Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

6^{th} set of exercises, 19.3.2003

1. Let u(x,α), x ∈ ℝ^{3}\ \overline{D} be the solution of the scattering problem by a sound-soft obstacle

D,

u(x,α) = e^{ikα·x}+ u_{sc}(x,α),

and A(x̂,α) the corresponding far field pattern. Prove the reciprocity relation

A(x̂,α) = A(−α,−x̂).

Hint: Starting from the Helmholtz representation for u_{sc}, show that

A(x̂,α) = \frac{1}{4π} ∫_{∂D}( u_{sc}(y,α)\frac{∂u}{∂n}(y,−x̂) − u(y,−x̂)\frac{∂u_{sc}}{∂n}(y,α)) dS

and similarly,

A(−α,−x̂) = \frac{1}{4π} ∫_{∂D}( u(y,−x̂)\frac{∂u_{inc}}{∂n}(y,α) − u_{inc}(y,α)\frac{∂u}{∂n}(y,−x̂)) dS.

Deduce the result from the above representations.

2. Prove the above reciprocity relation for the far field pattern of the solution of the Lippmann-Schwinger equation,

u(x,α) = e^{ikα·x}− ∫_{D}Φ(x − y)q(y)u(y,α)dy.

Hint: Use the same formulas as above, integration being over the surface of a sphere |x| = R.

3. Derive the following two-potential formula: Let A_{j}(x̂,α), j = 1,2 be the far field patterns corresponding to the compactly supported potentials q_{j}, j = 1,2, respectively, and denote by u_{j}(x,α) the corresponding acoustic fields with incoming plane wave with the direction α. Show that

A_{1}(x̂,α) − A_{2}(x̂,α) = \frac{1}{4π} ∫ u_{1}(y,α)(q_{1}(y) − q_{2}(y))u_{2}(y,−x̂)dy.

This formula is central in inverse scattering theory. Hint: Write

\begin{aligned}A_{1}(x̂,α) & = \frac{1}{4π} ∫ e^{−ikx̂·y}q_{1}(y)u_{1}(y,α)dy \\ & = \frac{1}{4π} ∫ (u_{2}(y,−x̂) − u_{2,sc}(y,−x̂))q_{1}(y)u_{1}(y,α)dy, \end{aligned}

and a similar formula for A_{2}(x̂,α) = A_{2}(−α,−x̂) and subtract.