Electromagnetic Waves in an Inhomogeneous Medium
In the previous chapter, we considered the direct scattering problem for acoustic waves in an inhomogeneous medium. We now consider the case of electromagnetic waves. However, our aim is not to simply prove the electromagnetic analogue of each theorem in
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The Helmholtz Equation
Studying an inverse problem always requires a solid knowledge of the theory for the corresponding direct problem. Therefore, the following two chapters of our book are devoted to presenting the foundations of obstacle scattering problems for timeharmonic acoustic waves, i.e., to exterior boundary value problems for the scalar Helmholtz equation. Our aim is to develop the analysis for the direct problems to an extent which is needed in the subsequent chapters on inverse problems. In this chapter we begin with a brief discussion of the physical background to scattering problems. We will then derive the basic Green representation theorems for solutions to the Helmholtz equation. Discussing the concept of the Sommerfeld radiation condition will already enable us to introduce the idea of the far field pattern which is of central importance in our book. For a deeper understanding of these ideas, we require sufficient information on spherical wave functions. Therefore, we present in two sections those basic properties of spherical harmonics and spherical Bessel functions that are relevant in scattering theory. We will then be able to derive uniqueness results and expansion theorems for solutions to the Helmholtz equation with respect to spherical wave functions. We also will gain a first insight into the ill-posedness of the inverse problem by examining the smoothness properties of the far field pattern. The study of the boundary value problems will be the subject of the next chapter.
2.1 Acoustic Waves Consider the propagation of sound waves of small amplitude in a homogeneous isotropic medium in IR3 viewed as an inviscid fluid. Let v = v(x, t) be the velocity field and let p = p(x, t), ρ = ρ(x, t) and S = S (x, t) denote the pressure, density and specific entropy, respectively, of the fluid. The motion is then governed by Euler’s equation 1 ∂v + (v · grad) v + grad p = 0, ∂t ρ D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, DOI 10.1007/978-1-4614-4942-3 2, © Springer Science+Business Media New York 2013
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2 The Helmholtz Equation
the equation of continuity ∂ρ + div(ρv) = 0, ∂t the state equation p = f (ρ, S ), and the adiabatic hypothesis ∂S + v · grad S = 0, ∂t where f is a function depending on the nature of the fluid. We assume that v, p, ρ and S are small perturbations of the static state v0 = 0, p0 = constant, ρ0 = constant and S 0 = constant and linearize to obtain the linearized Euler equation 1 ∂v + grad p = 0, ∂t ρ0 the linearized equation of continuity ∂ρ + ρ0 div v = 0, ∂t and the linearized state equation ∂p ∂ f ∂ρ = (ρ0 , S 0 ) . ∂t ∂ρ ∂t From this we obtain the wave equation 1 ∂2 p = Δp c2 ∂t2 where the speed of sound c is defined by c2 =
∂f (ρ0 , S 0 ). ∂ρ
From the linearized Euler equation, we observe that there exists a velocity potential U = U(x, t) such that 1 v= grad U ρ0 and ∂U . ∂t Clearly, the velocity potential also satisfies the wave equation p=−
1 ∂2 U = ΔU. c2 ∂t2
2.1 Acoustic Waves
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