A hybrid integral equation method for time harmonic free surface flow

APR. i'a

_Lab.

_y.

Scheepsbouwbrnk

R

CHIEF

oceedings of. the Firs t Intenationa i Conference

Tech nische Hog eschool

on Numerical Ship Hydrod;nanícs

_Deift

Gaitherebug., MD.., Oct., 19'5

A -YERtD h\!TEGtAL-EQUATlON MET1OD FOF

TML.HARMONIC FREFE-SURFACE FLOW

Rcnd V. Vuncy

Mat!s

tittt

of

Tcsooqy

Cmbrdp, Twxsettz G23 U.S.A.

ABSTRACT

The two-dirensional boindary-value problems for

time-harLinic f reesurface flow in water

of arbitrary bottom

topography are csnsidered.

A novel and versatile approach

for the solution of these types

of problems is presented

in this

paper.

The inner region of changing topography is

descrbcd by an integral relation

while the outer regions

of constant: depths are characterized

by eigenfunctions.

Pressure and flux fields of

the two representations are

¡rtched at the radiation boundaries.

The resulting "hybrid"

integrai equation is solved

numerically.

The method is

applied to

a variety

of radiation and scattering problems.

Test computations agree

well with

existing

results.

New

results

for a number of realistic geometries are

also

given

to illustrate the versatility

of the method.

]

Introduction

Hydrodyr.amic-force coefficients for two-dimensional

ohjeo.Ls in or

near a free surface

ar

of parti.cuiar relevance

to ship-mo don or platform-motion theory.

The term radiation

problem applies to situations

where the body oscillates in

otherwise calm water, while

scattering problems us'ially refer

to csìs Thee

t

body is helu

n

LLeIn

of JnLide't

of an object depend not only on its own geometry, but also

strongly on the physical features of the environment. _The

advent of supersize vessels and huge offshore structures has made the consideration of effects of restricted water

impor-tant. This paper presents a new technique of solving the radiation and scattering problem in water of complicated boundary configuration, a subject which has not received much attention in the literature.

A brief review of existing solution methods for the type

of problems being considered is in order. The classical work

is due to Ursell (1948) who solved thn heaving problem of _a

circular cylinder in an infinite fluid using a sequence of

singularities located at the origin. _{A more versatile}

formu-lation for general-shaped bodies consists of distributing the

so-called wave-sources on the body contour. _{The strength of}

these sources is obtained by solving a Fredholm integral equation of the second kind [Lebreton and Margnac (1966),

Frank (1967)]. This approach is still applicable when

the water depth is not constant, in which case the unknown strength in the integral equation will also include that part

of the bottom boundary that is varying. If the depth changes

from one limiting value to another, on the opposite side, _the

problem is considerably more complicated. A variety of

approaches have been used by previous workers to tackle scat-tering problems associated with certain simple geometries. Bartholomeusz (1957) obtained a long-wave solution for propa-gation over a step of finite height by considering an integral equation which was arrived at by matching the potential above

the step. Newman (1965) provided results for the transmission

and reflection coefficients over an infinite step by matching

wave-maker solutions for the two regions. Miles (1967)

considered a more exact solution of the unequal-depths problem

by using schwinger's variational formulation. _{Hilaly (1967)}

treated the same problem by matching eigenfunction expansions.

non-step-like transition can be handled, in principle, by exploit-ing a Green function, if available, that satisfies the bottom

conditions on both sides of the transition region. Evans

(1972) in fact derived this function. The resulting

expres-sion is quite complex. To this author's knowledge, no

appli-cation of it to practical problems has yet been made. Bai

and Yeung (1974) investigated two new approaches for treating

the problem of general bottom topography. One method utilizes

a variational formulation with the fluid domain represented by finite elements; minimization of a certain functional then

leads to the solution of the problem. The other approach

con-sists of applying the simple source function for an infinite fluid to a finite fluid domain, resulting in an integral

equation along the fluid boundary that has to be solved. Chan

and Hirt (1974) used an initial-value problem formulation and a finite-difference mesh to tackle the problem.

The method described in this paper stems from the author's earlier work [Yeung (1973), also Bai and Yeung

(1974)1 in which case, the inner region where all the geo-metrical changes occur is described by Green's theorem using

the source function for an infinite fluid. However, instead

of applying a radiation condition at a large distance from the origin of disturbance, we now match the flux and pressure fields at the "radiation boundaries" with solutions in the

outer fields represented by eigenfunction expansions. The

location of the radiation boundary can now be chosen arbi-trarily close to the body, thus shortening the length of the

contour which bounds the inner field. The method is found to

be highly efficient from the computation viewpoint.

2. The Boundary-Value Problems

Consider the two-dimensional motion of an iriviscid fluid with a free surface, as shown in Figure L The bottom

topo-graphy can be arbitrary, but must approach some asymptotic

-/ TI-/f

So (I) N5+1 SB 5=NT i-1 S = SouSFt)Z+uE.

Figure 1. Coordinate System and Notation

N2+1 1 SB________ ,],_,, N3+1 \n. Segment

Figure 2. Subdivision of Boundary Contour

generated by the oscillation of the body or by an incident wave

system impinging on the body. If the flow is assumed

irrota-tional and time-harmonic of angular frequency 0 , the fluid

motion can be described by a velocity potential function,

[q?(x,j.)

_cL(k)J , (2.la)

where

(2.lb)

with ott. () being the amplitude of the body motion.

is the unknown complex-valued spatial function which satisfies Laplace's equation in the fluid:

= O

(2.2)

and the following linearized boundary conditions:

q.cp = o

C) -f(s) (2.3a) (2.3b) (2.3c)

where = _{being the acceleration of gravity,}

,

is the undisturbed free surface, the bottom surface, and

the wetted portion of the body in its equilibrium position.

If j istheunitinwardnormal of the body with components fl), the functions associated with the inhomogeneous

boundary condition,

4c)

for sway, heave and roll motion of the

body are p , , and

(X-rI)

respectively. Aside fromthe

conditions (2.3), a radiation condition must be imposed in

order to render the problem unique. This may be stated as that

all propagating waves must be outgoing.

Once is known, the linearized hydrodynamic force or

0

time-dependent Bernoulli's equation and are usually written in terms of components in phase with the body acceleration and the

body velocity. For instance, for the heave motion we have

(2.4)

where and are the added-mass and damping coeff i-.

cients for heave and are defined in terms of the solution of

(x,y), corresponding to f(s) = by

-(2.5)

Coupling coefficients also exist if more than one degree of freedom of motion occurs.

If one is concerned with the diffraction problem, the

definition used in (2.1) is still applicable provided we choose

a1 =0 and a2=l/a . In this work, it will be more convenient

to solve for the total potential known usually as the scattering potential defined by

=

+ P

_(2.6)

where is the incident-wave spatial potential, defined by

q'1 -

1t:

VCVC)

eX,

(2.7)

and PD is the so-called diffraction potential. Hence, for

the incident-wave problem, the scattering potential defined by

(2.6) satisfies (2.2), (2.3a) and (2.3b) but

= O

on (2.8)

The inhomogeneity will come from the radiation boundaries as detailed in the next section.

The free-surface elevation (x,y) for either type of

-and

where

a

the free surface and is given by -.L5k i

cxe)=-R

1'r.)e

i

(2.9)

Nothing has been said so far about how the two types of

problems are related. Such relations can be most conveniently

obtained by the application of Green's second formula. This

was indicated in a recent work by Newman (1975) pertaining to

the case of equal asymptotic-depths. The results can be

gener-alized for the unequal-depths case. For this latter case, there

exist two sets of transmission and reflection coefficients, corresponding to the two different directions of incidence of

the propagating waves. The two sets are somewhat related as

indicated by Kreisel (1949) and Newman (1965). If _(R1Tj

(T )

denote the reflection and transmission coefficients

due to incident waves from the left and the right respectively,

i.e. 'p, r

4l.X

f

+ R2

eJ Kq),

T2 e°

ç

-(2.11a) (2.11b) (2. lic) t' -r

Z' ±

%

-+ R1

e

il K(p

L n X

T

,_-%co

with

Ycx) = v4(x,o)

+ £

(2.10)

then RIE

TT*(D+/cr)

R

f1 -i-

i r

(D/D-' ) =

_I

=

I,I = IR3]

=

where

4 -

mht

-

2Pi

- t

4

These are useful relations for checking consistency of the

numerical solution. The well-known Haskind's relation remains

basically the same:

-

DY

(2.17)

where is the exciting force or moment in the j-th

dírec-tion due to a unit-amplitude incident wave and is the

com-plex wave-amplitudes due to the forced oscillation of the body or any part of the bottom contour (i.e., if we are interested in the force on that particular portion) in the j-th direction.

Finally, if _E denote the radiation wave-amplitudes of two

independent solutions (or experiments) of the radiation problem,

J

AB..

-T

-

A B

A ft:

_{(A ß - A 5) D/D}

which can be used to obtain the coefficients ( ).

(T , ) then follow from (2.13), (2.14) and (2.15).

(2.18) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16)

3. The Method of Solution

We illustrate the method of solution by applying it to the

forced-motion or radiation problem. The treatment for the

scattering problem requires only slight modification. First

let , .,, be two artificially chosen vertical boundaries

subdividing the fluid domain into an inner region of changing

geometry and two outer regions of constant depths. These are

denoted by regions I, II, and III as shown in Fig. 1. Let P be

a field point in Region I, and Q be a variable point on the

boundary of this region. Application of Green1s second

ideri-tity to and _/ being the distance between P

and , then yields

where

On

$j:'

2Tp(F)

S)J0

+d$C)-F

-

--being variables of integration along

Z .

Here

denotes an infinitesimal arclength element along the boundary.

I 4C)--

)c

,1

¿

2L

Ji$(Q)

(3.1)

» .,, j

and the

normal

derivative of

_Q

is

either

known

or expressible in terms of itself. Hence

21rC?'r

₌

_f

CpQ)-J1))

-f- J

+1

and

Consider now regions II and III. Since the fluid depth is

constant, the solution written in the form of

1969], viz.,

£pC%,)

A e"

.Ln?OX 0L nC (tk)

-with the coefficients

eigenvalues m0,

'k

cendental equation:

in either one of these regions may be eigenfunction expansions [Wehausen,

+)

.

c;

CoS

_CorJÇ

and = for

X(3.3)

e

vÇ c:os y}' for

being unknown. The

are roots of the

trans-(3.5)

where _m represents either or The functional

forms of (3.3) and (3.4) have been chosen so that Ç represents

outgoing waves to the right and left in Regions II and III,

respectively. If we now make use of these representations of

and substitute them into (3.2), i.e., if we match at

and the potential (which is proportional to the fluid

pressure) and the normal velocity of the inner regions with those of the outer regions, we may rewrite (3.2) as:

2W f q

cÖ

-tA-1

+

r

The superscripts in (3.6) are used to designate the wave number

and depth _h on the appropriate side. Physically, one may

think of the last four terms of (3.6) as a vertical distribution of sources and normal dipoles of certain assumed forms on the

"radiation boundaries'. The functional form of this

distribu-tion is chosen to satisfy Laplace's equadistribu-tion. This idea follows

from one type of treatment used by Yeung (1973), in which case,

an exponentially decaying distribution was used on the

radia-u

tion boundary to represent the behavior of an outgoing wave

in water of infinite depth. The present expansion is evidently

a more complete treatment. In practice, only a finite number of

terms in the series of (3.6) will be taken; we denote this by

NR and NL

Referring to (3.6) we see that the velocity potential, (e

will be known everywhere if is known on the boundary

u , and if the coefficients of the expansions are

known. To obtain an equation for the

() ,

we simply let

-

_A:

¿

j

+ _kI

; e'

1r( (3.6)

where the functions and are defined

by the following integrals:

m(j+)

(

,Ç1Î?ì' c_o

mJ

G

C-

, )

-h

.Pöj Ì2) .: ₎ c&

m('7)

(

ioI \)

:4

(1,7)

(3.8)

the field point approach the aforementioned boundary. _The fol-lowing "hybrid" integral equation then results.

-Ti-cpcp

tj

F

+Ae[ -G:i

N,..

-mL[L}

5

Icro

5 .f '1

t -LR(

ccsh Ng .j. .sGko,4vj

tAe

¡-.11 o _{co5JiWt Jçi} CLSTh W

-e

° L

Io

=

J

4()ö

U for

_{F'E Z}

(3.10)

= for

F

0uSvg

(3.9)

If we now place the point P at _{( NRtI )} distinct locations

on and (

t4+

) distinct locations on , we shall

have enough equations to determine the unknown coefficients of

the expansions. Whence, making use of (3.3) and (3.4) on the

left-hand side of (3.6) and observing that the 's give a

e

with a simìlarly modified expression for PE _ . It was

found that the choice of these so-called control-points on L1.

or L.. , with everything else held constant, have no effects

on the numerical solution of the problem. Of course, one must

not choose two points to be so close that they become

numeri-cally indistinctive. (3.9) and (3.10) thus are a rational

system of equations for the solution of the problem under

con-sideration. One caution. however one should take in solving

(3.9) and (3.10) is that it is more desirable to absorb the

exponential factor following Ç by defining new coef

fi-cients, say ± , representing the product of the two.

This will prevent certain column elements of the matrix of the linear equations representing equations (3.9) and (3.10), which will be discussed later, from becoming excessively small if

or _XL have a large value.

For the scattering problem, we observe that the

represen-tation of (3.3) and (3.4) is incomplete. If the right-moving

incident wave is of amplitude

A1

in Region III, the scattering

potential is

c,p=

A1e

+Ae

cj

_1_) osrv'1(l.) (3.11)

The quantity in (3.3) is flow proportional to the

ampli-tude of the transmitted wave. By (2.8) the third integral of

(3.2) iow vanishes. Transposing all unknown quantities to the

left-hand side of the equation, we obtain the following equation

as the analog of (3.9)

-irqcp)

qøqh.c(

+5 cPJeríJc$

-1--t-

-)1

tA:e

Í2tL

T ₊

Z

e

'13

'0 -

kI

* See page 584.

-A

-Ln

[3G:

I +

c

em

G

T KI

mL

-(3.12)

The inhomogeneity now occurs on the left radiation boundary. The analog of (3.10) can also be obtained, in a similar fashion. Once the system of equations is solved, the complex-valued transmission and reflection coefficients are simply given respectively by

-(3.13)

R - Ai/AI

It is worthwhile to note in passing that in classical el-liptic boundary-value problems, one normally specifies either

the potential or the normal derivative on the boundary. In our

present approach, an

indirect

specification of both cpand cp

on and

Z

is accomplished by requiring them to be

expand-able in the y-direction as a sequence of orthogonal functions

with unknown coefficients. Although such a type of "boundary

condition" is hardly discussed in the theory of partial differ-ential equations, we have experienced no difficulty in finding a stable numerical solution, which will be discussed next.

4. Solution by the Method of Discretization

By the method of discretization, the integral equations (3.lO) and (3.1) for the unknown continuous function

can be reduced to a set of linear algebraic equations. First,

a set of grid points is chosen along the contour _S

and in Region I. Let these be denoted by (

.

, ) , as

defined in Figure 2. For a sufficiently fine grid, the unknown

lip".-.-approximated by a constant function . If equation (3.l)

is evaluated now at a set of control points denoted by

on the boundary, the following discretized representation is possible:

.[-1rj ±)cp

± A0 _$Ai' t

z. c s =

i

,

for

j= I,Z,

N1

(4.1)

.t.= Ñ1I)

_-. I4

where

&.

is the Kronecker delta and the values of in the

first sum are identical to those of i The influence

coef-ficients C... and are defined by integrals as follows:

-Ir

Q..

=

_{= , N4;}

_(4.2)

-foi. =

. =N5+t ...

with Q.

and being line integrals over a small segment

P

₍

;j ,+,)

=

J J0 I ()±-f J

(4.3)

Ç

(4.4)

and =

[ ±(X.,.)-

LGC,L)j

(4.5)

n

with the and defined by (3.8) and (3.). In Equation (4.5), the top and bottom signs are to be associated with the right and left radiation boundaries respectively.

If the control points are located on , Equation

(3.11) is to be used. The corresponding form for it is:

.

C..

± A T

-f

6 T::

_{A0 S}

+

CS

f_FX)

for

where for obvious reasons,

N3-N= N

The

T

are defined

by:

Jm(h)

₊

_LG(o,)J

(4.7a)

T = e

[-h+

o

T =1_r

_Cih

₊

equations for

(X. ,.)

on can also be written

in a similar manner. Therefore combining these with (4.1) and

(4.6) we have altogether i\Lr linear equations for N,- unknowns

which can be solved simultaneously. Of course, much of the

efficiency and feasibility of the method will obviously depend

on our

ability,to handle integrals defined by (4.3), (4.4),

(3), and

(3.i). The integrals and

_G.:j

can be eval-uated analytically in a rather straightforward manner, but the

's and C- 's have no simple closed-form solution. While does not pose any problem for numerical quadrature,

requires more careful consideration because of the oscillatory

behavior of its integrand, particularly when K is large. It

can be shown that these source and normal derivative-integrals can be written in terms of standard exponential integrals with

596

(46)

complex arguments. Namely, define the complex numbers then

=m(+i),

₌ rn

[thJ

,

o

F (

,

:

= - m

ìn(c)

c,

rn(-di)

+

-5m

f

e

E1(2) t

e

E

IQ

ih

- -jmk

z

,1e

e

'E1)e

7 mJ rnC

k)Jc{1+ (-7}

-h

n Y2

=

Y[Oi-] -

-1TcP(t)e

-

[ e() - eE,(-)J

(4.10)

+Jtee

Eu,)-where the exponential-integral special function E,() is

defined to be

E()

, j9j K

(4.11)

A preponderance of techniques is available in the literature

for efficient evaluation of this special function.

5. Results and Discussions

The present method of solution has been applied to a nurn

ber of radiation and scattering problems. In all cases where

existing results are available and are known to be correct, the

present computation yields very good agreement. As an example,

the heave added-mass and damping coefficients (or wave ampli-tude ratio) for a circular cylinder as well as a rectangular cylinder in water of finite but constant depth are shown in

Figures 3. For the rectangular case, these are compared

with Lebreton and Margnac (1966) who used the Green-function

integral-equation technique to solve the problem. For the

cir-cular case, results are compared with Bai and Yeung (1974)

ob-tained by two different methods. The present calculations are

consistent with the author's previous calculations using the

fundamental-singularity distribution method. For a circular

cylinder in water depth of h/a = 2.0, the current calculations yield a limiting value of added-mass coefficient equal to 0.500

as the frequency goes to zero. This value is found to be

sistent with that obtained by an alternative approach of con-sidering the limiting boundary-value problem (Newman and Yeung,

1975). The corresponding limiting value of the rectangular

cylinder is 0.822. Yeung (1973) showed that for a fluid with

different depths on each side, the damping coefficient is re-lated to the asymptotic wave-amplitudes by

2

This equation provides a consistency check of the integral of the potential on the body, Equation (2.5), and the asymptotic wave-amplitude defined by the first term of the eigenfunction

expansion, Equations (3.3) and (3.4). In all results shown,

Equation (5.1) is satisfied very accurately.

Figure 4 shows the hydrodynamic coefficients of a bulbous

598

+

t h=2 -b=1

-'fl/f

11: Rectangle

/

Rectangle

-S-/

-,

/

4<

/,

/ 0.5 1.0

/

/

4-0 Circle ___.*_____.s___*__ Circle Fig. 3

Heave added-mass coefficients and wave-amplitude

ratios of circular and rectanqular cylinders

-- } Present Calculations

X

Lebreton & Margnac (1966)

h= 2 S-- -- S- S--'1ff /7/F/Fl ff711/f/f f/f/f f //f/1f/IFI////l f, I I 1.5 2.0 1.0 0.8 0.6 0.4 0.2 0. 4-l, J Bai & Yeung (1974)

cylinder undergoing sway or heave motion in water of varying

depth. A sinusoid is used to represent the transition region. Results for the constant depth case with the depth being the

average of the two sides are also shown. For the particular

configuration considered, forces associated with the heave mode are much less sensitive to the change in bottom topography as

compared to those of the sway mode. Coefficients associated

with the sway mode also become practically indistinguishable

for the two cases when )..'a. ',o.2. This corresponds to a mean

depth/wave-length ratio of approximately 0.463. We observe

that the phenomenon of radiationless frequency for heave occurs

also in water of finite depth. Furthermore there exists a

frequency rañge in which the sway added mass becomes vanishingly

small and even slightly negative. Similar occurrences of

nega-tive added-mass coefficients were observed by Ogilvie (1963) when he considered the swaying force on a submerged circular cylinder near the free surface.

Turning our attention now to scattering problems, we show in Figure 5 the total wave amplitude function YT(x) due to a

right-moving wave propagating over a sinusoidal hump. The

results obtained earlier by Bai and Yeung (1974) using a

finite-element method have been incorrectly given. The complex wave

function YT(x) is defined as

= 2e

_Ec)LuYzc%)

where the first term after the first equality sign represents the incidentwave and the second the diffracted waves.

The scattering characteristics of a finite-size step are

given in Table 1 and are compared with Hhlaly (1967) . By

Equation (2.11):

lY*1zIYI

₌

(5.2)

.35 30 .25 .15 .10 .05 o 0. 0.1 0.2 _{0.3 0.4} 0.5 0.1 _{0.2 0.3} 0.4 _0.5 Fig. .54 .52 .50 I .48 .46 .44 .42 .40

1.2

0.6

- 0.4

0.2

Fig. 5 Amplitudes of diffracted and total waves

for a sinusoidal hump. (m0h=0.5)

602

-4.0 -2.0 0.

mx

o

where and Y , being the resulting wave amplitudes

cor-responding to unit-amplitude incoming wave, are essentially the transmission and reflection coefficients respectively.

Here, ' evidently represents the group-velocity ratio.

The

agreement as can be seen is quite good

Our method is next applied to determine the reflection

ability of a steep hump located in front of a step. The

speci-f ic geometry ospeci-f the hump is shown in Figure 6, with the results

presented also therein. Transmission characteristics of a

finite-size step without the hump are also plotted for the

pur-pose of comparison. A considerable increase in the reflection

coefficient occurs. However, one would still consider the

performance of the hump as a submerged breakwater to be rela-tively poor.

6. Concluding Remarks

We have examined in this paper a very versatile method for

solving the two-dimensional time-harmonic free-surface flow

problem for a fluid of arbitrary bottom topography. The method

is quite stable. Results obtained always satisfy consistency

and energy checks very well. Generally speaking, the numerical

accuracy is not as good for scattering problems when the ref

lec-tion coefficient is vanishingly small. This corresponds to

situations in which the wave frequency is fairly high. For

low to moderately high frequency, the technique discussed here

represents a simple yet potent tool for investigating the

hydro-dynamic interaction of bodies with their physical environment which can be made as realistic as possible.

Table 1. Reflection and Transmission Coef-ficients of a Finite-height Step

*TaJçfl frc Hilaly (1967). His phase angle cx is related to the present

notation by: cx = -(ri + arg R)

vh R1 T1 Eqn (5.3), R1I arg R1(°) arg T (°) L.H.S.= 0.03395 .4269 -163.14 .5799 4.99 1.00025 .4273* _{_163.26*} 5797*

497*

_1.00001 0. 04365 .4242 -160.86 .5846 5.64 1.00016 .4246 -161.00 .5846 5.61 1.00001 0.05813 .4201 -157.86 .5918 6.46 1.00011 .4205 -158.02 .5917 6.44 1.00005 0.06644 .4134 -153.73 .6035 7.56 1.00002 .4138 -153.93 .6034 7.53 1.00007 0.12115 .4016 -147.68 .6244 9.04 .99980 .4021 -147.94 .6244 9.02 1.00008 0.20031 -137.97 .6679 11.07 .99938 .3777 -138.32 .6681 11.04 1.00031 0. 39260 .3158 -120.25 .7803 13.05 .99851 .3162 -120.83 .7810 13.03 1.00023 1.09093 .1458 -90.67 1.0395 7.61 .99760 .1470 -91.99 1.0407 7.69 1.00025

1.2

o

0.8

o h

.02

.04

/

r

I, h+ '7

y(x)= _h+[2.5_.cos(Tx/h+)J

y+(x).. _h+[1_.cosCnx/h+)]

's

---"s

N

N N

Energy Trans. Factor,

K

5%_

Fig. 6

Reflection and Transmission Coeffs. of a

hump

>

, i

I'vh

i I i J J,

0.1

0.2

0.4

0.6

1.0

2.0

4.0

,

0.5

0.4

0.3

0.2

0.1

0.0

hump case step case

h/h

2.5

Acknowledgement

This work was carried out under the Naval Ship Systems Command General Hydrodynamics Research Program administered by

the Naval Ship Research and Development Center, Contract Number

N00016-75-C-0236. The author is also grateful to partial

support by the National Science Foundation, Grant Number

GK-43886X. Acknowledgement is made for the valuable assistance of Mr. Y. H. Kim, Research Assistant in the Department of Ocean Engineering, MIT.

REFERENCE S

Bai, K. J. and Yeung, R. W., "Numerical Solutions to Free-Surface Flow Problems," 10th S7np. on Naval Hydro., Cam-bridge, Mass., 1974.

Bartholomeusz, E. F., "The Reflexion of Long Waves at a Step," Proc. Carnb. Phil. Soc., vol. 54, 1958, pp. 106-118.

Chan, R. K. and Hirt, C. W., "Two-dimensional Calcula-tions of the Motion of Floating Bodies," 10th Syrnp. on Naval Hydro., Cambridge, Mass., 1974.

Evans, D. V., "The Application of a New Source Poten-tial to the Problem of the Transmission of Water Waves over a Shelf of Arbitrary Profile," Proc. Canib. Phil. Soc., vol. 71, 1972, pp. 391-410.

Frank. W., "Oscillation of Cylinders in or Below the Free Surface of Deep Fluids," Naval Ship Research and Develon-ment Center, Report 2375, 1967, vi + 40 pp.

Hilaly, N., "Diffraction of Water Waves over Bottom Discontinuities," Univ. of Calif. Berkeley, Rept. HEL 1-7,

Sept. 1967, 142 pp.

Kreisel, G., "Surface Waves," Quart. Appi. Math., vol. 7, 1949, pp. 21-44.

Lebreton, J. C. and Margnac, M. A., "Traitement sur Ordinateur de Quelques Problems Concernant l'Action de là

Houle sur Corps Flottants en Theorie Bidimensionelle," Bulletin du Centre de Recherches et d'essais de Chatou, No. 18, 1966.

Miles, J. W., "Surface-Wave Scattering Matrix for Shelf," J. Fluid Mech., vol. 28, 1967, pp. 755-767.

Newman, J. N., "Propogation of Water Waves over an Infinite Step," J. Fluid Mech., vol. 23, 1965, pp. 399-415.

Newman, J. N., "Interaction of Waves with

Two-dimen-sional Obstacles: a Relation between the Radiation and

Scat-tering Problems," to be published in J. Fluid Mech., 1975.

Newman, J. N. and Yeung, R. W., "On the Low-frequency

Limit of Added Mass of Cylinders," in preparation, 1975.

Ogilvie, T. F., "First- and Second-Order Forces on a Cylinder Submerged under a Free Surface," J. Fluid Mech., vol. 16, 1963, pp. 451-472.

Ursell, F., "On the Heaving Motion of a Circular

Cylinder in the Free Surface of a Fluid," Quart. J. Mech. Appi.

Math., vol. 2, 1949, pp. 218-231.

Wehausen, J. V. and Laitone, E. V., "Surface Waves," Handbuch der Physik, vol. 9, pp. 446-778, Springer-Verlag, Berlin, 1960.

Yeung, R. W., "A Singularity-Distribution Method for Free-Surface Flow Problems with an Oscillating Body," Univ. of Calif. Berkeley, College of Engineering, Report NA 73-6, Aug.,