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Helsinki University of Technology \hfill Somersalo/Bingham \\
Mathematics

\section*{Mat-1.198 Scattering Theory}

\subsection*{6$^\textrm{th}$ set of exercises, 19.3.2003}

\vfill

% \emph{%
% By doing the exercises, you can earn points that will be taken into account 
% when determining your grade for the course:
% Do these exercises in advance, mark them as solved
% on the list at the exercise session and be prepared to present
% your solution on the blackboard.
% }

\begin{enumerate}


\vfill
\item 
Let $u(x,\alpha)$, $x\in \R^3\setminus\overline D$ be the
solution of the scattering problem by a sound-soft obstacle $D$,
\[
 u(x,\alpha)=e^{ik\alpha\cdot x}+u_{\rm sc}(x,\alpha),
\]
and $A(\widehat x,\alpha)$ the corresponding far field pattern.
Prove the reciprocity relation
\[
 A(\widehat x,\alpha)=A(-\alpha,-\widehat x).
\]

\vfill
{\em Hint:} Starting from the Helmholtz representation for $u_{\rm sc}$,
show that
\[
 A(\widehat x,\alpha)=
 \frac{1}{4\pi} \int_{\partial D}\left(u_{\rm sc}(y,\alpha)
\frac{\partial u}{\partial n}(y,-\widehat x) -
u(y,-\widehat x)
\frac{\partial u_{\rm sc}}{\partial n}(y,\alpha)\right)dS
\]
and similarly,
\[
 A(-\alpha,-\widehat x)=
 \frac{1}{4\pi} \int_{\partial D}\left( u(y,-\widehat x)
\frac{\partial u_{\rm inc}}{\partial n}(y,\alpha) -
u_{\rm inc}(y,\alpha)
\frac{\partial u}{\partial n}(y,-\widehat x)\right)dS.
\]
Deduce the result from the above representations.


\vfill
\item 
Prove the above reciprocity relation for the far field
pattern of the solution of the Lippmann-Schwinger equation,
\[
 u(x,\alpha)=e^{ik\alpha\cdot x}-\int_D\Phi(x-y)q(y)u(y,\alpha)dy.
\]

\vfill
{\em Hint:} Use the same formulas as above, integration being
over the surface of a sphere $|x|=R$.


\vfill
\item 
Derive the following two-potential formula: Let
$A_j(\widehat x,\alpha)$, $j=1,2$ be the far field patterns
corresponding to the compactly supported potentials $q_j$,
$j=1,2$, respectively, and denote by $u_j(x,\alpha)$ the corresponding
acoustic fields with incoming plane wave with the direction $\alpha$.
Show that
\[
 A_1(\widehat x,\alpha)-A_2(\widehat x,\alpha)
 =\frac{1}{4\pi} \int u_1(y,\alpha)(q_1(y)-q_2(y))u_2(y,-\widehat x)dy.
\]
This formula is central in inverse scattering theory.

\vfill
{\em Hint:} Write
%\begin{eqnarray*}
\begin{equation*}
\begin{split}
 A_1(\widehat x,\alpha) &= 
 \frac{1}{4\pi} \int e^{-ik\widehat x\cdot y}q_1(y)u_1(y,\alpha)dy
\\
%\noalign{\vskip4pt}
 &= 
 \frac{1}{4\pi} 
 \int \big(u_2(y,-\widehat x) - u_{2,{\rm sc}}(y,-\widehat x)\big)
 q_1(y)u_1(y,\alpha)dy,
\end{split}
\end{equation*}
%\end{eqnarray*}
and a similar formula for $A_2(\widehat x,\alpha)=A_2(-\alpha,-\widehat x)$
and subtract.


\vfill

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