THE UNIVERSITY OF READING Department of Mathematics PhD Thesis

# The University of Reading Department of MathematicsPhD Thesis

A thesis submitted for the degree of Doctor of Philosophy
September 2006
###### Abstract

In this thesis we study four problems in the area of scattering of time harmonic acoustic or electromagnetic waves by unbounded rough surfaces/unbounded inhomogeneous layers. Specifically the four problems we study are:
i) A boundary value problem for the Helmholtz equation, in both 2 and 3 dimensions, modelling scattering of time harmonic waves due to a source that lies within a finite distance of the boundary and which decays along the boundary, by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes. In particular, in the 2D case, the boundary value problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting unbounded rough surface, with the magnetic permeability a fixed positive constant in the media, in the transverse electric polarization case;
ii) a boundary value problem for the Helmholtz equation with an impedance boundary condition, in 2 and 3 dimensions, modelling the scattering of time harmonic acoustic waves due to a source that lies within a finite distance of the boundary and which decays along the boundary, by an unbounded rough impedance surface;
iii) a problem of scattering of time harmonic waves by a layer of spatially varying refractive index at the interface between semi-infinite half-spaces of fixed positive refractive index (the waves arising due to a source that lies within a finite distance of the layer and which decays along the layer). In the 2D case this models the scattering of time harmonic electromagnetic waves by an infinite inhomogeneous dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces, with the magnetic permeability a fixed positive constant in the media, in the transverse electric polarization case;
iv) a boundary value problem for the Helmholtz equation with a Dirichlet boundary condition, in 3 dimensions, modelling the scattering of time harmonic acoustic waves due to a point source, by an unbounded, rough, sound soft surface.

We study problems i), ii) and iiii) by variational methods; via analysis of equivalent variational formulations we prove these problems to be well-posed in the following cases: For i) we show that the problem is well-posed for arbitrary rough surfaces that are a finite perturbation of an infinite plane, in the case that the frequency is small or when the medium in the layer has some energy absorption; and when the rough surface is such that the resulting domain has the property that if is in the domain then so to is every point above , we show the problem to be well-posed for arbitrary large frequency with certain restrictions on the rate of change of the refractive index; for ii) we show that the problem is well-posed for arbitrary rough Lipschitz surfaces that are a finite perturbation of an infinite plane, in the case that the frequency is small; and when the rough surface is the graph of a bounded Lipschitz function, we show the problem to be well-posed for arbitrary frequency; for iii) we establish that the problem is well-posed under certain restrictions on the variation of the index of refraction.

We study problem iv) via a Brakhage-Werner type integral equation formulation, based on an ansatz for the solution as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green’s function. We establish, in the case that the rough surface is the graph of a bounded Lipschitz function, that the problem is well-posed for arbitrary frequency.

An attractive feature of our results is that the bounds we derive, on the inf-sup constants of the sesquilinear forms in problems i), ii) and iii), and on the inverse operator associated with the single- and double-layer potentials in problem iv), are explicit in terms of the index of refraction, the geometry of the scatterer and the other parameters of the respective problems.

### Chapter 1 Introduction

#### 1.1 Preamble

Consider, if you will, the following problem: An aeroplane flying above the surface of the earth, generates some noise. This noise travels through the air striking the rough surface of the earth beneath it. The noise bounces off or scatters from the surface. Given that we know the exact nature of the noise produced by the aeroplane, and given that we know the exact shape of the earth’s surface beneath it, can we predict the resultant propagation of noise as it strikes the earth’s surface and scatters?

The above problem is a typical example of what are known as rough surface scattering problems. Rough surface scattering problems arise frequently in the natural world and the study of these problems has been borne out of research in many diverse areas of science. The above example shows their importance to the science of sound propagation and noise control; on a much smaller scale, in the field of nano-technology, they are relevant in the study of the scattering of light from the surface of materials; and in the technology of solar heating, their understanding is important for the correct choice of solar paneling; in addition these problems crop up in medical imaging and seismic exploration.

It is the overall aim then, of the mathematical and engineering community, to resolve these problems. This thesis is intended as a contribution to this subject. We are concerned primarily with the initial, mathematical and theoretical questions that should – to a mathematician at least – be answered, in this field, prior to the implementation of numerical and computational techniques that will simulate the process of rough surface scattering, and ultimately give answers to the problems stated above.

Thus, in what follows, we are concerned with the mathematical aspects of rough surface scattering problems. In particular we are interested in the correct mathematical formulation of these problems and we intend to analyze under which conditions they are well-posed. Specifically we study four rough surface scattering problems. Three of these we study by variational methods – these are acoustic scattering by an impedance surface; electromagnetic scattering by inhomogeneous layers above a perfectly conducting rough surface (the Transverse electric polarization case); and the transmission problem – and one by integral equation methods: acoustic scattering by a sound soft surface. We will shortly take a more precise look at what these problems are and look at the mathematical models that govern acoustic and electromagnetic propagation.

We wish to end this first section by introducing some nomenclature and notation used throughout and by setting the scene of our scattering problems. In accordance with the terminology of the engineering literature, we use the phrase rough surface to denote a surface which is a (usually non-local) perturbation of an infinite plane surface, such that the whole surface lies within a finite distance of the original plane.

Let denote a point in , , and let so that . Further, for let and . We will denote the region of space in which the acoustic or electromagnetic waves propagate, i.e. the air above the earth’s surface in the aeroplane problem we described above, as . Thus , and will be assumed to be a connected open set or domain. Moreover we’ll assume there exist constants such that

 Uf+⊂D⊂Uf−.

We let denote the boundary of , i.e. the rough surface. The unit outward normal to will be denoted . Finally, we define .

This definition of is rather complicated – see the picture below. It may help the reader, in coming to terms with this abstract description to keep in mind the following case, included in our definition of . This is the case where the scattering surface is the graph of some bounded continuous function. For example in the 3D case, we have that for a bounded and continuous function ,

 Γ:={x=(x1,x2,x3)∈R3:x3=f(x1,x2)}, (1.1)

after which the domain is given by

 D={x=(x1,x2,x3)∈R3:x3>f(x1,x2)}.

Of course, in general one wishes to consider rough surfaces that are not simply the graphs of functions, but of more complex geometry that one encounters in reality. It is a major point of this thesis that we establish well-posedness results for wave scattering by rough surfaces that are not the graphs of functions, under certain other restrictions (low frequency of the waves, for example). Nevertheless a great deal of the well-posedness results we establish in this thesis are restricted to this setting, where the rough surface is the graph of a bounded continuous function.

We aim in the next sections to turn our attention to the mathematical modelling of these problems. We’ll begin by looking at acoustics; then we’ll take a look at electromagnetics.

#### 1.2 The mathematical description of the scattering problems

##### 1.2.1 Acoustics

The wave equation is the classical model that describes acoustic propagation. Let denote the perturbation of pressure at a point in and at a time . It is this that we seek to find. In other words, for example, it represents the resultant sound distribution that we desired to predict in the aeroplane example earlier. Let denote the source of acoustic disturbance, i.e. the noise produced by the aeroplane, which we suppose we know or are given. Then the inhomogeneous wave equation relates the two via,

 1c2∂2U∂t2−ΔU=−G, in D×R. (1.2)

Here is the speed of sound in the medium. We may wish to assume that is a constant, which is appropriate if the region is occupied by one medium, such as air which is at rest; but also we will consider the case when the medium in varies throughout , in which case depends on position. We note also at this point that the density perturbation of the air, also satisfies (1.2), though with a different function , and, provided the wave motion is initially irrotational, then, the velocity, , of the air is the gradient of a scalar field the velocity potential, which also satisfies the wave equation, with a yet different function . Further, the relationships between these three quantities are given by

 v=∇Ψ, U=−ρ0Ψt, U=c2ρ, (1.3)

where denotes the density of the unperturbed state. We will suppose that our waves are time harmonic. This means we will assume that for a function and then look for solutions to the wave equation in the form for some function . Here denotes the angular frequency of the waves. On making this assumption one sees that equation (1.2) reduces to

 Δu+k2u=g, in D, (1.4)

where , is the wavenumber. Equation (1.4) is known as the inhomogeneous Helmholtz equation. We should mention at this stage that a slightly different model for acoustical scattering that takes into account the effect of dampening in the region , and which takes as it’s starting point the dissipative wave equation, leads once again, with similar manipulations to those above, to the inhomogeneous Helmholtz equation (1.4), but this time with being a complex valued function (see [32] pages 66-67). Thus the case when is complex valued in (1.4) is also of interest in acoustics.

Thus given and given our aim will be to find satisfying (1.4). That (1.4) be satisfied in a classical sense, requires that we should find belonging to the space Alternatively we may, to get a handle on solving the problem, seek to find a solution satisfying (1.4) in a weaker sense, for example a distributional sense, in which case it would be more appropriate to look for a solution in perhaps. When we precisely pose our problems, later on, it will be important to know in what exact function space to look for . This will depend on the function space setting of . For the meantime though we wish to brush over such issues and set up the basic problem. However we should mention here something about the nature of . In our motivating problem of the aeroplane it is appropriate to view as a function of compact support. In fact, in our variational formulations of the problem, will be given more generally as a function in although its support will lie at a finite distance from the boundary . This means that the support of will lie in for some .

Another source of acoustic excitation that we will consider in this thesis, is that due to a point source. Here – a delta function situated at . The interest in this sort of excitation again stems from the wish to study wave sources of compact support: any such source can be represented as superpositions of point sources located in the compact support.

Finally we should mention one important type of acoustic incidence, not covered in the work of this thesis. This is plane wave incidence. Generally the analysis that we apply in this thesis requires that sources should decay along the boundary; as such none of our results apply to scattering of plane waves.

###### Boundary conditions.

Finding a unique solution to equation (1.4) will not be possible without requiring the solution to satisfy an appropriate boundary condition. We will look at two such boundary conditions: the Dirichlet boundary condition and the impedance boundary condition.

###### Dirichlet boundary condition.

Here we require that on the boundary . In this case is said to be a sound soft surface. Physically it corresponds to there being no pressure on the surface. It is appropriate to assume this when there is a huge jump in pressure across a surface. An example of this arises in underwater acoustics: If we imagine a submarine beneath the sea emitting sound and if we wish to know just how this sound propagates, then, the problem domain would be the region occupied by the sea and the rough surface would be the surface of the sea. Given the large drop in pressure as one moves from the sea to the air above it, then, it would be appropriate in this case to assume that the pressure is zero on this rough surface .

###### Impedance boundary condition.

In order to motivate this boundary condition let us just note that the relation between pressure and velocity potential given in (1.3), can be simplified, on making our time harmonic assumptions that

 U=Re(u(x)e−iωt) and Ψ=Re(ψ(x)e−iωt)

for . The new relation between and is then

 u=iωρ0ψ. (1.5)

Now, the classical Neumann boundary condition, that

 ∂ψ∂ν=0

on the boundary , assumes that the component of fluid velocity normal to the surface vanishes. This makes sense for a rigid surface. But for more general surfaces, the normal velocity is non-zero and the quantity , defined by

 Zs=u∂ψ/∂ν (1.6)

is finite on the boundary. is called the surface impedance (e.g. [58]). In general depends on the variation of the acoustic field throughout the medium of propagation. Often however depends only on the properties of the boundary surface (and on the angular frequency , but we will assume that this is constant in our study of the impedance problem): specifically, for a given stretch of surface, for example a concrete road surface, the ratio is constant; and then for a different stretch of surface, for example that covered by a field, it assumes yet a different constant value. In this thesis we will always assume that is independent of the distribution of the acoustic field, in which case we say that the boundary is locally reacting.

Thus defining as

 β=ρ0cZs (1.7)

we see that, from the above discussion, is a function of the boundary , and we will suppose that . Using (1.7) and (1.5), equation (1.6) can be rewritten as,

 ∂ψ∂ν=ikβψ or ∂u∂ν=ikβu, on Γ,

and we have arrived at the impedance boundary condition, also known as the Robin boundary condition or the third boundary condition.

It can be shown – see for example [28] page 24 – that if the ground/boundary is not to be a source of energy then a necessary condition on is that

 Reβ≥0.

In chapter 3 we’ll make – and to some extent justify – some extra assumptions on .

The solution to (1.4) will still not be unique even when we impose one of the boundary conditions. A radiation condition is also required, many authors referring to this as an extra boundary condition at infinity. The role of the radiation condition is to pick out the physically realistic solution.

In this thesis we make use of a radiation condition called the upward propagating radiation condition, (UPRC). To state this we introduce the fundamental solutions to the Helmholtz equation (1.4) in the case when the wavenumber is a positive constant i.e. . This is , given by

 Φ(x,y)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩i4H(1)0(k+|x−y|),n=2,exp(ik+|x−y|)4π|x−y|,n=3,

for , , where is the Hankel function of the first kind of order zero. is a solution to the Helmholtz equation (1.4) with in the special case when and , a point source located at . The (UPRC) then states that

 u(x)=2∫ΓH∂Φ(x,y)∂xnu(y)ds(y),x∈UH, (1.8)

for all such that the support of is contained in .

The (UPRC) was proposed in [14]. In the case that the wavenumber has imaginary part, one can derive this representation for the solution of the Helmholtz equation in , under mild assumptions on the growth of the solution at infinity: see [13].

In the case that we can rewrite (1.8) in terms of the Fourier transform of . For , which we identify with , we denote by the Fourier transform of which we define by

 Fϕ(ξ)=(2π)−(n−1)/2∫Rn−1exp(−i~x⋅ξ)ϕ(~x)d~x,ξ∈Rn−1. (1.9)

Our choice of normalization of the Fourier transform ensures that is a unitary operator on , so that, for ,

 ∫Rn−1ϕ¯ψd~x=∫Rn−1^ϕ¯^ψdξ. (1.10)

If then (see [16, 7] in the case ), (1.8) can be rewritten as

 u(x)=1(2π)(n−1)/2∫Rn−1exp(i[(xn−H)√k2+−ξ2+~x⋅ξ])^FH(ξ)dξ,x∈UH.

In this equation , when .

Equation (LABEL:uprcstar) is a representation for , in the upper half-plane , as a superposition of upward propagating homogeneous and inhomogeneous plane waves. A requirement that (LABEL:uprcstar) holds is commonly used (e.g. [34]) as a formal radiation condition in the physics and engineering literature on rough surface scattering. The meaning of (LABEL:uprcstar) is clear when so that ; indeed the integral (LABEL:uprcstar) exists in the Lebesgue sense for all . Recently Arens and Hohage [7] have explained, in the case , in what precise sense (LABEL:uprcstar) can be understood when so that must be interpreted as a tempered distribution. Arens and Hohage also show the equivalence of this radiation condition with another known as the Pole Condition.

In summary, our acoustic problems will be to look for a solution to the Helmholtz equation (1.4), satisfying the radiation condition (LABEL:uprcstar), and satisfying one of the boundary conditions: if it is the Dirichlet boundary condition, then we will refer to this problem as the Dirichlet problem for the Helmholtz equation, or simply the Dirichlet problem; in the case where we use an impedance boundary condition, we will refer to the problem as the impedance problem for the Helmholtz equation or simply the impedance problem.

##### 1.2.2 Electromagnetics

In classical electromagnetics, Maxwell’s equations relate the electric field intensity , the magnetic field intensity , the electric displacement and the magnetic induction , to the cause of electromagnetic excitation, namely the charge density function and the current density function . Maxwell’s equations are (see [65] pages 1-9):

 ∂B∂t+∇×E=0 in D, (1.12)
 ∇.D=ρ in D, (1.13)
 ∂D∂t−∇×H=−J% in D, (1.14)
 ∇.B=0 in D. (1.15)

Note that equations (1.13) and (1.14) can be combined to give

 ∇.J+∂ρ∂t=0. (1.16)

Making the assumption that the current density and charge density are time harmonic, i.e. that

 J(x,t)=Re(exp(−iωt)^J(x))

and that

 ρ(x,t)=Re(exp(−iωt)^ρ(x))

for known and , and that also

 E(x,t)=Re(exp(−iωt)^E(x))
 D(x,t)=Re(exp(−iωt)^D(x))
 H(x,t)=Re(exp(−iωt)^H(x))
 B(x,t)=Re(exp(−iωt)^B(x))

for unknown functions , , , and , we obtain the time harmonic Maxwell’s equations:

 −iω^B+∇×^E=0 in D, (1.17)
 ∇.^D=^ρ in D, (1.18)
 −iω^D−∇×^H=−^J % in D, (1.19)
 ∇.^B=0 in D. (1.20)

We now reduce these 4 equations down to 2, eliminating the quantities and , by supposing there hold two constitutive laws that relate and to and , respectively. These laws depend on the matter in the domain occupied by the electromagnetic field. In this thesis we suppose that the material occupying is inhomogeneous, that is, it is a composition of different materials (e.g copper, air etc.); that the material is isotropic, in other words the material properties do not depend on the direction of the field; and also we assume the material is linear. It then follows that the constitutive equations are ([65] page 5)

 ^D=ϵ^E (1.21)

and

 ^B=μ^H, (1.22)

where is positive and bounded and is known as the electric permittivity; whilst is the magnetic permeability, and is assumed to be a constant. One further constitutive equation is that

 ^J=σ^E+^Ja, in D (1.23)

where is non-negative and is called the conductivity and the vector function is the applied current density. Regions of where is strictly positive are termed conducting. Where the material in is termed dielectric.

Now using the constitutive relations (1.21), (1.22), (1.23) and also using equation (1.16) in its time harmonic form,

 ∇.^J−iω^ρ=0 in D,

in the time harmonic Maxwell’s equations (1.17), (1.18), (1.19), (1.20), we derive that

 −iωμ^H+∇×^E=0 in D, (1.24)
 ∇.(ϵ^E)=1iω∇.(σ^E+^Ja) in D, (1.25)
 −iωϵ^E+σ^E−∇×^H=−^Ja in D, (1.26)
 ∇.(μ^H)=0 in D. (1.27)

We remark that by supposing the constitutive relations to hold, equations (1.25) and (1.27) are now redundant; they can be derived by taking the divergence of (1.26) and (1.24) respectively. Moreover we can eliminate the variable by substituting (1.24) into (1.26), to arrive at one equation for :

 ∇×(∇×^E)−ω2μϵ[1+iσωϵ]^E=iωμ^Ja in D, (1.28)

recalling that is assumed to be constant. Letting , and letting

 k2=ω2μϵ[1+iσωϵ] (1.29)

we see that (1.28) becomes

 ∇×(∇×^E)−k2^E=G. (1.30)

In this thesis we assume that the problem is two dimensional: precisely we suppose that the electric permittivity , the magnetic permeability and the conductivity are invariant in the direction. Moreover we only study the Transverse Electric (T.E.) case. In the T.E. case we seek the electric field intensity in the form where is supposed to be independent of the variable and also we assume that . On making these assumptions we see that (1.30) becomes

 ΔE+k2E=g in D. (1.31)

It is the solution to this equation that we will seek to find when we are given . We see that we have once more arrived at the inhomogeneous Helmholtz equation. As in the last section it must be supplemented by boundary and radiation conditions.

We will look at two, two-dimensional problems involving the scattering of electromagnetic waves. The first involves scattering by a perfectly conducting rough surface. In this case the appropriate boundary condition is that

 ν.^E=0 on Γ.

Since we are assuming that , this means we should require that on . In addition we then impose the radiation condition (LABEL:uprcstar); for this it’s necessary to assume that outside a neighbourhood of the boundary the quantity in (1.31) takes on a constant positive value . We will call this problem, that of scattering by an unbounded rough inhomogeneous layer.

The second electromagnetic problem we wish to study will be known as the transmission problem. Here the domain of electromagnetic propagation is assumed to be the whole of . As such no boundary conditions are required, but rather we impose the radiation condition both in the upward and downward directions, assuming that the function in (1.31) assumes positive constant values and above and below a strip of finite height within which may vary. We will return to to this idea and elaborate on it when we come to study the transmission problem in chapter 4.

#### 1.3 Hadamard’s criterion and numerical implementation

It is our general aim to solve the problems that we posed in the last section. It is instructive to consider a little why we cannot find explicit solutions to these problems, via mathematical techniques. There are essentially two answers to this question. One is that, put simply, these problems are too difficult. Another involves the complex nature of the problem: for if, in the earlier aeroplane example, we consider the scattering surface to possess a complicated, that is realistic, geometry, and if the source of acoustic waves is similarly of a complex and realistic nature, then one can hardly expect that the scattered acoustic field will have such a simple form as to be able to be described by an explicit mathematical function. Indeed, such complicated scattered fields are best described by pictures generated on a computer.

Thus to solve such problems numerical and computational techniques are essential. On the theoretical side we should ensure that the mathematical problems we set are well-posed, in that they satisfy Hadamard’s criterion. This states that for a given mathematical model:
1)there should exist a solution;
2)the solution should be unique;
3)the solution should depend continuously on the data. For example any solution to equation (1.4) should satisfy an inequality

 ∥u∥X≤C∥g∥Y,

where is a constant and and are normed spaces to which and respectively belong.

In this thesis we are primarily concerned with showing that the problems we state are well-posed and not with the approximate solution to these problems using numerical techniques. However, our approach to our problems is geared toward the ultimate goal of numerical computation of the solution: specifically, the variational formulations that we derive from our problems in chapters 2, 3 and 4 should be suitable for finite element implementation; similarly the boundary integral equation that we derive from our problem in chapter 5 should be suitable for solution via boundary element methods. In particular the explicit bounds we establish, on the sesquilinear form, in chapters 2, 3 and 4, and on the inverse operator associated with the double- and single-layer potentials in chapter 5, should prove helpful in the analysis of the numerical implementation.

#### 1.4 Literature review: Overview

We conclude this introduction with a broad review of the literature.

The problem of rough surface scattering has long been studied and there have been many contributors to the subject. Principally research has focused on the use of numerical methods to solve these problems. The review of Warnick and Chew [77], summarises numerical strategies, implemented over the past 30 years or so, that seek to simulate the scattering of electromagnetic (and also acoustic) waves by rough surfaces. Warnick and Chew roughly group these numerical strategies into three categories: differential equation methods, boundary integral equation methods and numerical methods based on analytical scattering approximations. A critical survey of scattering approximations is carried out in the review of Elfouhaily and Guerin [40].

In the review [71], Saillard and Sentenac are interested in formulating rough surface scattering problems from a statistical point of view. Here, the rough surface is not a known quantity in the problem, but rather, one only has information on certain statistical properties of the surface, so that the shape of the rough surface is described by a random function of space coordinates and time. The problem is then to determine the statistical properties of the scattered field, such as it’s mean value and mean intensity, as functions of the statistical properties of the surface. Obviously such a problem is of interest, since in reality, one often will not know the precise shape of the rough surface. In this thesis however, we always assume that the rough surface is known. Saillard and Sentenac then proceed to describe approximate methods for solving these problems numerically. They point out, however, that few authors have undertaken a rigourous mathematical study of the problem.

In [61] Ogilvy reviews research in this area, again with an emphasis on random rough surfaces and on numerical techniques. See also the books by Voronovich [76], Petit [63] and Wilcox [78] and the review of DeSanto [34].

Finally we should make mention of the closely related field of scattering by bounded obstacles. A very complete theory of this class of problems has been developed, especially by use of boundary integral equation methods, see for example [32].

We will return to the literature review, with a much closer scrutiny of the papers that are related to the work in this thesis, as we tackle the various problems in chapters 2, 3, 4 and 5.

## Part I Variational Methods

In the next three chapters we apply variational methods to three of our scattering problems. An excellent introduction to the theory of variational methods can be found in Lawrence C. Evans’s book ‘Partial Differential Equations’ [43] chapters 5 and 6.

There are two main theorems that we will require:

###### Theorem 1.1.

Lax-Milgram. [e.g. [65] Lemma 2.21.] Let be a Hilbert space, with norm and inner product given by , respectively. Suppose that is a bounded sesquilinear form such that for some it holds that

 |b(u,u)|≥α∥u∥2,u∈H.

Then for each there exists a unique such that

 b(u,v)=G(v)v∈H,

and

 ∥u∥≤α−1∥G∥H∗,

where denotes the norm of .

###### Theorem 1.2.

Generalized Lax-Milgram Theorem. [e.g. [47] Theorem 2.15.] Let be a Hilbert space, with norm and inner product given by , respectively. Suppose that is a bounded sesquilinear form such that for some the inf-sup condition holds:

 α:=inf0≠u∈Hsup0≠v∈H|b(u,v)|∥u∥∥v∥>0; (1.32)

and the transposed inf-sup condition holds:

 sup0≠u∈H|b(u,v)|∥u∥>0. (1.33)

Then for each there exists a unique such that

 b(u,v)=G(v)v∈H,

and

 ∥u∥≤α−1∥G∥H∗.

### Chapter 2 Scattering by unbounded, rough, inhomogeneous layers

#### 2.1 Literature review

In this chapter we study, via variational methods, a boundary value problem for the Helmholtz equation modelling scattering of time harmonic waves by a layer of spatially-varying refractive index above a rough surface on which the field vanishes (we called this the problem of scattering by unbounded, rough, inhomogeneous layers in chapter 1 – see the electromagnetics section). We recall from chapter 1 that in the 2D case this problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting rough surface in the transverse electric polarization case. Moreover it is a model, in 2 and 3 dimensions, of time harmonic acoustic scattering by a rough surface in a medium in which the wavespeed varies with position or in which there is dissipation.

We commence with a thorough survey of the literature on this problem. In fact this problem, in which we study the Helmholtz equation (1.4) with a function of position, seems to have received little attention with the exception of [20]. Mainly this problem has been studied in the special case when is constant throughout the region , (in which case the problem reduces to what we have called the Dirichlet problem for the Helmholtz equation in chapter 1.) Thus, let us begin this survey by looking at contributions to this problem when is assumed constant.

The pioneering paper on this subject seem to be the uniqueness proof of Rellich [67]. Here Rellich assumed that the rough surface roughly resembled a paraboloid. In [60] Odeh proves uniqueness of solution in the case that the rough surface is smooth and is either a cone or approaches a flat boundary at infinity. Willers proves the existence of a unique solution to this problem, in [79], making the assumption that the boundary is and is flat outside a compact set.

In another, somewhat related body of work existence of solution to the Dirichlet problem is established by the limiting absorption method, via a priori estimates in weighted Sobolev spaces (see Eidus and Vinnik [39], Vogelsang [75], Minskii [57] and the references therein.) The results obtained are still however limited in that one must assume that the rough surface approaches a flat boundary sufficiently rapidly at infinity and/or that the sign of is constant on outside a large sphere, where denotes the unit normal at .

The most recent, and indeed most complete results in this field, have been developed by Chandler-Wilde and his collaborators. Principally Chandler-Wilde et al have concentrated on employing boundary integral equation (BIE) techniques to settle the question of unique existence of solution to these problems. However, the loss of compactness of the associated boundary integral operators in the case when the boundary is infinite, meant that the theory of boundary integral equations for scattering by bounded obstacles did not translate easily to the problem of rough surface scattering (c.f. the literature review in chapter 5). As such generalizations of part of the Riesz theory of compact operators have been developed - see the work of Arens, Chandler-Wilde and Haseloh, [4], [5] - requiring only that the associated boundary integral operators be locally compact, and ensuring that existence of solution to the boundary integral equation follows from uniqueness.

In [11] Chandler-Wilde and Ross prove a uniqueness theorem for the Dirichlet problem, in an arbitrary domain with the assumption that . The same authors in [12], then derive some existence results in 2D for mildly rough surfaces, using BIE techniques.

In [16], Chandler-Wilde and Zhang show uniqueness to the Dirichlet problem in a non-locally perturbed half-plane with piecewise Lyapunov boundary. This time the wavenumber is assumed real and the problem is formulated with a radiation condition. Moreover an integral equation formulation is proposed and existence of solution is established, for mildly rough surfaces, in 2 dimensions, by using the results of [12]. Finally, in [14], Chandler-Wilde, Ross and Zhang show existence of solution for domains with Lyapunov boundary, in 2D, but this time with no limit on the surface slope or amplitude. They do this by employing novel solvability results on integral equations, contained in the paper. See also [83], where similar results are obtained but with an alternative integral equation formulation.

Recently Chandler-Wilde, Heinemeyer and Potthast looked at the Dirichlet problem in 3 dimensions, again by an integral equation approach, and were able in [22], [23], to establish that the problem was well-posed in the case that the boundary is the graph of a Lyapunov function. We will in fact extend these results, to the case when the boundary is the graph of a Lipschitz function in chapter 5. Indeed, see the literature review in chapter 5 for further details on the use of BIE techniques.

In other recent work [25] Chandler-Wilde and Monk adopted a different approach to this problem. Using variational methods they were able to establish the well-posedness of the Dirichlet problem, in both 2 and 3 dimensions, for much more general boundaries: specifically, for small wavenumber they showed the problem to be well-posed for totally general boundaries, that were not the graphs of functions and requiring no regularity; and for arbitrary wavenumber they established the same for those domains having the property that if then so too is every point above . To prove their results they first reformulated their boundary value problem as an equivalent variational problem on a strip; they then analyzed the variational problem and made use of the Lax-Milgram and generalised Lax-Milgram theorem of Babuška, to prove well-posedness. A key ingredient in their proofs was the derivation of an a priori bound on the solution.

The purpose then of this first chapter is to extend these results to the problem of scattering by rough inhomogeneous layers; indeed we will make use of a lot of the results contained in [25] and mimic their methods throughout.

We should mention some papers, that are related, in terms of the methods they employ, to the paper of Chandler-Wilde and Monk. These are [49] by Kirsch, and [41] by Elschner and Yamamoto, who study the Dirichlet problem; and the papers [8] of Bonnet-Bendhia and Starling and [72] of Szemberg-Strycharz who study the diffraction grating or transmission problem as we’ve called it here. In all of these papers a variational approach is used. All of the authors begin by reducing their problems to a variational problem on a strip; however the assumption made in all of these papers, that the scattering surface/diffraction grating is periodic and that the source is quasi-periodic, leads to a variational problem over a bounded region, so that compact embedding arguments can be applied and the sesquilinear form that arises satisfies a Gårding inequality which simplifies the mathematical arguments. However we should say that the approach adopted in [25] – and the one that we adopt here – is very similar to the one adopted in [49], [41], [8] and [72]; in particular the use of the Dirichlet to Neumann map (see (2.3) and (2.10) below) and the exploitation of its properties was done in [49], [41], [8] and [72].

An attractive feature of our results and indeed of those in [25] is the explicit bounds we obtain on the solution in terms of the data , which exhibit explicitly dependence of constants on the wave number and on the geometry of the domain. Our methods of argument to obtain these bounds are inspired in part by the work of Melenk [54], Cummings and Feng [33], Feng and Sheen [45] and Chandler-Wilde and Monk [25].

Finally we should also discuss the paper [20] of Chandler-Wilde and Zhang; the authors here deal with the same problem as we deal with here, (i.e. the wavenumber varies throughout the domain) and employ an integral equation approach to establish well-posedness in 2D when the surface is flat so that the domain is a half-space. This paper would appear to present the best results on this problem to date. We should point out in what ways our results are an improvement on these:
1) Our results work in both 2 and 3 dimensions;
2) our rough surfaces are much more general: Specifically, when the maximal value of is small or has strictly positive imaginary part we show the problem to be well-posed for totally general boundaries, that are constrained to lie in a strip and which require no regularity; and for arbitrary subject to assumption 1 (see below) we establish well-posedness for those domains having the property that if then so too is every point above ;
3) the assumption (see assumption 1 below) that we make in order to prove theorem 2.3 is slightly more general than the assumption 2.4 that is used in [20].

Finally we should mention some other spin-off papers of [25]: these are [27] in which the same authors apply similar methods to scattering by bounded obstacles; and [30] in which Claeys and Haddar use the same approach to tackle the problem of scattering from infinite rough tubular surfaces.

Note that the results contained in the section ‘-ellipticity of the sesqulinear form’ were presented at the Waves 2005 Conference.

#### 2.2 The boundary value problem and variational formulation

In this section we introduce the boundary value problem and its equivalent variational formulation that will be analyzed in later sections. For () let so that . For let and . Let be a connected open set such that for some constants it holds that

 Uf+⊂D⊂Uf−, (2.1)

and let denote the boundary of . The variational problem will be posed on the open set , for some , and we denote the unit outward normal to by .

Let denote the standard Sobolev space, the completion of in the norm defined by

 ∥u∥H1(D)={∫D(|∇u|2+|u|2)dx}1/2.

The main function space in which we set our problem will be the Hilbert space , defined, for , by , on which we will impose a wave number dependent scalar product and norm, .

Recalling the basic model from chapter 1, we will make the assumption that the variation in is confined to a neighbourhood of the boundary. The following then is our exact formulation:

The Boundary Value Problem. Given , and such that for some , it holds that the support of lies in , and that , , for some , find such that for every ,

 Δu+k2u=g in D

in a distributional sense, and the radiation condition (LABEL:uprcstar) holds, with .

###### Remark 2.1.

We note that, as one would hope, the solutions of the above problem do not depend on the choice of . Precisely, if is a solution to the above problem for one value of for which and , then is a solution for all with this property. To see that this is true is a matter of showing that, if (LABEL:uprcstar) holds for one with and , then (LABEL:uprcstar) holds for all with this property. It is shown in Lemma 2.1 below that if (LABEL:uprcstar) holds, with , for some , then it holds for all larger values of . One way to show that (LABEL:uprcstar) holds also for every smaller value of , say, for which and and , , is to consider the function

 v(x) := u(x)− 1(2π)(n−1)/2∫Rn−1exp(i[(xn−~H)√k2+−ξ2+~x⋅ξ])^F~H(ξ)dξ,x∈U~H,

with , and show that is identically zero. To see this we note that, by Lemma 2.1, satisfies the above boundary value problem with and . That then follows from Theorem 2.3 below.

We should give some motivation for looking for a solution such that for every . In the special case when the boundary is flat, so that is a half-space, say, is constant and is smooth and has compact support we can explicitly construct the solution. Suppose . Then if denotes a point source located at , with , then a solution to the problem, find , such that

 Δu+k2u=δy in D,

and , is given by

 u(x)=G(x,y):=Φ(x,y)−Φ(x,y′),

where is the reflection of in the boundary , and is the Green’s function for . Note that

 |u(x)|≤C(1+x3)(1+y3)|x−y|2 (2.2)

for some , (see for example chapter 5, (5.35)).

Moreover, a solution to the problem, find such that

 Δu+k2u=g in D,

and

 u=0 on Γ,

is, for compactly supported and smooth , given by

 u(x)=∫DG(x,y)g(y)dy,x∈D.

It follows from the bound (2.2) that for every , where (one can deduce this by using the techniques of section 5.4 of chapter 5, for example). Further, by an application of Green’s theorem it follows also that , for every . This motivates, that in the general case, we seek a solution such that , for all .

We now derive a variational formulation of the boundary value problem above. To derive this alternative formulation we require a preliminary lemma. In this lemma and subsequently throughout the thesis, we use standard fractional Sobolev space notation, except that we adopt a wave number dependent norm, equivalent to the usual norm, and reducing to the usual norm if the unit of length measurement is chosen so that . Thus, identifying with , , for , denotes the completion of in the norm defined by

 ∥ϕ∥Hs(ΓH)=(∫Rn−1(k2++ξ2)s|Fϕ(ξ)|2dξ)1/2.

We recall [2] that, for all , there exist continuous embeddings and (the trace operators) such that coincides with the restriction of to when is . In the case when , when may not be the boundary of (if part of coincides with ) we understand this trace by first extending by zero to . We recall also that, if , , and , then , where , , , . Conversely, if and , , then . We introduce the operator , which will prove to be a Dirichlet to Neumann map on , (see (2.10) below), defined by

 T:=F−1MzF, (2.3)

where is the operation of multiplying by

 z(ξ):=⎧⎪ ⎪⎨⎪ ⎪⎩−i√k2+−ξ2if |ξ|≤k+,√ξ2−k2+for |ξ|>k+.

We shall prove shortly in Lemma 2.2 that and is bounded. We now state lemma 2.2 from [25]. For completeness we include the proof.

###### Lemma 2.1.

If (LABEL:uprcstar) holds, with , then , for every ,

 Δu+k2+u=0 in UH,

, and

 ∫ΓH¯vTγ+uds+k2+∫UHu¯vdx−∫UH∇u⋅∇¯vdx=0,v∈C∞0(D). (2.4)

Further, the restrictions of and to are in , for all , and

 ∫Γa[∣∣∣∂u∂xn∣∣∣2−|∇~xu|2+k2+|u|2]ds≤2k+Im∫Γa¯u∂u∂xnds. (2.5)

Moreover, for all , where denotes the restriction of to , (LABEL:uprcstar) holds with replaced by .

###### Proof.

If then, as a function of , for every and . It follows that (LABEL:uprcstar) is well-defined for every , and that , with all partial derivatives computed by differentiating under the integral sign, so that in . Thus, for and almost all ,

 F(u|Γa)(ξ) = exp(i(a−H)√k2+−ξ2)^FH(ξ), (2.6) F(∂u∂xn∣∣∣Γa)(ξ) = i√k2+−ξ2exp(i(a−H)√k2+−ξ2)^FH(ξ), (2.7) F(∇~xu|Γa)(ξ) = iξexp(i(a−H)√k2+−ξ2)^FH(ξ).

Therefore, by the Plancherel identity (1.10), , with

 ∫Γa|u|2ds=∫Rn−1|exp(2i(a−H)√k2+−ξ2)||^FH(ξ)|2dξ≤∫ΓH|FH|2ds

and

 ∫Γa|∇u|2ds≤∫Rn−1[|k2+−ξ2|+ξ2]|exp(2i(a−H)√k2+−ξ2)||^FH(ξ)|2dξ, (2.8)

while

 ∫Γa[∣∣∣∂u∂xn∣∣∣2−|∇~xu|2+k2+|u|2]ds=2∫|ξ|

and

 Im∫Γa¯u∂u∂xnds=∫|ξ|

Thus (2.5) holds and

 ∫UH∖Ua|u|2dx≤(a−H)∫ΓH|FH|2ds. (2.9)

Further, from (2.8) it follows that

 ∫UH∖Ua|∇u|2dx ≤ (a−H)k2+∫|ξ|k+ξ21−exp(−2[a−H]√ξ2−k2+)√ξ2−k2+|^FH(ξ)|2dξ ≤ ∫Rn−1(2(a−H)k2++√2|ξ|)|^FH(ξ)|2dξ,

since for and for . Thus if . That is clear when , and