.0

3.18

7.0

3.01

6.0

2.96

5.7

2.85

6.0

2.96

5.8

2.92

5.7

2.97

5.7

3.1

5.7

2.9

5.5

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-where amplitude A and phase F are given by ~

=

AeiF.

(If the field ~ satisfies the HeLmholtz-equation (8l, it

follows that

s r .

'ilG=O,i.e. orthogonality of F and G, which

provides another test of validity for the parabolic approximation).

Configuration 11 has been studied by Flokstra and Berkhoff (1977},

and Bettess and Zienkiewicz (1977}, using a finite element elliptic

methode In figures 6 and 7, the relative wave amplitudes on the

line -of symmetryI y=y m ' are presented.

It appears that the minimum near the rear end of the shoal cannot

be represented properly by the solution of scheme 11. This is

caused by the occurrence of branchpoints, for which

A=O.

In the vicinity of such points, the phase is a multiple valued

function, and the energy flux lines are closed. So, the application

of scheme II then results in a smoothed so~ution, which has better

convergence properties, and which is preferabie when the accuracy

requirements are not too high.

7. Summary and conclusions

For the propagation of periodic surface waves in shoaling water,

a parabolic wave equation (18) has been derived, based on the splitting

technique of Corones (1975). This method yields a pair of coupled

equations for the transmitted and reflected fields, and the

parabolic equation results from neglecting the reflected field.

In the case of an area with straight and parallel bottom

contour-lines, the asymptotic form of the solution for high frequencies

is compared with the geometrical opties approximation. There is

20:

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In the presence of a caustic, there is a reasonable agreement

provided the angle of incidence is close enough to 90°.

Otherwise the coupling hetween the transmitted and reflected

wave fields cannot be neglected, and systematic corrections

snould be applied, if one wants to recover the complete wave field.

Finally, two numerical solution techniques are presented in the

form of finite difference schemes, each based on a different

form of the parabolie equation. As an example, wave propagation

over a circular shoal is considered, where the geometrical opties

approximation prediets a cusped caustic line. For two hottom

configurations, .the results are compared with similar calculations

in literature, showing a reasonable agreement. Which solution

technique is preferabie depends upon the required accuracy and

the available computer capacity.

The parabolic equation method may be applied to short wave

propa-gation in large coastal areas of complex bottom topography.

Acknowledgement

The author wishes to thank ir. W.F. Volker of the SWA-department,

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Appendix Motion and spreading of a wave packet

The asymptotic character of the wave field bebind a shoal

of finite extension, in water of constant depth, is governed

in tlie parabolic approximation by equation (3Ql, with n=l.

Let '1'

=

I + e: Xl

=

kox

y'

=

koY

i then the

dis-turbance e: satisfies the Schrödinger-equation (omitting'l:

2 · de: _ 0~

-_

dX (Al)

Assuming E (and derivativesL sufficiently square integrable,

the following quantities are defined:

- the norm of the wave packet

(A2)

- the norm of the derivative

(A3)

- the mean position of the packet

<y> = -

_N

I

f

00_{_00} yl E12 dy (A4)

- the mean velocity d

<v>

=

dx <y> (AS)

- the spreading

S

=

/«y _ <y» 2>

=

l<y2> ._ <y>2 (A6)

22 •.

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and < v> are constants of the motion:

dN

=

0

dx-

,

dMdx -

=

0

d

dx

<v> - 0 CA7)

For the spreadinq S, one f~ds

(AB)

Inteqratinq

(Aal

twice, one o.bta~s the result, that for large x, S'increases linearly:

s

(x »co) ::: /~ - <V>2 •x (A9)

It follows then from (A21and (A9.L, that the magnj.tude

[e

I

of the wave packet approaches zeroI as

1/

IX

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References

Berkhoff, J.C.W., 1972 Computation of combined refraction-diffraction,

Proc. 13 th Coastal Engrg. Conf., Vancouver,

ASCE 1, pp. 471-490, 1973.

Berkhoff, J.C.W., 1976 Mathematical models for simple harmonie linear

water waves, Report on mathematical investigation, Delft

Hydraulics Laboratory, Report W 154-IV •

Bettess, P. and Zienkiewicz, O.C., 1977, Diffraction and Refraction of

surface waves using Finite and Infinite Elements,

Int. J. Num. Metli.Engng. 11(8)_,1271-1290 •

Biésel, F., 1~72, Réfraction de la houle avec diffraction modérée,

Proc. 13 tfi Coastal Engrg. Conf., Vancouver, ASCE 1,

pp. 491-501, 1973.

Booij, N. 1978 A proposed equation for refraction and diffraction of

water waves under the cambined influence of current and depth,

Manuscript, Delft University of Technology •

Corones, J., 1~75, Bremmer Series that Correct Parabolie Approximations,

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24.

Flokstra, C. and Berkhoff, J.C.W., 1977, Propagation of short waves over a circular shoal, Report W154-V (in Dutchl, Delft Hydraulics

Laboratory.

Ito, Y. and Tanimoto, K., 1~72, A method of numerical analysis of wave propagation - Application to wave diffraction and refraction,

Proc. 13 th. Coastal Engrg. Conf., Vancouver, ASCE 1, pp. 503-522, 1973

Jonsson, I.G. and Brink-Kjaer, 0., 1973, A comparison between two reduced wave equations for gradually varying depth, Inst.

Hydrodyn. and Hydraulic Engrg., Tech. Univ. Denmark, Progr. Rep. 31, pp. 13-18.

Langer, R.E., 1~37, On the Connection Formulas and the Solutions of the Wave Equation, Physical Review 51, 669-676.

Lundgren, H., 1976, Physics and Mathematics of Waves in Coastal Zones, Proc. 15 th. Coastal Engrg. Conf. Hawaii, ASCE, pp. 880-885.

McDaniel, S.T., 1975, Parabolic approximations for underwater sound propagation, J. Acoust. Soc. Am. 58, 1178-1185.

Radder, A.C., 1977, Propagation of short gravity waves in shoaling water, Refraction and diffraction, parts I and II, Rijkswaterstaat,

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Richtmyer, R.D. and Morton, K.W., 1967, Difference methods for

initial-value problems, Intersc. PubI., New York.

Schönfeld, J.Ch., 1972, Propagation of two-dimensional short waves, Manuscript (in Dutchl, Delft University of Technology •

Seckler, B.D., and Ke lIer, J.B., 1959., Asymptotic Theory of Diffraction

in Inhomogeneous Media,

J.

Acoust. Soc.

Am~,

31, 206-216.

Smith, R. and Sprinks, T., 1975, Scattering of surface waves by a

conical island,

J.

Fluid Mech. 72, 373-384.

SVendsen, I.A., 1967, The wave equation for gravity waves in water of

gradually varying depth, Coastal Engrg.·Lab. and Hydraulic Lab. Tecn. Univ. Denmark, Basic Res.-Progr. Rep. 15, pp.

2-1.

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Fig. 1

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26 •.

Y!Lo-4

2 3 4 ~ e 7 ij 9 10 :----X/LO

x/Lo;'7

:~

.

I 234 ~ 8 T

i

~y/Lo

X/Lo ...6

~--~~---r--~~--~--~

234 ~ ij 7 8 _Y/Lo

._

-

-

-

-

--

---

-

-

--

--

_

.

_-

.._._.

comparison of relative wave amplitudes for bottem

confiquration I, between results of the schemes I and II (continuous curves,

6x/Lo=1/8 )

and results of Ito and Tanimoto (19121.and of Flokstra and Berkhoff (19771

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Fig. 2

Contourlines of the amplitude for confiquration

I,

I

_---_.

accordinq to scheme I

-- --

U

we

--

/

-

Lo

---

=

--

1/8).

-- --- ---

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Fig. 3

Contourlines of tne amplitude for confiquration I,

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Ir.

(

~

/

Lo

=

1

/

8

).

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29.

1.20 210 200 190

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---

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--

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---

--

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--

_

.._---

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-Fig.

·

5

Contour~ines of the phase for configuration I, according

to

scheme

Ir

Cillc/Lo =

1/8).

-I

31.

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.

.

.

.

10 15

--

--,-

---_

..

_

----

,

--

-

--

-

-25 X/L4 20 --- ---_.

Fig. 6 Comparisonof re~ative waveamplitudes for configuration II,

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on the.line of symmetry,between resul ts of schemeI

(continuous curves}and resul.ts of Flokstra and Berkhoff

(lS77L(circlesl

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t

_{2.60 .}

t

₂₂₀ :~ -00

o

--.---.25 --- X/Lo

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--- ---~---_._---_. -

-Fig. 7 Comparison of relative wave amplitudes for configuration

::rI,on the li.neof symmetry, between resu~ts of scheme TI

(continuous curvesL and results of Flokstra and

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