On the solvability of a two-dimensional water-wave radiation problem

NATIONAL TECHNICAL UNIVERSITY OF ATHENS

DEPARTMENT OF NAVAL ARCHITECTURE AND MARINE .ENGINEERING

On the solvability of a two-dimensional water-wave radiation problem by G. A. Athanassoulis (REVISED VERSION)

October 1983 Report Number 11-1983

Address: Department of Naval Architecture and Marine Engineering Naval Architecture Laboratory

TABLE OF CONTENTS

Page

ABSTRACT 1

Introduction 1

Notation and terminology 4

Formulation of the radiation problem

and some preliminary results ₈

Background theory ₁₅

The weak solvability of the radiation problem 19

Regularity of solution 28

APPENDIX ₃₀

ACKNOWLEDGMENTS 31

ABSTRACT

The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on

the free surface is established for all but a discrete set of oscillation frequencies. The body-boundary is taken to be o 'class (see §2) and the body-boundary condition is satisfied in L2sense. The proof relies on an expansion theorem (Athanassoulis _[1] ) and on the property of the associated

water-wave multipoles to be a Riesz basis of [(-n,o),a fact which is

established in the present paper.

Under stronger geometrical restrictions on the body-boundary it is

proved, using a method due to Ursell [10] , that the weak solution is

actual-ly a classical one, that is the velocity field is continuous up to and including the body-boundary.

1. Introduction

This paper is concerned with the study of the boundary-value problem arising when a two-dimensional body floating on the free surface of an

unbounded, infinitely deep, incompressible and inviscid fluid,performs time-harmonic oscillations of small amplitude about a fixed mean position. This problem is to be studied under the additional assumption that energy is radiated in both directions towards infinity,in which case it is usually called a radiation problem.

The solvability of the water-wave radiation problem for either two or three-dimensional floating obstacles of general shape has been studied by John [2] , Beale [31 , Lenoir and Martin

[4] , [5] , and Lenoir E6]

John [2] reduced the problem to a Fredholm singular integral equation of the second kind and treated the three-dimensional case assuming uniform,

finite depth; however, his results are also valid ir the two-dimensional case for either finite or infinite depth. He established the existence and uniqueness of a classical solution when the wetted surface of the floating

2 body satisfies certain geometrical conditions, namely it is of class

it intersects the free surface perpendicularly and is intersected by every vertical line in at most one point. Beale [3] proved the unique weak

solvability of the same problem for all but a discrete set of oscillation frequencies removing John's geometric restrictions. Lenoir and Martin [4]

[5) treated the case of three-dimensional bodies floating in a fluid of infinite depth. Using the limiting absorption principle they proved the existence of a generallized solution for all oscillation frequencies for which uniqueness holds. They also provided a general uniqueness theorem which, however, as pointed out by Martin and Ursell, was not correct (see

the discussions of a paper by Euvrard et al [27] ). Lenoir [6] treated, by the same method, the two-dimensional case for either finite or infi-nite depth.

For the two-dimensional case with infinite depth Athanassoulis [i) showed, using a method due to Ursell [7] , _[8] and the conformal mapping technique,

that,if a solution of the radiation problem exists, it can be expanded in

an infinite series of special functions, called water-wave multipoles.

In the present paper this multipole expansion is to be used to establish the weak solvability of the radiation problem for all but (possibly) a dis-crete set of oscillation frequencies (Theorem 5.3) provided that the body-boundary is sufficiently smooth, i.e. it belongs to the class t (see § 2), and, as a consequence, it intersects the free-surface perpendicularly. It is worth mentioning that in our approach John's convexity condition is no more necessary. Subsequently, under stronger geometrical restrictions on the

-3-body-boundary,it is proved that the unique weak solution is,..in fact, a clas-sical one, i.e. the velocity field is continuous up t and including the body-boundary. Morecvr, it is established that the multipole expansion coef-ficients of the velocity field are of order

O(iI'i?)

_{(Theo'rerrvi 6.1.).}

The idea of using relevant multipole expansions to prove the solvability of water-wave radiation problems goes back to Ursell (see Ursell _[9] , [10]

and Yu and Ursell [ii] _{) who has treated the case of a semi-circular floating}

body. _{Ursell reduced the question of the solvability of the radiation problem} to that of an infinite linear system and studied it using Fredhoim theory for compact operatm-s. In our treatment the proof of the weak solvability theorem,presented in §5, makes use of the basis property of the water-wave multipoles, which is also established in the same paragraph,whilst the proof of the regularity theorem,presented in § 6, is along _{lines due to Ursell}

jio].

In establishing the basis property of the water-wave _{multipoles we}

need some elements of the spectral theory of compact operators _analytically dependent on the spectral parameter. The pertinent background material along with the needed elements of the theory of bases in Hilbert spaces are summa-rized in § 4.

2. Notation and terminology

A Cartesian coordinate system Oxz.3 is introduced with xaxis on the mean free surface, x3-axis directed vertically upwards, i.e. in direction opposite to that of the acceleration of gravity, and the center

0

_{inside the floating body. A point in the (x2X3)-plane is} denoted by

_{=xr3) or}

_{in complex notation.}

The mean fluid domain is defined by _{D where}

D is a compactum in the lower half-plane, intersecting the

_x-axis

and representing the floating body. The mean free surface of the fluid,

i.e. the part of D-axis lying outside D, is denoted by'D, _and

the mean wetted surface of the floating body, i.e. the common boundary of _D and D, is denoted by 1DS. The symbols

D, D

denote the pointsets which are symmetric to

t, D

_{with respect to D.-axis;} see figure 1. We also define

D= DU

D*IU

_DUF

KF

-4

(

'D

K5

K

D

T

-5

The class of quadratically Lebesque-integrable, complex-valued functions, defined almost everywhere in (c,6) is denoted by

L(cI).

The scal:r product and the norm in are denoted by () and

($,) ,respectively.

2

The space

L(-nc)

is decomposed into two orthogonal subspaces: the subspace ,

spanned byjco9

and the subspace

in which forms a complete orthogonal sequence.

Some smoothness requirements are needed for the body-boundary _D8.

We shall say that8 belongs to the class

.ifbis a simple,

closed curve, described parametrically by equations

x2=x2W), x3=X3(Q))

E-nn]

and the functions

X(0),X3(e)

have the following properties:

They are continuously differentiable and their first derivate sat-isfy a Holder condition with exponent o(o,L)

Either

c>LI2

or the first derivaties of X2(9) and G) are of bounded variation.

((9))(d.))2o

for every

The above conditions are needed to ensure the validity of the Lemma 3.1 below.

More restrictive assumptions on the body-boundary will be introduced in §6, where the regularity of the radiation problem will be studied.

-6

A4=A24

4acncL

v.44

=

= O+jO

V I\f

2 =

Lç.= i4O

=(2?)+çw),

ecCj

=

in _{the definition of the zero and unit elements}

(2.2)

(2.3)

(2..4)

(.5)

(2.

6')

C2.

(.8)

I 4

Two noninteracting imaginary units 3 and t are used, making neces-sary the distinction between the

j

-complex numbers c3c(i-j ()e 1k))

and the _{-complex numbers}

_{(c1) .Products of}

j-

_{and %-complex}

numbers also occur, leading to objects of the form

O(4Je+S

(ijji'

which are called ij-complex numbers. The sets of and j-complex numbers are denoted by

,

and C1. respectively. The basic no-tions and operano-tions in _{are defined below.}

Let _{be U_complex numbers. They can be represented}

in the form A= +jw t

s4jW,5

_{weTe Z)W,}

,v/$

ct.

Then, we define

Equality

Addition Zero element Multiplication -Unit element Scalar multiplication Absolute value

The subscript

and the absolute value will be omitted in the sequel whenever no confusion is likely to occur.

The system

(12+)

'

O 4)

_{is a comrriutative ring, which)} however, is not a field (there exist non-zero, nonivertible elements).

Furthermore,

_.

_{is a norm, and the set t.1;}

_,

_{equipped with the}

ope-rations (2.3), (2.5), (2.) and the norm (2.9), becomes a comutative Banach algebra. For a rather complete study of the algebraic and topo-logical structure ofjj see Athanassoulis [1] , appendix i.

-7-Finally, if A=+ +i.'+1,jS ELj

we define

RetA

+3

LA

=

-8

3. _{Formulation of the radiation problem and some preliminary results}

Under the assumptions made in 1, the fluid motion is described by a velocity potential

where

cr),

thea-complex amplitude of

c (x-)

, satisfies the Laplace equation

+c)O

x...eTt,

(3d')

and the boundary conditions

KD)_c1$()=O) _/

t)

_LC),

C13

_(3.3)

>0 S=3

(3.4)

and

4(x)

jKO(x)- QxcD)

_x1I'oô.

(3.5)

Here and below, w is the frequency of oscillation, _{is the} accelera-tion of gravity and

/'r

denotes differentiation along the normal

of directed outwards with respect

to

the fluid. Introducing the j-complex potential

where

W(xis

the i-complex amplitude of the stream function

.4) (?Fc),

equations (3.1)-(3.5) are tranformed to

F (w') b ;,_

a-

ki

C

VO

_C3

I

cLF(w

.

(w)

c)

I

ctF(w)

cund

f(w)

LW

_t12"

-9

->

CD,

-

CO1

(3.io

respectively.

Re-Eo-G (3.5) and (3.10) are two fonns of the radiation _condition. For a discussion of this condition see, e.g., Stoker [12] _{, §4.3, 6.7,} Athanassoulis[1] _, 2,4.

A conformal mapping W=f()is now introduced, transforming the ex-terior of the unit circle [: jSt>i _{in the s-plane onto the domain}

in the W-plane. Such a function always exists, provided that

has at least two boundary points (Riemann's mapping theorem) and _{may be} chosen so that the pointsets

K

SI1O)

Ski-KF-LP1,SO)

Sti

and

KIm<O) ii>i1

in the 3-plane to be tranformed to the pointsets

and

D

in the w-plane)respectively (see figure 1).

Finally, we define

K=ImO

KJ\K

If D belongs to the class , then:

The function (3) can be extended in the region S:

es-tablishing a one-to-one and bicontinuous correspondance between

the regions [S 3I'iand EcUM (Osgood-Caratheodory's theorem). The function is continously differentiable and its derivative

10

-3. The function is expanded in a Laurent series of the form

2-t

c4>o,

t_-L,

where all CS are real numbers since the domain is symmetric with respect to 2-axis.

4. The equation

Using the change of variable _{we may transform the equations} (3.6)-(3.10) into the equations

F') b. i-ccdy1jc

,

Ij

+

L

ciS

K0

cl$

F

0)

36 KF)

Re

J

cLF(S)/d3

1

)-

0 E1j ± ) where

F,(3)

circLs

4-cr

F(--(S))

It should be noted that the complex potential

_(s)

_{depends also on the}

3)

Re±o01

(3J3')

(3A4')

w(-(e')

(3.L2)

realizes a parametric representation of the boundary

I5

such that

the functions _and

satisfy the conditions (i), (ii), (iii) introduced _{in the definition of}

"C

parameter K0=W/. To emphasize this fact we shall often write instead of (3).

The three sets of equations (3.1)-(3.5), (3.6)-(3.1O) and (3.13)-(3.17) are mutually equivalerrL under the assumption In the present paper we shall work with the third formulation and, for the sake of brevity,

the equations (3.13)-(3.17) will be collectively referred to as the radiation xroblem P(<0)..

We shall now state an expansion theorem for the ware potential

1SK0),

which is of fundamental importance for the subsequent treatments. This theorem provides an extension of the multipole expansion introduced by Ursell [26]

for body-boundaries symmetric with respect to the vertical axis.

Theorem 3.1 ( The expansion theorem; Athanassoulis [1] , § 4). A function

çi<

satisfies the conditions (3.13), (3.14), (3.16) and (3.17)

if

and only if it may be represented in the form

F (.K')

=

t\6 (&);

(-?-(S; K0)

-t-II_ d'm Mw(

K:))

(3.:L)

')

where

AO)A.)cL.

cE1

-1

_(3.i)

tW

6'0(W i<) =4(i-j)

-j-

(3.2d.')

fl

(. ao

-

<c

W

Gj

(w3

K)=- (iu)

_{K0 e}

₊

P (w Ko))

2fl

w

_iK0W(

_{_.s} 1(OU.

(W c) e

tt

e.

cLud)

S

C3. 21

0

(WK0)

*i<J(w;c)

cjWo

cLW

it follows that and

_(wC)/Lv'l

12

-and the path of integration in the last integrals is taken to be in the lower half plane.

Remark 3.1.1. The functions G0(w;)and

E(w;Ko)

represent free surface flows which are, respectively, symmetric and antisymmetric with respect to the axis

Ox3

Remark 3.1.2. The function

(w

K0 can be written in the following alternative form

F(W;

K0)

e

cow')

-of KGQ..

and they remain bounded as

K0> o

for each

Introducing now (3.18) into (3.15) and differentiating term-by-term we obtain

IJ \c

2-e-i

where is Euler's constant. With the aid of the above expression we see

that F(W,K)is,for each WQJ an analytic function of K0 in

where Q. is any open,bounded, simply connected region of .containing a part of the positive real axis but not containing the origin. Moreover, the

function K0F(WKo) tends to zero as O

_, for each fixed v14D. Since

;:

s=iIa)

are also analytic functions

= v(e) eE-[-n,o]

,

'4'e're

s_c.)

13

-The termwise differentiation in (3.22) is justified, in the sense of

pontwise convergence, if the series EJ2-l.Q and

1'ct

converge. The convergence of the first series is a consequence of the body-boundary smoothness assumption (see Lemma 3.1 below). The convergence of the second series will be considered in § 6, where will be established that

d= O(iIw3),

provided that is represented by a finite Laurent series and that is a simple,closed curve with continuous tangent.However,

if the left hand side of (3.22) is interpreted in a limiting sense, that

is as the limit when l3)-i+, the weaker condition _{is sufficient} to make (3.22) valid in L2-sense. This point of view will be adopted in § 5, where the weak solvability of the problem will be proved, without restricting to be a finite Laurent series.

Setting now

1\D

cycL

mlt

)2)3J..)

(3.

we can rewrite the equation (3.22) in the form

DC)

GEn)O.

(3.2C)

The central question is now the following : Under what conditions and

in what sense may the function

V(f3)

_{expanded with} respect to the sequence

0

It would be noted that the forcing term

V()

may be also depended upon the parameter K0,as, for example, in the diffraction problem. But this fact does not introduce any further complication.

14

-We now state a lema on the conformal mapping coefficients of a

boundary, which is often used in this work.

4,C(

Lemma 3.1 If

DG

and

{c

are the Laurent expansion coefficients

of

the conformal mapping fw'ction _{(see equation (3.11)), then}

CQ

_{1 oo.}

i

The proof of the above lemma is based on two classical theorems on the absolute convergence of Fourier series (Zygmund [13] , Theorems 3.1 and 3.6,

pp.

240-241) and the Denjoy-Lusin Theorem on the absolute convergence of the series of Fourier coefficients (ibid.p.232).

Finally, we collect, in the form of a theorem, _{some properties of the} functions ')(c)S=Oji tV _(Lko'))tn 23J...) and

which will be needed in § 5.

Theorem 3.2 Suppose that Then

i) 65((s)) <o))

ci, are continuously differentiable with respect to _G."< for any fixed Theyare also analytic functions of K0Q,for each SEK

ii.. are uniformly bounded with respect to

'w

_{and continuously} differentiable with respect to

S{s:

31>.Lj,the infinite series contained

being terimise differentiable

iii) (G 0')1So,i, are continuous with respect to

_OE-fl16]

for any fixed K0EQ. They are also analytic functions of K0E, for each

_eE-noT1.

uniformly bounded with respect to ' and continuous with respect to E'

E-n111].

15

-4. Background theory

In this section we present some definitions and theorems underlying the proof of the solvability theorem, given in the next section. The materials presented concern the spectral theory of compact operators analytically depen-dent on the spectral parameter and the theory of bases; they are given for the Hubert space case, although can be properly generalized for Banach _{or even,}

in some cases, for linear topological spaces (See I-Iarazov 122] , Singer [14] ).

Let E be a separable Hubert space and be The set of all bounded linear operators in

E.

By (.,.) and _{(. II} are denoted the inner product and the induced norm in

E

, respectively.

Definition 4.1.

Let T()(resp.

) be an operator-valued (resp.

yector-valued) function defined on a simply connected region G.Cc1,with values in

B(E)

(resp.

E ).

We shall say that

T(x)

(resp. ) is analytic in

O.

if the scalar-valued function (T(yX)\J) (resp. ((.x'X))is analytic in

_Q

for any

XJEE.

This apparently uweakllnotion of analyticity is actually equivalent to the strong" one, based on the existence of a Fréchet derirative with respect to

X.

(See,e.g., Taylor

[24],

p. 205, or Rule and Phillips [2.5], p.93).

Theorem 4. 1. (Gohberg and Krein [23], p. 21). Let T(.t) be an operator-valued function, analytic in an open, simply connected region

QCC,

and sup-pose that all values

of T()

are conipact operators. Then,for all points

where is a set

of

isolated points

of Q,

the number t(2)

of

linearly independent solutions

of

the equation

T())xO,

eE

16

-In particular, if t()Q for at least one point in _Q.. , then for all

the operator I-T() has a bounded inverse.

Definition 4.2. Let be two sequences with elements in

E

.

Then, the two sequences X y,,will be said to be quadratically near each other if

o3 _(4.L')

the two sequences çxj3l.j,,,3 will be said to be equivalent if there exists an automorphism

T

, i.e. an isomorphism of

_E

onto itself, such that

Theorem 4.2. (Higgins [25] _, p. 75). Suppose that and are two equivalent sequences in

E

. Then, I is a (Schcruder, Bessel, Hilbert, Riess) basis of

E if

and only if is a (Schcruder, Bessel, Hubert, Riesz) basis of

E.

We shall now state and prove a theorem providing a criterion for the equi-valence of a sequence in a Hubert space

_E

_{to a complete orthonormal system} of

E

. _{This theorem can be considered as an extension of some relative theorems}

of Bary (see Bary [16] or Kato {17] , pp. 246-266).

Theorem 4.3. Suppose that is a complete orthonormal system of a

Hilbert space E , and

()3J

eQCIJ is a family of sequences in

E

_defined

by

=

(4)

where

q())y=i,21..J

are analytic rector-valued functions of JEQ,with values

in

E.

Suppose also that

(z1.3)

-

17

-where

B

is a positive constant, independent of

1)

and that

_E.l).kLfor

at least one point

of a.

Then the sequences

{x

fjj))

are equivalent for all

Q,where

is a set of isolated points of

G..

Moreover,

fy)J

is a Riesz basis of

E

for

Remark 4.3.1. The set . may be contable, finite, or even empty.

Remark 4.3.2. _{Condition (4.3) is equivalent to the quadratic nearness of} the sequences

[X.n3

and

To prove Theorem 4.3 the following Lemma is needed.

Lemma 4.1. Suppose that {x

is

a coirrplete orthonormal system

of

_E

and

is

a sequence in

E

such that I

foOThen

the operator T E

E

defined on {xJ by TDc,, can be linearly extended on the whole

_E

as a compact operator.

Proof. To start with, let us extend the operator

T

on the whole E.If only a finite number of in the expansion

c(X

of an element

zeE

are different from zero, then we define

T=Io

_{,Tx,=.c*',.The}

_{same definition} remains valid when the seies

_oçX

_{has infinite terms, provided that} the series'o( converges in

E

. This is indeed the case since (N>rJ)

IIc4,

II

1c1)). >IIfl

'T\T1

and the series LQ(T) and converge. The compactness of the operator

T

follows from the condition

<

See, e.g. Smirnov _[18] , § 138.

Proof of the Theorem 4.3. We introduce the operator

T(>)

defined on by

T)x=q()

and extend it on the whole

E

by means of the definition

18

-According to Lemma 4.1,T(A) is a compact operator for each G) . Furthermore,

it is analytic with respect to

Consider now the operator

A()I+LT()

_{. Obviously,}

thus the equivalence of the sequences {x and is equivalent to the invertibility of operator

AG).

Since,

LIT()IJ = i:;1

II TO)

II

tIB,

it follows that there exists a point

_X=X,

_{in which (j.&0T(X)Hi; this} ensures the invertibility of A(>).Accordingly, applying Theorem 4.1, _we conclude that

A)

is invertible for all 2kQ\'.Finally, invoking

Theorem 4.2, we see that _{is a Riesz basis of}

_E

_{for all}

XQ\.

This completes the proof of the theorem.

04

2.

2

Remark 4.3.3. Since

A(o(.J (),where

_1l4.)xand

jo(.)=JJxi)

od,

e=f

the invertibility of the operator

A(X)

is equivalent to the condition

2

4ct=O

or

e

yefl\(

(A)

i.e. the e-iinear independence of the sequence _{$)),1.In the application to} the radiation problem condition (4.4) _{is, in fact, a uniqueness theorem.}

Then, )

2.

ii) Il2e)c

i(+i-a)Oi

_=-

cx3

_sa)

whereby it follows that

19

-5. _{The weak solvability of the radiation problem}

The main tool in establishing

the solvability of the radiation problem '(K0)is the basis property of the sequence

(K) }

in the space

L(_n,o).

Let us note that the functions

(;K)

depend analytically upon the Parameter k0Cj,which effects as the _{parameter in the general theory} of the preceding section.

1,

Lenvna 5.1. _{Suppose that}

and consider the multipoles

The proof of the above _{results is staightforward,}

so it will be omitted.

According to (5.3) the _sequence

is quadratically near to the system _{which is an orthogonal}

basis of the Hubert _space

L28. However, we cannot deduce _{any completeness or basis property}

of from this fact, since, in _general

_(&K0')

To proceed, we shall modify _{the multipoles}

'(& ç)

_{in such} a manner that the modified multipoles, _say (4)

(. <) :

_{j) belong .o}

2

ii) satisfy a condition _{similar to (5.3).}

20

-t)(O; K0')

3(O '(s)

'ifl=,3

and try to determine the constants by means of relations

(KA(K)f\

(K0)=

0)

00 01

(KoA(Ko)=O)

where

=

(CO5(5))

(O;

_{K0) )} . Relations (5.5) are

equivalent to the condition (G; K0)

L

The system (5.5) is uniquely solvable for each

rn'),

provided that

' (

°<°

(), (s.G)

(In any case, since

i(K0)

is an analytic function of 1it may have only

isolated zeros in

G..

Consequently, we can modify, if necessary, the region Q,,exc1udinq from it the zeros of (K0')) . Then, the constants can be

expressed in the form

o4 <Q 0)4

A=A (Ko)=

,(Ko)) M2J)

where

(-©

(rf)

() 1L\(<))

Om 44 11-Joj

C<o) =(-O

_O'Yfl

o)+4(%c)')

_{/ LØK0),}

34O

e.4(j) __tt

_(e2

(2-2)

₂

and-S .

(5.4)

(5.5'

(*)

_{The geometrical meaning of this condition is that the projections of}

the functions (Q;Ko) and (e;o) in the subspace

i

The infinite series in (5.9) converges, provided that . From

the equations (5.8) and (5.9) it is easily seen that

I 0)4 I o4

'ni=2,3...,

We can now state and prove the following

Lenna 5.2. Suppose

DBE4, Mo)*O)

and consider the modified multipo lee

where 21

-40q (;Ko),

cJn

K0)

j2Q)ceSiY

-

J-.

(5.{o)

(5j{%)

;;;ç 1(K0)G;K0).

(5.ia

Then,

1) 9(;Ko)cC([-n,o) arnL

K0)

1Zoo

_,

et-n1o1&,

(5J)

2.

ii) 41(;K0)

_£LB

I I

(;

K0)

B

(5.

i4)

where a positive constant independent of K0

Proof. The proofs of i) and ii) are straightforward, hence will be omitted. To prove iii) we observe that

22

-U

c0) fl £.'- II

(2-Q

_{SY) (4'-2)O II}

-

_{O) I}

I k°')11

'

e

Using now (5.2), (5.10) and Theorem 3.2-(iii) we obtain

II(;K)ft

whereby it follows the desired result (5.14), and the proof of the lemma is completed.

Let us recall that the set QCcCis a bounded, _{open, simply connected}

region containing a part of the positive real axis but _{not containing the origin.} Since,however, inequality (5.14) remains valid

_{as K0*O (see Remark 3.1.2),}

we can take as a region containing a point

_K,

_{such that}

_tKc,=lZ

in which case . tKO

<

j

. Defining

ccj

in this way and using Lemma

5.2 and Theorem 4.3 we arrive at the following

Theorem 5.2. Supppose that

*j (0)KOEQC.Then, for

all

k'L,

where is a set

of

isolated points

of a)

the sequence

is

a (nonorthogonal) Riesz basis

of

the Hilbert space

B'

equi-valent to the orthogonal basis

Remark 5.1.1.

KE

if and only if there exists a sequence of scalars

{c(ynJ such that

(5.i5)

(See Remark 4.3.3). Clearly (5.15) is _{a nonuniqueness condition for the} ra-diation problem

V(K0)

In the remaining part of this paper we shall simplify the notation by negle-ting _{K0 from the arguments of various functions. Accordingly, we shall}

oW

We cave the basis property of the sequence

Theorem 5.2. Suppose that

D6

to

_,

_cod-

_K0

Then, the sequence { is a (nonorthogonal) Riesz basis

of

the

Hubert space

0

Proof. Let

V()be

an arbitrary element of

t.(-n16).

In the first place we seek two scalars

CLO) oh.,

such that

Condition (5.16) is equivalent to the system

cL

+

+ L4

-(V5= (ccSsG),\'(G))), which is always solvable defining uniquely _c[c, and

cLL)

since

/x z/: 0

Since _1J(e)

L B

and the sequence

2

is

d.

Riesz basis of

_LB

(Theorem 5.1), there exists a unique sequence of scalars such that

U)

=1()

i-B,

&cL

-

Q3.

(5.i

Using (5.16) and (5.18) we obtain the following expansion of terms of where 04

V(Y_

co)

_(-n,°);>

_lb

c1

-

23

24

-The series in the right hand side of (5.t)is absolutely _convergent.

For, according to (5.10), (5.18) and Schwarz inequality, it follows

(M>)

\

Thus,the proof of the theorem has been completed.

We shall now state and prove the main result of this work. Theorem

5.3.

(The weak solvability theorem). Suppose that

o, U(e)

cL

KOEQ\. (In the physical

problem

K0(o

_{oo) ) Then, there exists a wiique weak solution}

_of

the radiation problem

'(K0)

_;

that is, there exists a unique conrplex

potential J() such that:

it is continuous throughout

K={S: Im3c S1

_L

_,

it satisfies conditions (3.23), (3.14), (3.16),

(3.27)

and

if

is

the limit

i

F(re.)

'c>

L+

(4)

exists for almost all

&E-n,oJ

and defines a function

F(o) 1(-n,o),

satisfying the boundary condition

(3.25)

in

Proof. belongs to

2

L-sense.

V()LD(

_PiO

It should be noted that the values of the functions tX(-) and

oO.

(5.2o)

Remark 5.3.1. _{In fact, the limiting behaviour described in iii) remains} valid as approches along any path lying in

K

which is not tangent to the unit circle.

Under the stated hypotheses the function

a

L (-n,o)

and the sequence is a Riesz basis of La(fl,O) _{. Therefore, there exists a unique sequence of scJoxs {'D°such} that

25

-as well -as the values of the scalars

_D

are 3-complex numbers. Accordingly,

H

(5.222)

Consider now the function

(3)

₌

4c))+ D1(s))

(5.2

i)1(3)is continuous on

K

. To establish this fact it suffices to prove the continuity of the function

M(5),

3K.

C5.a')

Since the sequence

MS)

is uniformly bounded on

K,

the series (5.24) is dominated by the series which is absolutely convergent (see equation (5.20)). Accordingly, the series (5.24) converges uniformly on

K

defining a continuous function there.

1()

satisfies the conditions (3.13), (3.14), (3.16) and (3.17) because of the Expansion Theorem 3.1.

(4)

It remains to prove that the function

f.

(S)

zd()/ctS

converges for almost all

GEfl,cJ

as S)-'i-I-, defining an

-function which satis4ies the boundary condition (3.15). Difecentiat-ing (5.23) we obtain (1)

-'i-)

(.25)

where (a)

cL)

{1)

_D

_kM(S)

(5.26)

H

(3)

can be also written in the form

(.2)

where

First, we study the function

4 (e )

96\flifl]

--9

According to (5.21)

> oe

₆

L (_n,n')

whilst(DI)1L,y(eie)

represents a continuous function, as

be proved in a manner similar to that used for proving the continuity

(i') iO ₂

of -1() in i). Consequently,

\ (

_{) is an L -function on the unit}

circle.

(41

Now we study the limiting behaviour of the function _{i-t (') as}

(S1-+i+. Let us note that this function is univalent and i-analytic

throughout the open domain

si>i3.

Furthermore, we have

?2fl

(2

2

LD

=

lJ DO

s-13Tn

' 2. where

rry

-

26

-2

±

Using the above relations we find

1) (t) e oO 2

a

S t)9(D\

I

_(+i)

1. 00 oQ _'I

-(ti#2)

_)&

=>-,>_ (+i)

_'=a

1=2

(5.2)

C5.3o)

c)(\2\)2Qo (5.M)

it can

27

-whereby, in conjunction with Abel's theorem (see, e.g., Goldberg [19] § 9.6) we obtain

1 ti)

f

)

(j

.

5. 32)

t')

The last condition characterizes the function _{as a Hardy function} of class

_62;

_{see Hoffman [20], p.39 and Walsh [21]}

, § 6.10, 6.11.

U)

Since

H ('t.°)

_{is harmonic and satisfies (5.32) and}

'tj

(e )

is an

I-function of the unit circlewe conclude, applying Fatou's theorem "

(Hoffman [20] , ch.3) , that the limits ..irni

) exist for almost all

OE-flfl]

and the equality

(i)

ry4 (tre )

.'.t- 4_+

holds almost everywhere in . In fact, (5.33) is valid as

the point approaches along any path in the open set

{:

>t3

which is not tangent to the unit circle.

Using now (5.25), (5.26) and (5.33) we see that the limits

(i)

[ (;,') exist for almost all

ee-n, o]

and

Q;

S)

=

for almost all

E[-nO]

as

S

approaches -nontangentially.

(.1)

G

The function ( )satisfies the boundary condition (3.15) in

L2-sense since, by construction,

ot

ee (e)_D)

and, by the definition of

vc9)=>

₀

6. Regularity of solution.

Let us, introduce the following geometrical assumption

(A) The body-boundary is such that the Laurent series of the

con-formal mapping function has only a finite number of terms.

In the present section we shall prove,using a method due to Ursell

[io]

the following regularity result

Theorem 6.1. (The regularity theorem). Suppose that is a simple, closed curve with continuous tangent, satisfying the assumption ('i). Suppose

also that and that

#o

K0eQ\L.Then,

the unique

weak solution

of

the problem (K0) given by the expansion (5.23), is actually a classical one, i.e. the velocity field is continuous thoughout

The coefficients

_D

of

the expansion (5.23) are of order

0

(5.jrw) Remark 6.1.1. Under the assumptions stated the curve is actually an analytic one.

Proof. According to Theorem 5.2 (see equation (5.18))

2

where

O() V(LO9()

1(G))

V(G) = U(G)

cke

/cLS

and &to,ckj are given by (5.17). Equation (6.1) is equivalent to

where LJ5=(co5(seU(0)) and q . Multiplying (6.2) by

Os %1

S we obtain

*sUs=ss+

(6.3)

It can be proved, with the aid of the assumption (A), that

n

-

28

-a

D

S-23)...)

dswi

I

oO

q

2

1 I

29

-(see Appendix). Moreover, since _{is analytic and}

e)c(C-n1oJ), u(e)

_kas continuous second derivative, from which it follows

_{that UO(I!?). Thus}

E

2

Consider now (6.3) as a functional equation in _{P. . Because of (6.4),}

2

the solution of equation (6.3) must be in ; that is

(6.5)

Using (6.3), (6.4), (6.5) and Schwarz's _{inequality we obtain}

IsUs-st)5I

£4Ko>1Ijs1a

00

52.'

from which it is concluded that

_{O(i/S2 )}

. _{The termwise}

differentia-bility of the expansion (5.23) and the continuity of

_c-j.,f(5)JL3

throughout

¼YOFUB

now at once. Thus, the proof of the theorem is completed.

Remark 6.1.1. Ursell

[ioj

_{bas studied the existence, uniqueness and} regularity of solution of the radiation (heaving) _{problem for the semicircular} boundary based directly on equations (6.2) and (6.3),i.e. _{withour resorting to} the basis properties of the water-wave multipoles,developed in § 5. This more effective approach can be easily extended for body-boundaries satisfying the assumption (A), and this was essentially done in the _{present section. However,} the author was not succeded to prove inequality (6.4) when the nonzero conformal mapping coefficients are infinite.

Here we shall sketch the proof of inequality (6.4). Since, in the present case, it follows that we obtain 5's APPENDIX 30 -FJ

+#','4

0

(-i)

_{c45-2)(e+-s-2) -}

_{___}

ss1

(A

-

pyy2

Using (5.10) and the inequality

rn=i

a

22

2

_{a-eI .ICej}

2

(+i-a) 5

tT tn(.is-a)t(.Pi_s_a)

a4

where A(N),K and A are positive constants. Moreover,

_{() and}

(.) have continuous second derivatives ; hence _=O(L/.S2)

ooI

iSo(

consequently

11sj2Ac'c

z

J

52

Each term in the sum appearing in the right hand side of (AZ) dominated by the quantity

S2.

Ba

(+(5+)a

+

cx=c,L

is

(A3)

+ K

_A

51 2.

(Aa)

31

-where . Following Ursell (rio\

, p. 295) we can easily show that

4

4i1

d

Using now

(/z) (A3)

and

(t.'1)

we obtain

A(t'J)

(M

'= a

which is the required inequality. Since AL(1I))oQ as 'J> o3 this proof breaks down if the number of nonzero coefficients is infinite.

ACKNOWLEDGMENT

The author is grateful to Professor F. Ursell and to the reviewer for their suggestions concerning the possibility of obtaining regularity results applying Ursell's method. The author would also like to express his deepest graditute to his friend and colleague P.D. Kaklis who gave much of his time on reading a preliminary draft of this paper pointing out errors and deficien-cies and suggesting improvements.

32 -REFERENCES

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An expansion theorem for water-wave potentials,

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f2] F.John,

On the motion of floating bodies II,

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Eigenfunction expansions for objects floating in

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M.Lenoir and D.Martin,

An application of the principle of

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Surface waves on deep water in the presence of a

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[8

F. Ursel 1,

The expansion of water-wave potentials at great

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[io]

F. Ursell, A probtem in the theory

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K.Hoffman, Banach spaces of analytic functions, Prentice Hall Inc., Englewood Cliffs, N.J., 1962

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34

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D.F. Harazov, On the spectrum

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