A Function-Theoretic Approach to a Two-Dimensional ...

APfllkoltl• A,..frrll, Vol. )4, Pf. l U-lOl lhpn.,. 1u1l1blt -.1y from lloo publilllor l'l101oc:.>pyl o1 Pl""- by I~ ot1ly

0 1994 Oonloo Md 8,....h Sclcocc l'ubliahcn S.A. PnMed ,. Mala)'fia

A Function-Theoretic Approach to a Two-Dimensional Transient Wave-body Interaction Problem Communicated by R. P. Gilbert

'

GERASSIMOS A. ATHANASSOULIS and GEORGE N. MAKRAKIS

National Technical University of Athens, Department of Naval Architecture and Marine Engineering, P.O. Box 64070, 15710 Zografos, Athens, GREECE. AMS: 7681S. 3SCIO. 3SA22, 44AIO

Abstract. The linearized initial-boundary value problem describing the liquid motion in an ocean of infinit·e extent, caused by a two-dimensional floating body is investigated. Introducing a time-dependent complex velocity potential F(w, t) and performing an analytic continuation in the upper half plane, it is shown that the solution of the above problem can be exj,ressed in terms of the solution of an initial-value problem for F(w, t). The latter problem is solved explicitly by means of an appropriately defined Laplace transform, acting on time-dependent w-aoalytic functions. The simultaneous presence of the space complex variable w = x 2 + ix 3 and the transform complex variable p = .,µ, v E R }

(3.3)

FUNCTION THEORETIC APPROACH

289

The basic algebraic and topological structure of C; 1 has been presented in (3, Appendix 1J. Here we shall present in short some further results concerning the ij-complex numbers, wh ich will be of importance for the subsequent analysis. First, we recall the definition of the absolute value in C;,

I A,,,= (1e2 +

..\2+µ2+112)1/2,

(3.4)

and the definition of the fou r projection operators Re;A

= 1t + jµ ,

Re, A

=

IC+ i..\, lm,A =A+ jv, lm 1 A =µ+iv. Using these operators we can realize the i- and j- decomposition of an ij-complex number as follows

A= Re,A + i /m;A

= Re; A + jlm;A.

(3.5)

Recall also that C,, is a commutative ring but not a field , since there exist nonzero elements having no multiplicative inverses. The set of noninvertible elements is given by the union I;, = l/j U l ;j, where

I,,± = { B

ij = -l ±2-w,

w E C; } = { B = -I ±-ij p, p E C; } .

(3.6)

2

Note that each of I;j, I;~ forms a two-sided proper ideal in C ;;. Moreover, C ;; equipped with the norm {3.4) becomes a commutative Banach algebra over C; (or over C;). On the basis of the identities

=O

I+ ij I - ij 2 2 we can prove that C;,

I

1 + ij -2-

= I~ ffi l,j

l - ij

+ -2- =

[l ± 2

I I

ij]n = l ±2 ij

1

(3.7)

where ffi denotes the direct sum (cf. [27, p.207}).

The following representations of an i-complex (j-complex) variable in terms of its j-complex (i-complex) replica are very useful for the subsequent analysis I - ij I + ij _ I - ij 1 + ij _

w

=-

- w;

2

where w E C ; and w1 = Re;w

+-

-w;, p 2

+ j /m;w , w1 =

while p E C, and p; = Re;p + j lm;p,

=-

-

2

p;

+-

-

2

p;,

Re;w - j/m;w EC;,

p; = Re;p- jlm;p E C, .

(3.8)

(3.9} {3.10)

An important subset of C ,; is the following

C;;= {:4= pw: Pe c,,w e c,}.

(3.11)

C;,

T he subset is a multiplicative (but not a n additive) subgroup of C ;; which is characterized by the following PROPOSITION 3.1 An ij-complex number A =

C,,

1t

+ i..\ + jµ + ijv E C ;;

to the subset iff 1e11 = ..\µ. I The elements of C;; possess the follow ing property,

belongs

290

G.A. ATHANASSOULIS AND G.N. MAKRAKJS

l+ij( 1-ij( 1-ij( 1-ij( ( pw )n = -2pw, )n +-2pw;)n = -2p;w )n + -2p;w )n '

(3.12)

where w1 , w1 ,p.,p, are given by equations (3.9). The proof of (3.10) is immediate by ta.king into account the identities (3.7) a.nd (3.8). Let us now turn our attention to the study of the Laplace transform images of time-dependent w-a.na.lytic functions. At first we state the following PROPOSITION 3.2 (Cf. 145, p.2571). Let F..,(t) = F(w , t) belong to S'(-oo < t < oo) for each w E D, and suppqse that F•(w) =< F(w , t) , ef>( t ) > is a.na.lytic in D for each E S. Let also exp( -O't )F111 (t) E S( -oo < t < oo) for O' E (O'., 0' 2 ) ~ R . /\

Then , the two-sided Laplace transform F (w,p) of F(w,t) defined by (3.1), is p-analytic inn for each w E D (and w-a.na.lytic in D, for each p En). I The Laplace-transform image of (ESn) (and, in fact , of any similar linear evolution equation with constant coefficients; see a.lso Section 6) is an ordinary -valued coefficients of the differential equation in the i-complex domain, with form up'\ u E C;,p E C,. Accordingly, its solutions can be expressed, at least locally, in terms of power series of the form

c;;

/\

= E d,,.(f(p)wr =:F(f(p)w), ao

F (w,p)

dm EC;,

(3.13)

M:I

-

/\

where f(p) is a polynomial of p with values in C;;. Clearly, F (w, p) is not only a function of two arguments (w,p) E D x 0 ~ C, x C ;, but a.lso a. function of the simple argument (!(p EC;; . This fact will be essentially exploited subsequently.

)w)

After obtaining the solution :F(f(p )w) of the transformed equation, two main operations are to be performed. First, thew-analytic extension of :F(f(p)w) throughout a.n appropriate domain containing the liquid domain, and second, the Laplace inversion of :F(f(p)w) back to the time domain. In performing these operations the i-decomposition or the j-decomposition of :F(J(p)w) is indispensable. The following proposition provides a systematic a.nd bandy way for obtaining these decompositions. PROPOSITION 3.3 Every function

C,

1

g :C;1 -+

C;;, defined in some subdomain of

through a power series of the form ao

= E c.,,, (qzr , q EC;,

EC;

(3.14)

Q(q:i) = -2-Q(qz;) + -2-Q(qz,),

(3.15)

Q(qz)

z

m=O

can be represented either in the form 1 + ij

or in the form

Q(qz)

1 - ij

ij ij • = -1 +2 -Q(q;z) + -1 -2 -Q(q;z),

(3.16)

FUNCTION THEORETIC APPROACH

291

where z,, z,, q., q, are defined as in equations (3.9a, b). If c,,, E C; (resp. c.,.. E C1) then g(qz, ), 9 ( qz,) (resp. g(p,z ), 9(p,z)) are ordinary analytic functions with respect \o q (resp.z) in some subdomain of C, {resp.C;), and the former (resp. the later) representation realizes the i-decomposition (resp. the j-decomposition) of the function g(qz). I Proof. The representations (3.13) and (3.14) are straightforward consequences of the identities (3 .10) and the power series representation (3.12).

C,,

Note that the latter proposition permits us to express analytic functions on in terms of ordinary analytic functions of one complex variable (either C, in or in C ;). (In fact, such a reduction is also possible for analytic functions OD c,,). Thanks to this fact many useful properties of ordinary analytic functions, concerning e.g. integral repre~entations and asymptotic expansions, can be transferred to analytic functions on C;,.

C.;

We shall now consider some special functions on (or C;;). As a first example we define the exponential function on C;;, by using the standard power series representation: co

exp(A)

A"

= E-,, "=O n.

A=

1t

+ i~ + jµ + ijv EC;;.

(3.17}

All the usual properties of exponential function remain valid for the function exp(A), A EC,,. (Cf. (8, Chap.II}. For example, exp(qz) = exp(az} {cos(bz) + jsin(bz)}, p =a+ jb EC; , z EC;.

(3.18)

Moreover, using (3.7) and (3.15), the following identity can be easily proved

1 + ij 1 - ij ] [- 2- vi + - 2- V2 1 + ij 1 - ij :: - -exp(v1) + - -exp(v1) , v1,v, EC; or C,. 2 2

=

exp

(3.19)

We can now define th~ logarithm on Ci, as the inverse function of the exponential , through the relation exp(log(·)) =(-).Then, using (3.17), we can prove that log(qz}

= -I +2 -ij log(qz;) + -1 -2 -ij Iog(qz;) = ij ij = -I +2 -log(q;z) + -l -2 -log(q;z).

(3.20) (3.21)

where z,, z;, q; , q; are defined as in equations (3.9a, b). The usual algebraic pre>perties of the logarithm can be proved with the aid of the representations (3.18). For example log(qz)

= log(I q I z) + jarg(q) = log(q I z'I) + iarg(z).

(3.22)

292

0 .A. ATHANASSOULIS AND G.N. MAKRAKIS

Clearly, the logarithm as defined above is a multiple-valued function. For later use (see Section 5) we now define an exponential-integral function on Let q E CJI b, w E C;, a E R , and r. be a contour connecting the points ooexp(ia) and w, and lying in an appropriate sector O; of the plane C, (see below). Then, we define (all integrals appearing below are contour integrals along

C.r

r, ) £1 (qbw)

= =

-1"'

ooe•ca

1"' u- exp(-q;ln.t)du 1"' u- exp( - q;ln.t)du =

+ij -12 -1 - -ij 2

=

u-• exp(-qln.t)du d~

I -

+ ij 2

I

(3.23)

(3.24)

ooe 10

1

ooe• 0

_

I - ij

-Ei(q;bw) + -

2

-E1(q;bw),

(3.25)

where E 1 (z) = - J:C, u- 1 exp( - u)du = ft° u- 1 exp(-zu)du, z E C;, is the standard exponential integral [l, p.228J. The integral representations of E1 (q;bw) and £ 1 ( q;bw) are valid provided that Re; {q;bw} and Re; {q;bw} are positive. Thinking of q and bas parameters , and w as a variable, we can easily fix the sector 0 1:

Il;

=

{wEC; :iw l>0,-1r/2+(1argp l- argb) 0, (5.23) Now, combining (5.17) and (5. 18) we obtain the formula (5.15). This completes the proof of the lemma. T HEOREM 5.2. The unique ftlndamental solution of (AWE) vanishing fort 5 0, is continuous for -oo < t < oo and is given by

S1(w t) '

igt) t = ( -2w

(1- ·-· 3 igt2

)

2' 2 ' 4w '

(w, t) E Ci x [O, oo) .

1

(5.24)

Proof: Observe that for arg w = -i we have z = i( ¥,; )112 > 0, and thus we can apply (5. 15). Using this relation and the fact that D 2.,(-z ) = D 2., (z), we find

Using (5.21 ) and (5. 15 ) we can write (5.14) in the form

FUNCTION THEORETIC APPROACH

= =

(igt) exp (igt 2w 4w

2 )

~

299

(!; ~ ; igt

2 )

2 2 4w

=

(igt) ~ (!. ~- igtl) 2w 2'2'4w' arg w

=

-11"

/2, t

> 0.

(5.26)

Obviously, (5.25) can be analytically continued in the domain c; x (O, oo), and therefore we obtain (5.24). This, in conjunction with Theorem 5.1, completes the proof of the theorem. It should be noted that in the case g( w) = w, the functions S1 ( w, t) and 5 11 , (w,t) solve the classical Cauchy- Poisson problems, corresponding to an impulsive initial free-surface elevation and an impulsive initial pressure distribution on the free surface, respectively. ( It can be shown that the fundamental solution (5.21) is the same with the Cauchy-Poisson potential given by Lamb (22, pp.384-386J. apart from a multiplicative factor). Thus, the content of this section may be also considered as a new method for solving the classical Cauchy-Poisson problems.

6. DISCUSSION In this paper a transient WBI problem has been studied, modeled as an initialvalue problem for the complex velocity potential F( w, t ) (see (2.10),(2.11 )). Initialvalue problems (IVP) for w-analytic time-dependent functions F(w, t) will be referred to in this section as w-analytic IVP. (Note that we do not require analyticity with respect to ti{Ile variable t) . The only general method for solving such problems known to the authors 1 is the reduction to operator equations for which contraction mapping principles either in scales of Banach spaces of w-analytic functions [29J, [30J or in Banach spaces with a weighted supremum time-dependent norm [36J (See also (391), are applicable. (Moreover, if we postulate analyticity with respect to time we have at our disposal the classical version of CauchyKovalevskaya theorem). Although these methods are fairly general in some respects (for example, they do work for nonlinear w-analytic TVPs), they have two serious drawbacks. First, the obtained solutions are valid only in the small with respect to time, and second the space variable w is restricted in bounded domains. A Laplace-transform method has been developed in this paper for treating linear w-analytic IVPs in any (possibly unbounded) domain D ~ C; . This technique can efficiently be combined with function-theoretic techniques for solving the corresponding problems in the complex.frequency domain (See, e.g., [21 ]). The method developed herein can be applied to a broad class of linear equations with time-independent coefficients (or even with coefficients polynomially dependent 1 An efficient alt.emative method is available whenever the problem is self-similar. In this case the equation for F( w, t) can be reduced to an ordinary difl'ereoiial equation with respect to the nondimentional variable gt 2 /w which can be solved directly (32]

300

G.A. ATHANASSOUUS AND G.N. MAKRAKJS

on t ime, cf. (1 0, p.2621). Some further, apart from equation (2. 10), examples of w-analytic wave equations, related to the linearized water-wave theory, are the following:

oF((, t)/8( - i/3/(()F,u {(, t)

= ~(t)/(()Cn

(6.1)

where F ((, t ) = F (f((),t), and/((} is the conformal mapping function realizing the transformation of the exterior unit disk onto the exterior domain D ,

= G(w,t), (6.2) F,u (w , t) + ig8F(w, t)/8w - i-rif F(w, t)/8w 3 = 0 (6.3) Equations (6.1 ) comes from (2. 10) after the substitution w = / ((). This equation (8/8t - U8/8w) 2 F(w,t) + ig8F(w,t)/8w

might be of value for obtaining series expansion representation for Sn( w , t) convergent up to the body boundary 808, if the latter is not a semi-circle (cf. l4J. 121]). Equation (6.2) corresponds to a WBI problem similar to that studied in this work but with the floating body undergoing a steady horizontal translation with velocity U, apart from its transient motion. Finally, equation (6.3) corresponds to the WBI problem studied herewith, with surface-tension effects being taken into account. The versions of (6.2), (6.3) after the transformation w = / (() may be also of interest. It seems now proper to make some comments on the physical origin of equations (2.10) and (6.l)-(6.3). In classical water-wave theory [41], l34J, l26J, the field equation is the Laplace equation, and the wave character is introduced by the free-surface condition. In the function-theoretic treatment presented herein the field equation is replaced by the condition of w-analyticity of the complex potential F(w, t) and, after a suitable analytic continuation, the free-surface condition becomes the governing equation of the problem. Thus, in this model, the elliptic and the evolutionary character of the transient WBI problem are naturally coupled in F(w, t). On the other hand, on the basis of a real-variable formulation, one finds that the transient WBI problem can be reformulated as a nonlocal abstract wave equation, supported on the free surface, which has the form (see, e.g., [16J , [lSJ, ll4J, [9])