Generalized eigenfunction expansions for linear water waves C. Hazard and F. Loret, Laboratoire de Simulation et Mod´elisation des ph´enom`enes de Propagation (URA CNRS 853), France, E-mail: [email protected], [email protected]. SUMMARY The present paper is devoted to generalized eigenfunction expansions for scattering problem. A new method for establishing such expansions is proposed and applied to scattering of linear water waves. 1.

Introduction

  Z Ls2 (R) = v : R → C; (1 + x2 )s |v(x)|2 dx < ∞ , R

In this paper, we show a general way of establishing eigenfunction expansions for linear scattering problems, and illustrate it in the context of linear water waves. From a physical point of view, such an expansion provides the connexion between transient and time-harmonic motions. More precisely, it leads to decompose the state of the system at every time on a continuous family of time-harmonic states which represent the response of the system to a family of timeharmonic plane waves. How can one obtain eigenfunction expansions? In the absence of scatterer, the Fourier transform is the very tool for this work (§2.). A natural idea for dealing with scatterers consists in considering the problem as a perturbation of the former free situation. This leads to consider the eigenfunctions of the free problem as incident time-harmonic waves, and to search those of the scattering problem by adding a perturbation term representing a scattered time-harmonic wave (§3.). The first application of this approach is due to Ikebe [4] for the Schr¨odinger equation. In hydrodynamics, the scattering of linear water waves by a fixed body was first studied by Beale [1], and then by different authors (see [3]). The common feature of these studies is that most proofs are highly “problem-dependent”, that is, a slight change in the definition of the perturbed problem requires to adapt most proofs. The purpose of the present paper is to show a more synthetic approach which has the advantage to allow general perturbations of a free water wave problem. In fact we shall see how to construct generalized eigenfunction expansions for any “compact perturbation”: we shall give the precise meaning of this compactness property. For the sake of simplicity, we shall illustrate the method with the 2D scattering (in a half space) by a fixed rigid immersed body, but the method easily extends to more involved situations such as the seakeeping problem for an elastic floating body. The general ideas are described in [2], and the detailed application to linear water waves is the object of a forthcoming paper. We shall use the following notation, for s ∈ R :

(with the particular case L02 (R) = L2 (R)), which al2 (R) as dual spaces in lows to consider Ls2 (R) and L−s the scheme 2 (R) if s > 0. Ls2 (R) ⊂ L02 (R) ⊂ L−s

R

This means that the integral R u v can be seen as the scalar product (· , ·) in L02 (R) as well as the duality 2 (R) and L2 (R). product h· , ·i between L−s s 2.

The “free” problem

This paper deals with localized perturbations of the ˜ = linear water wave equations in the half-space Ω  X = (x, y) ∈ R2 ; y < 0 delimited by the free surface F˜ = {x = 0} (the tilde character will be used for all quantities related to this free situation). Without ˜ outer excitation, the velocity potentiel ϕ˜ = ϕ(X,t) satisfies (1) (2)

˜ ∆ϕ˜ = 0 in Ω, ˜ ∂t2 ϕ˜ + ∂y ϕ˜ = 0 on F,

together with the initial conditions (3)

˜ ˜ ˜ ϕ(0) = g0 and ∂t ϕ(0) = g˙0 on F.

These equations are easily solved using a horizontal Fourier transform. We give in Proposition 1 below an abstract interpretation of this well-known result, with is the basis of our perturbation approach. Let A˜ denote the operator formally defined by A˜ = ˜ |F˜ where H˜ is the “harmonic lifting” from F˜ to (∂y H) ˜ i.e., for v˜ defined on F, ˜ the function ψ ˜ = H˜ v˜ is the Ω, solution to ˜ ˜ = 0 in Ω, ∆ψ ˜ ˜ = v˜ on F. ψ Setting ϕ˜ = H˜ u, ˜ this allows us to rewrite (1)–(2) as an abstract wave equation (4)

˜ |F˜ . ∂t2 u˜ + A˜ u˜ = 0 with A˜ = (∂y H)

It may be easily seen that A˜ actually defines an un˜ and bounded positive selfadjoint operator in L02 (F),

the solution to (4)–(3) writes n o ˜ 1/2 (5) u(t) ˜ = Re e−iA t u˜0 , where u˜0 := g0 + iA˜ −1/2 g˙0 . The Fourier transform actually provides a diagonal form of this expression in a spectral basis of A˜ defined by the functions (6)

w˜ λ,k (X) =

eλ(ikx+y) √ for λ ∈ R+ and k = ±1, 2π

which are time-harmonic solutions to (1)–(2): they √ represent plane surface waves of frequency λ which propagate towards k × ∞. In the sequel w˜ λ,k will denote either the above functions or their restrictions to 2 (F) ˜ Note that w˜ λ,k ∈ L−s ˜ if s > 1/2. F.  Proposition 1 The projection on the family w˜ λ,k

 ˜ (s > 1/2), (7) (U˜ v) ˜ λ,k := v, ˜ w˜ λ,k F˜ ∀v˜ ∈ Ls2 (F) defines (by density) a unitary transformation from ˜ to the spectral space L02 (F) n o 2 R L2 (R+ ×{±1}) = uˆλ,k ; R+ ∑k=±1 uˆλ,k dλ < ∞ . ˜ = Moreover U˜ diagonalizes A˜ in the sense that f (A) ∗ + ˜ ˜ U f (λ) U for every bounded function f : R → C, which can be written more explicitely (8)

Z

˜ v˜ = f (A)

R+

f (λ)

∑ hv,˜ w˜ λ,k i w˜ λ,k dλ.

k=±1

Z

∑ hu˜(0) , w˜ λ,k i w˜ λ,k e−i

R+ k=±1

√

λt

dλ.

We claim that a similar expansion hold for every compact perturbation of the free water wave problem (with a possible additional discrete contribution due to possible trapped modes, see e.g. [5]). 3. A simple perturbation Consider the problem of scattering of water waves by an immersed fixed rigid obstacle. We denote by ˜ the domain exterior to its boundary Γ (so that Ω⊂Ω ∂Ω = F˜ ∪ Γ). The equations satisfied by the velocity potential are now given by (9) (10) (11)

∆ϕ = 0 in Ω, ˜ ∂t2 ϕ + ∂y ϕ = 0 on F, ∂n ϕ = 0 on Γ,

(12)

± w± ˜ λ,k +Wλ,k . λ,k = w

These functions correspond to time-harmonic solu± tions to (9)–(11) if Wλ,k satisfies (13)

± ∆Wλ,k = 0 in Ω,

(14)

± ± ˜ = 0 on F, − λWλ,k ∂yWλ,k ± ∂nWλ,k = −∂n w˜ λ,k on Γ.

The sign +, respectively −, is assigned to outgoing, respectively incoming, waves, which is specified by means of the standard radiation condition at infinity: Z 2 ± ± (15) lim ∓ iλWλ,k ∂|x|Wλ,k dy = 0. R→+∞ |x|=R

To be sure that both families (12) actually define generalized spectral bases for A, we shall make use of an abstract framework. 4.

The latter formula is the generalized eigenfunction ex˜ For f (λ) = exp(−iλ1/2t), it yields pansion of f (A). the diagonal form of (5): u(t) ˜ = Re

as well as initial conditions similar to (3). Exactly as for the free problem, these equations can be expressed as an abstract wave equation of the form (4) which involves the perturbed selfadjoint operator A = (∂y H)|F˜ ˜ where H is the perturbed harmonic liftinstead of A, ing (obtained by inserting the Neumann condition on Γ). How can one construct a spectral basis for A? Simply by considering two kinds of perturbations of the plane waves w˜ λ,k , written in the form

Abstract Perturbation Result

For the sake of simplicity, we keep the particular functional spaces introduced in the previous sections to present some general results. We denote by A˜ and A ˜ (contrary two bounded selfadjoint operators in L02 (F) to A and A˜ which are unbounded). We assume that  2 (F) ˜ of A˜ in the we know a spectral basis w˜ λ,k ∈ L−s sense of Proposition 1 Let us first show an intuitive construction of a generalized spectral basis for A considered as a perturbation of A˜ . We denote their difference D := A − A˜ . The idea is to search solutions wλ,k to (A − λ)w = 0 in the form wλ,k = w˜ λ,k + pλ,k . Using the above definition of D and the fact that (A˜ − λ)w˜ λ,k = 0, we see that the perturbation term pλ,k must satisfy (16)

(A − λ)pλ,k = −D w˜ λ,k .

But this equation is ill-posed if λ belongs to the spectrum of A , which is precisely the situation we are interested in. A way to solve it consists in replacing first A − λ by A − ζ for some ζ ∈ C \ R+ . Indeed the resolvent (17)

R (ζ) := (A − ζ)−1

˜ since the specdefines a bounded operator in L02 (F) + trum of A is contained in R . Then, setting C± = {ζ ∈ C; ± Im ζ > 0}, we can consider both one-sided limits p± λ,k := −

lim

C± 3ζ→λ∈R+

R (ζ)D w˜ λ,k ,

which formally satisfy (16). The study of the behavior of R (ζ) near R+ is the object of the so-called limiting absorption principle. Noticing that

R (ζ) = R˜ (ζ)(Id + D R˜ (ζ))−1 , (which is easily deduced from the definition of D ), we see that the existence of R (λ ± i0) depends on one hand, on a limiting absorption principle for the free problem, i.e., the existence of the limits R˜ (λ ± i0), on the other hand, on an additional property which ensures the invertibility of Id + D R˜ (λ ± i0). More rigorously, let us introduce the following Definition 2 The free operator A˜ is said to satisfy a “strong limiting absorption principle” if R˜ (ζ) := ˜ to (A˜ − ζ)−1 considered as an operator from Ls2 (F) 2 ˜ L−s (F) has one-sided limits (18)

R˜ (λ ± i0) := ±lim R˜ (ζ) ∀λ > 0, C 3ζ→λ

5. The strong limiting absorption principle 5.1

One-sided limits

We now prove the free operator A˜ satisfies the onesided limits property of definition 2 consequence of the regularity properties of {w˜ λ,k }λ,k with respect to λ > 0. For a fixed λ, consider the partial projection ˜ → C2 defined by U˜ λ v˜ = hv˜ , w˜ λ,. i. Then the U˜ λ : Ls2 (F) ˜ diagonal expression of the resolvent R(ζ) which follows from (8) writes
Z ˜ u˜ , v˜ = R(ζ)

(19)

R+

(U˜ λ u˜ , U˜ λ v) ˜ C2 dλ , λ−ζ

with (U˜ λ u˜ , U˜ λ v) ˜ C2 = ∑k∈{±1} U˜ λ u(k) ˜ U˜ λ v(k). ˜ ˜ First notice that Uλ u(k) ˜ is the Fourier transform F u˜ of u. ˜ Then to proceed to the limit, we make use of ˜ in L1 (R) the continuity of the embedding of Ls2 (F) 2 and the h¨olderian continuity of F (Ls (I)) = H s (I), the classical Sobolev Space, where s > 1/2 and I R+ : lim

C± 3ζ→λ

5.2

0

∈R+

(U˜ λ u˜ , U˜ C2 v) ˜λ dλ λ − λ0 R+ ±iπ(U˜ λ0 u˜ , U˜ λ0 v) ˜ C2

˜ u˜ , v) (R(ζ) ˜ = PV

Z

Finite energy property

and these limits satisfy the following property: if  ˜ then Im R˜ (λ ± i0)u, ˜ u˜ = 0 for some u˜ ∈ Ls2 (F), ˜ (which means that a non-excited R˜ (λ ± i0)u˜ ∈ L02 (F) time-harmonic wave must have a finite energy).

We are going to stick to the second property of definition 2. To prove this property, we shall use an integral ˜ representation of the resolvent R(ζ) of A˜ together with the asymptotic behaviour of its Green’s function. ˜ and ζ ∈ C \ R+ we esFor a given function u˜ ∈ Ls2 (F) tablish from (19) and Fubini’s theorem, the following integral representation

Definition 3 A is called a compact perturbation of 2 (F) ˜ A˜ if D appears as a bounded operator from L−s 2 ˜ ˜ to Ls (F), and D R (λ ± i0) are compact operators in ˜ for every λ > 0. Ls2 (F)

(20)

˜ u(x) R(ζ) ˜ =

(21)

Gζ (x, x0 ) :=

Then we have (see [2])

Z F˜

Gζ (x, x0 )u(x ˜ 0 )dx0 ,

∑

Z

+ k∈{±1} R

w˜ λ,k (x) w˜ λ,k (x0 ) dλ , λ−ζ

hence Theorem 4 Assume that the free operator A˜ satisfies the strong limiting absorption principle of Definition 2. Then every compact perturbation A of A˜ (which is assumed to have no point spectrum) satisfies a similar limiting absorption principle, and the one-sided limits of its resolvent are given by

R (λ ± i0) = R˜ (λ ± i0)(Id + D R˜ (λ ± i0))−1 . Moreover both families w± ˜ λ,k , λ,k = (Id − R (λ ± i0) D )w are generalized spectral bases of A .

(22)

Gζ (x, x0 ) =

1 π

Z +∞ cos(rρ) 0

ρ−ζ

dρ

with r := |x − x0 | and where Gζ (· , ·) is the nothing but the trace on F˜ of the usual Green’s function of the seakeeping problem, noted here G2D ζ . As we search for an asymptotic behaviour of the in˜ ± i0), we have to proceed tegral representation of R(λ to the limit ℑm(ζ) → 0 in (20), on either side of the positive real axis. Denoting by Gλ±i0 the one-sided limits of the Green function, we classically obtain  1  Gλ±i0 (x, x0 ) = ±ie±iλr + ℜe eiλr E1 (iλr) , π

where E1 denotes the exponential integral. This function has the following asymptotic behaviors ln(r) + o(1) , r → 0 , π (24) Gλ±i0 (x, x0 ) ∓ ie±iλr = o(r−1 ) , r → ∞ . (23) Gλ±i0 (x, x0 ) ∓ ie±iλr = −

Now we have to establish the asymptotic behaviour of ˜ ± i0) f at infinity to conclude. u˜± := R(λ ˜ One shows simply using (23)–(24) for any f in Ls2 (F), the integral representation u˜± has when |x| → +∞, x uniformly in the direction kx = |x| , the following asymptotic expansion

Then using the definition of the harmonic liftings H˜ and H the action of D can be described by the following diagram

D : f 7→ Φ˜ 7→ ∂n Φ˜ |Γ 7→ Ψ 7→ Ψ|F˜ ˜ := H˜ A˜ ( f ) and Ψ := H D ( f ) satisfy where Φ ˜ ˜ = 0 in Ω ∆Φ ˜ + αΦ ˜ = f on F˜ ∂y Φ ∆Ψ = 0 in Ω ∂y Ψ + αΨ = 0 on F˜ ˜ on Γ ∂n Ψ = −∂n Φ

−s

˜ ± i0) f = ±2iπw˜ λ,±k (x)U˜ f (λ, ±kx ) + o(|x| ) R(λ x with s > 1/2. It is now easy to see that if for a given function ˜ f ∈ Ls2 (F), ˜ ± i0) f , f i = ±2πkU˜ f k2 0 = ℑmhR(λ then necessarily U˜ f (λ, ±kx ) = 0 which implies obvi˜ ously that R(λ ± i0) f ∈ L02 (F).

˜ |F˜ and Ψ are given by the integral And we know that Φ representations (25)

(26)

˜ |F˜ (x) = − Φ Ψ(X) =

6. The compact perturbation property Since definition 3 involves bounded operators, we cannot compare directly A˜ and A but an invertible bounded and real fonction of those operators as A˜ = ˜ R(−α) and A = R(−α) with α ∈ R+ . So the spectral analysis of the latter will provide those of A˜ and A. Moreover, to stick to the results exposed in §4., we consider the part of A spectrally absolutely continuous still denoted A . ˜ First, it can be easily seen that R(−α) satisfy the definition 2 noticing that
−1
˜ −1 − α) R˜ (ζ) = A˜ − ζ = −ζ−1 Id + ζ−1 R(ζ for all ζ ∈ C\]0, 1/α[. Then one shows the following technical result Proposition 5 The operator D := A − A˜ naturally ˜ to L02 (F), ˜ acts in fact from Ls2 (F) ˜ defined from L02 (F) 1/2 ˜ to Hs+ε (F), with ε > 0 such that 1/2 < s < 3/2 − ε . This proposition leads to the compact perturbation property noticing that the canonical injection 1/2 ˜ ˜ is compact, and noticing that Hs+ε (F) ,→ Ls2 (F) D R˜ (λ ± i0) can be considered as an operator acting 1/2 ˜ 2 (F) ˜ to Hs+ε from L−s (F). We are now going to describe briefly the way to get proposition 5. First of all we can notice that setting u˜ = A˜ f and u = A f is equivalent to ˜ u) ∂y H( ˜ + αu˜ = f on F˜ , ∂y H(u) + αu = f on F˜ .

Z Γ

R

F˜

f (y)G−α (x, y)dy

0 Ψ(X 0 )∂nX 0 G2D −α (X, X )

0 0 −∂nX 0 Ψ(X 0 )G2D −α (X, X )dX

for all X 0 ∈ Ω. Henceforth we have to extend the expression (25) to ˜ This is done using formula (23) and showing Ls2 (F). simply by integrations that G−α (x, x0 ) = O(1/r2 ) . 1/2 ˜ Then we must show that Ψ|F˜ ∈ Hs+ε (F). By virtue 2D of (26), we just have to show that G2D , ∂ x G−α and −α 2 2D ∂x G−α have the same behaviours on J := {(X, X 0 ) ∈ (R2− )2 ; y = 0 , X 0 ∈ Γ} as 1/r2 . References [1] J.T. B EALE, Eigenfunction expansions for objects floating in an open sea, Comm. Pure Appl. Math., 30 (1977), pp. 283–313. [2] C. H AZARD, Analyse Modale de la Propagation des Ondes, Habilitation Thesis, University Paris VI, 2001. [3] C. H AZARD AND M. L ENOIR, Surface water waves, in Scattering, edited by R. Pike and P. Sabatier, Academic Press, 2001. [4] T. I KEBE, Eigenfunction expansions associated with the Schr¨odinger operators and their applications to scattering theory, Arch. Rational Mech. Anal., 5 (1960), pp. 1–34. [5] M. M C I VER, An example of non-uniqueness in the two-dimensional linear water wave problem, J. Fluid Mech., 315 (1996), pp. 257–266.