Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

2^{nd} set of exercises, 5.2.2003

1. Consider the acoustic medium scattering problem of solving

(Δ + k^{2}n(x))u(x) = 0 in ℝ^{3},

where n(x) = c_{0}^{2}/c(x)^{2} ∈ C(ℝ^{3}) is the refractive index of the medium, and n = 1 outside a bounded domain D. We write

u = u_{inc}+ u_{sc},

where the incoming field u_{inc} is known and satsifies the Helmholtz equation in the whole ℝ^{3}, and the scattered field u_{sc} satisfies the Sommerfeld radiation condition at infinity. By using the Helmholtz representation formula, derive the Lippmann-Schwinger equation for u,

u(x) = u_{inc}(x) + k^{2}∫_{D}Φ(x − y)(n(y) − 1)u(y)dy.

2. Let the incoming field in the previous excercise be a plane wave propagating in the direction α̂

u_{inc}(x) = e^{ikα̂·x}, α̂ ∈ S^{2}.

(a) Write the far field pattern u_{∞} (x̂) for the scattered field.

(b) Consider the Born approximation of the solution,

u(x) ≈ u_{B}(x) = u_{inc}+ k^{2}∫_{D}Φ(x − y)(n(y) − 1)u_{inc}(y)dy.

What is the far field pattern of u_{B}?

3. Calculate the far field pattern of the Born approximation explicitly, when the refractive index is given as

n(x) = \{ \begin{matrix}1 + h, & |x| < R \\ 1, & |x| ≥ R \end{matrix}

where h > 0 is a constant.

How does the approximate far field pattern behave in the (a) forward scattering direction, x̂= α̂, and (b) the Backscattering direction, x̂= −α̂?

4. Let D ⊂ ℝ^{3} be a bounded domain containing the scatterer, so that outside D the field u satisfies the Helmholtz equation. As in the previous problems, let

u = u_{inc}+ u_{sc},

where the incoming field satisfies the Helmholtz equation in the whole ℝ^{3} and the scattered field satisifes the radiation condition. Show that then

u_{sc}(x) = ∫_{∂D}( u(y)\frac{∂Φ}{∂n(y)}(x − y) − Φ(x − y)\frac{∂u}{∂n(y)}(y)) dS,

i.e., in the Helmholtz representation formula, we may replace u_{sc} by u.