The Acoustic Scattering of a Layered Elastic Shell Medium

Gang Bao and Lei Zhang

1 School of Mathematical Sciences, Zhejiang University, Hangzhou,Zhejiang 310058, China

2 College of Science, Zhejiang University of Technology, Hangzhou,Zhejiang 310023, China

Abstract.This paper investigates the scattering of a three-dimensional elastic shell scatterer embedded in a layered unbounded structure.The background structure comprises a two-layer lossy media separated by an unbounded rough surface.The shell body is filled with an elastic material,the interior of which is vacuum.Given an incident acoustic point source,our aim is to determine the acoustic and elastic wave fields in the space.This problem is known as a fluid-solid interaction problem(FSIP).In this work,the uniqueness of the FSIP solution is proved by using the integral equation method.Based on the decay properties of Green’s function,the equivalence of boundary integral equation systems and the FSIP is established.Since the system of integral equations containing several integral operators on infinite intervals,we introduce the index theorem of integral operators and analyze the integral operators on infinite intervals to prove the system of integral equations is Fredholm.We then obtain the uniqueness of the system of integral equations and the corresponding existence and uniqueness result of the FSIP.

Key words: Fluid-solid interaction problem,Helmholtz equation,Navier equation,integral operator,index theorem,existence and uniqueness.

1 Introduction

As the shell structure is often used in underwater targets,the acoustic scattering problem of underwater elastic shells is the foundation of target reconstruction in the field of underwater acoustics.A large number of studies dealing with various aspects of the transient response of submerged and/or fluid-filled elastic shell structures with simple geometrical configuration (closed-form analytic solution for spherical or cylindrical shells) have been reported in the literature.The reader is referred to [7,8,14] and the references therein.

This paper considers a time-harmonic acoustic wave incident onto an interface of the two-layered medium from the above.The medium above the surface is supposed to be filled with some homogeneous fluid with a constant mass density,such as air,where the region below is occupied by another homogeneous fluid (such as water)containing a submerged shell with a general shape.An acoustic wave incoming through the interface from the air into the water is incident on the elastic shell,which generates an elastic wave inside the shell body.The process further yields acoustic waves scattered in the interior of the shell,which is the fluid-solid interaction on the shell between acoustic and elastic waves.

For the scattering of fluid-solid interactions,often a linear elastic bounded obstacle surrounded by an inviscid compressible fluid is considered.A Kirchhoff-type formula for transient elastic waves was originally sketched by Love in[10].Helmholtz and Kirchhofftype integral formulas were systematically derived for elastic waves in isotropic and anisotropic solids by Pao and Varatharajulu [13].In the region where the wavelength is of the same order of magnitude as the period and much greater than the depth of the grating,theoretical and numerical results concerning diffraction of ultrasonic waves on periodic liquid–elastic solid interfaces were presented[6].Various systems of boundary integral equations over the interface between the fluid and the solid were derived and analysed by Luke and Martin [11].Hsiao,Kleinman and Roach presented weak formulations of the coupled problem using the field equations in the solid and boundary integral equations in the fluid.They applied some appropriate function spaces and obtained that the weak formulation in terms of a sesquilinear form which is not strongly coercive but for which the Gärding’s inequality holds [9].Hsiao and Yin considered a regularization formulation of the hypersingular operator for the two-dimensional fluid-solid interaction problem [16].

There are also recent results on the related acoustic or elastic scattering of an unbounded rough surfaces.When an unbounded interface is considered,additional difficulties arise.Arens applied the integral equation method to prove the existence of a solution in elastic wave scattering by unbounded rough surfaces [1,2].More recently,homogeneous obstacle acoustic composite scattering problems have been considered.A boundary integral equation is proposed for the associated boundary value problem.Based on some energy estimates,the uniqueness to the solution or the scattering problem is established[3].However,little is known in mathematics or computation for the scattering problems in the presence of an elastic shell structure.In this work,the asymptotic behavior of the Green tensor is analyzed and a novel boundary integral equation is proposed for the associated FSIP.The index theory of systems of multidimensional singular integral equations is further implied to prove the well-posedness of the solution for smooth interfaces.

The paper is organized as follows.In Section 2,we introduce the model of the scattering problem for three-dimensional elastic shells scatterer immersed in a layered unbounded rough surface structure and present some asymptotic analysis for the Green function of the acoustic equations and the Green tensor of the elastic equations.Section 3 is devoted to the uniqueness of the scattering problem.The equivalent integral representations which satisfy the boundary integral equations are proposed in sections 4.Furthermore,we prove that the system of integral equations is Fredholm and has a unique solution.Consequently,the existence and uniqueness of the scattering problem are obtained.

2 Model of the scattering problem

In this paper,we study an acoustic point sourcepincincident on an elastic shell scatterer which is embedded in a two-layered lossy medium separated by an unbounded rough surfaceSin three dimensions.We first present the model of the scattering problem for describing the fluid-solid interactions.The problem geometry is shown in Fig.1.

Without the loss of generalities,letS={x=(x1,x2,x3)∈R3:x3=f(x1,x2)},wheref∈BC2(R2),i.e.,fand its first and second partial derivatives are all bounded continuously differentiable.Hence the surfaceSdivides R3into the upper half space Ω1and the lower half spacewhere

LetDbe a bounded shell embedded ini.e.,D⊂⊂Define Ω2=Denote by Γ±={x∈R3:x3=H±}the plane surface above the unbounded rough surface and below the obstacle,respectively.Define={x ∈R3:f(x1,x2)＜x3＜H+}and={x∈R3:H-＜x3＜f(x1,x2)}.The obstacleD=D1∪Γ2∪Ω3is composed of the bounded domainsD1and Ω3with smooth boundaries Γ1,Γ2∈C2,whereD1is filled with the homogeneous isotropic elastic material with Lamé moduliλandμ,Poisson’s ratioν=and mass densityρ,which satisfy

and Ωl(l=1,2,3) are filled with the homogeneous compressible inviscid fluid with densityρfl ＞0.Obviously,from (2.1),we have

The scattering by an obstacle with elastic shell in a two-layered medium is modeled by

whereω ＞0 is the angular frequency,the pair (p1,p2,p3,u) represents the acousticelastic total field andp1=pinc+psc,u=(u1,u2,u3).Here=(νj1,νj2,νj3),(j=1,2)denote the unit normal vector on the boundary Γjdirected into the exterior ofD1,νSdenotes the unit normal vector on the boundarySpointing from region Ω1to region Ω2,and

Wave numberkl:=ω(ω+iγl)/whereγlis the damping coefficient andclis the wave speed in Ωl,satisfyℜ(kl)＞0,ℑ(kl)＞0,(l=1,2,3).ω2ρs=(ω2ρ+iωγs),hereγs＞0 is the damping coefficient inD1,obviously,we have

We define a symmetric stress tensorσ(u) by

and satisfies

Then,we rewrite the elastic wave equation in the form

whereE(u)=I∈R3×3is the identity matrix,the superscript⊤denotes a transpose,tr(E(u))is the trace of the matrixE(u),and∇uis the gradient tensor given by

The scattering problem can be formulated mathematically as a boundary value problem with a suitable radiation condition,i.e.,(psc,p2) satisfy the radiation conditions

whereνdenote the unit normal vector on the boundary∂Brdirected into the exterior ofBr(O).HereBr(O) is a ball centered at the origin with radiusrand boundary∂Br=denote the hemispheres above and belowS,respectively.Next,we define the fundamental solution to the Helmholtz equation by

which satisfies

In the elastic domainD1,the fundamental Green’s tensor (Kupradze matrix) is defined by

which satisfies G(x,y)=G(y,x),in indicial notation

wherekpandksare the compressional and shear wave numbers defined as

Furthermore,we introduce a third rank Green’s tensor [13] as

Physically,Υ(x,y) represents the stress field atxgenerated by three mutually perpendicular concentrated forces aty.The role of Υ toσ(u) is like that of G tou.To distinguish between Υ and G,we refer to G as the Green’s displacement tensor and Υ as the Green’s stress tensor.

Assume the incident field is given by

wherexs ∈:={x∈R3:x3＞H+}and satisfies

For 0＜α＜1,C0,α(S) denotes a Banach space of real-or complex-valued bounded uniformly Hölder continuous functions decay sufficiently rapidly at infinity defined onS,with the norm

C1,α(S) denotes the set of bounded continuously differentiable real-or complexvalued functions with bounded and uniformly Hölder continuous first derivative and decay sufficiently rapidly at infinity onS,a Banach space with the norm

For surfaces Γ1and Γ2,the Hölder spaces are defined as usual,see,e.g.,[12].All these definitions generalize to 3-vectors by requiring all 3 components to be in the corresponding scalar set.In the vector case,the norms are to be understood as sums of the scalar norms of the components as

Now,we can formulate FSIP as the following boundary value problem.

We prove the solvability results to (FSIP) via the integral equation method and energy analysis.

Lemma 2.1([4]).For each fixed y ∈Ωj,j=1,2,the Green function gj have the asymptotic behavior:

Proceeding as in the proof of the Lemma 2.1 we can derive the following lemma.

Lemma 2.2.For each fixed y ∈the Green displacement tensor G and stresstensorΥhave the asymptotic behavior as follows

3 Uniqueness of a scattering problem

The following theorem presents the uniqueness results for solutions to the scattering problem.

Theorem 3.1.The scattering problem2.1with the radiation conditions(2.7)has at most one solution.

whereν=ν(x) stands for the unit normal vector atx ∈pointing out ofObviously,from Lemma 2.1,(2.7) and (2.14),we can obtain

Lettingr→+∞,we have from (3.1) and (2.7) that

whereνS=νS(x) denotes the unit normal vector atx∈Spointing from region Ω1to region Ω2.Similarly,we have

By combining (3.3),(3.4) and the continuous conditions onSin (2.3),we obtain

For each fixedx∈D1,applying the interface condition on Γ1,Γ2and the integral relationships touin the regionD1,we have

Taking the imaginary part of (3.9),we have

with the aid of＞0,ℑ(ρs)＞0 and (3.10),we have

which implies thatpjanduvanish identically in Ωj(j=1,2,3)andD1,respectively,ifpinc=0.

4 The existence and uniqueness of the equivalent boundary integral equation

We now prove the well-posedness of the scattering problem by the boundary integral equation method.

4.1 The equivalent boundary integral equation

Therefore,the following operators can be defined by applying the above potential operators.

On Γj(j=1,2),we define the single-layer potential operator:C0,α(Γj)→C1,α(Γj)and the double-layer potential operator:C0,α(Γj)→C1,α(Γj) by

Theorem 4.1.If(p1,p2,p3,u)is a solution of Problem2.1,then it has the integral representations

and the boundary integral equations hold

Proof.For each fixedx∈applying the Green’s second theorem top1andg1in the regionwe obtain

whereν(y) stands for the unit normal vector aty ∈pointing towards the exterior ofUsing Lemma 2.1,(3.2),(4.4) and the continuity conditions onS,lettingr→+∞,we have for each fixedx∈Ω1that

Similarly,using the continuity conditions on Γ1and Γ2,we have the integral representations

In (4.1a) and (4.1b),lettingx →S,respectively,from the jump relations and the continuity conditions onS,we have the boundary integral equation as

Note that the boundary integral equations (4.8) involve the unknownIt requires to take the normal derivatives of (4.1a) and (4.1b) onS,which leads to boundary integral equations with hyper-singular kernels.We combine the normal derivatives of (4.1a) and (4.1b) to avoid this issue.Taking the normal derivatives of (4.1a) and (4.1b) onSand adding them together,using the jump relations and the continuity conditions onS,we have the boundary integral equation

In (4.1b),lettingx →Γ1,from the continuity conditions on Γ1,we have the boundary integral equation

In (4.1c),lettingx →Γ2,from the continuity conditions on Γ2,we have the boundary integral equation

Integral formulas for the vectoruat any pointxcan now be obtained from equations (2.6) and (2.12).Taking the scalar product of Eq.(2.6) on the right by G and of Eq.(2.12) on the left byuand subtracting one from the other,we have

which gives

Note that

Hence,(4.13) can be rewritten as

Integrating Eq.(4.14) over the volumeD1and applying the divergence theorem to the left-hand side,we obtain

Combining the continuity conditions on Γ1and Γ2,we have the integral formula foru:

In(4.1d),lettingx→Γ1and Γ2,respectively,from the jump relations on Γ1and Γ2,we have

or in indicial notation forn=1,2,3

This completes the proof.

Proof.Forp1has the integral representation (4.1a),then we have

It follows from (2.15) and (4.20) that

Similarly,forj=2,3,we can obtain

Noting that for anyx∈D1andy ∈Γ1∪Γ2,we havexy.Foruhave the integral representations (4.1d),then we have

With the aid of (2.12) and (4.23),we can obtain

Hence,from(4.17)and(4.24),it is easy to verify that-=·σ(u)on Γ1and=·σ(u) on Γ2.Furthermore,with the help of Lemma 2.1 and (4.19),we derive

Combining (4.25),(4.26) andℑ(k1)＞0,we obtain

whereCis a positive constant independent ofr.Similarly,we can show that

Sincep2satisfies (4.22) and the above radiation condition,applying the Green’s second theorem top2andg2in the region Ω2,using the jump relations of the single-layer and double-layer potentials,respectively,we get the boundary integral equations

Hence,from (4.10) and (4.28),it is easy to verify that

Similarly,we can obtain that

In (4.1a) and (4.1b),lettingx→S,respectively,using the jump relations of the single-layer and double-layer potentials,we get the boundary integral equations

Then,combining the two equalities(4.29)and(4.30),we have the boundary integral equation as

From (4.8) and (4.31),it is easy to verify thatp1|S=p2|S.

Taking the normal derivatives of (4.1a) and (4.1b) onS,using the jump relations of the single-layer and double-layer potentials,we get the boundary integral equations

and

Adding them together,we have the boundary integral equation

From (4.9) and (4.34),it is easy to verify that

This completes the proof.

By using Theorem 4.1 and Theorem 4.2,we conclude that the problem 2.1 is equivalent to the systems (4.1a)–(4.1c) and (4.1d) which satisfy the boundary integral equations (4.8)–(4.11) and (4.18).

4.2 Solvability of the system of integral equations

Theorem 4.3.The system of boundary integral equations(4.2)is Fredholm.

invertible.Due to the weakly singular operators do not contribute for the symbol matrix,thus,we consider the system matrix

where all the remaining elastic terms can be evaluated atω=0.The calculation for the elastic part of the symbol matrix are similar in structure and for details we refer the reader to[15].With the abbreviationδ=for the system(4.35),we obtain the symbol matrix

where I∈R4×4is an identity matrix,and

Hereθj(j=1,2)is the angle a line in thee1-e2plane makes with thee1-axis whenx ∈Γj,respectively.Furthermore,from (4.36) and (4.37),it is obvious that the symbol matrix M is Hermitian satisfying

We note that the conditions on the Lamé constants in (2.1) and (2.2) make a Poisson’s ratio ofν=,1,∞impossible,which implies that detM(x,θ1,θ2)0 and satisfies

By the Corollary 5.1 in [15],the index of system (4.2) is zero.Therefore,the Fredholm theorems apply to that system.The system of integral equations (4.8)–(4.11) and (4.18) is Fredholm when the solid is elastic.

Consequently,from Theorem 4.3,a unique solution to (4.2) exists if the corresponding homogeneous system (4.40) has only the trivial solution.

Theorem 4.4.There exists a unique solution to the system of integral equations(4.2)in the product space X.

Proof.Let

denote a product space.We suppose that

is a nontrivial solution of the system (4.40).Define the following functions by,respectively,

which satisfy

Similarly,it can be seen thatare the smooth solutions of the Eq.(4.46)in their respective domains.Through the above Lemma 2.1 and Lemma 2.2,the proof is similar to that of the radiation conditions in Theorem 4.2,w1andw2satisfy the following radiation condition

Letx→Γ1from Ω2andD,respectively,then it follows from (4.40) and (4.42)that

and

In addition,since

inD,hence,it follows from (4.50) that=0 inD.In particular,=0 on Γ1.

From (4.42),we can also derive

We now respectively subtract (4.50) from (4.49) and (4.52) from (4.51) to obtain

In (4.44),letx→Γ1from Ω2andD1,respectively,then from the 5th to 7th row of matrix (4.40),we have

We now respectively subtract (4.54) from (4.55) and (4.58) from (4.57) to obtain

Similarly,we get

In (4.41) and (4.42),letx→Sfrom Ω1and Ω2,respectively,we have

Hence,it follows from the first row of the matrix (4.40),(4.61) and (4.62) that

Furthermore,we can also take the normal derivative of (4.41) and (4.42) onS,respectively,

Then,it follows from the second row of the matrix (4.40),(4.64) and (4.65) that

then from (4.63) and (4.66),we have

The proof is similar to that of the equations in Theorem 4.1,from (4.67),wj(j=1,2,3) andvsatisfy the following equation,respectively,

By taking the imaginary part of (4.68a) and (4.70) respectively,we have

It follows immediately from combining (4.71) and (4.72) that

which implies thatwj=0,∇wj=0 in Ωjandv=0,∇v=0,∇·v=0 inD1.In particular,we have

Ifωis not a Jones frequency andis not the eigenvalue of the homogeneous Helmholtz equation in their respective domains,then (4.53),(4.59),(4.60),(4.63)and (4.66) imply that

Thus,we have shown that the homogeneous system (4.40) has only the trivial solution.Consequently,from Theorem 4.3,there exists a unique solution to the system of integral equations (4.8)–(4.11) and (4.18).

Finally,from Theorem 4.1,Theorem 4.2 and Theorem 4.4,a unique solution to scattering problem 2.1 exists.

Acknowledgements

The work of GB was supported in part by National Natural Science Foundation of China (No.U21A20425) and a Key Laboratory of Zhejiang Province.The work of LZ is supported by National Natural Science Foundation of China (No.12271482),Zhejiang Provincial Natural Science Foundation of China (Nos.LZ23A010006 and LY23A010004)and the Scientific Research Starting Foundation(No.2022109001429).