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\begin{document}



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

\section*{Mat-1.198 Scattering Theory}

\subsection*{2$^\textrm{nd}$ set of exercises, 5.2.2003}

\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.
% }

\vfill
\begin{enumerate}
\item Consider the acoustic medium scattering problem of solving 
\[
 (\Delta + k^2 n(x))u(x)=0 \mbox{ in $\R^3$,}
\]
where $n(x) = c_0^2/c(x)^2\in C(\R^3)$ is the refractive index of the 
medium, and $n=1$
outside a bounded domain $D$. We write 
\[
 u= u_{\rm inc} + u_{\rm sc},
\]
where the {\em incoming field} $u_{\rm inc}$ is known and satsifies the
Helmholtz equation in the whole $\R^3$, and the {\em scattered field}
$u_{\rm sc}$ satisfies the Sommerfeld radiation condition at infinity.
By using the Helmholtz representation formula, derive the 
{\em Lippmann-Schwinger equation} for $u$,
\[
 u(x)=u_{\rm inc}(x) +k^2 \int_D\Phi(x-y)(n(y)-1)u(y)dy.
\]
   
\item
Let the incoming field in the previous excercise be a plane wave
propagating in the direction $\widehat\alpha$
\[
 u_{\rm inc}(x)= e^{ik\widehat\alpha\cdot x},\quad \widehat\alpha\in S^2.
\]
(a) Write the far field pattern $u_\infty(\widehat x)$ for the
scattered field.

(b) Consider the {\em Born approximation} of the solution,
\[
 u(x)\approx u_{\rm B}(x) = u_{\rm inc} +k^2 
\int_D\Phi(x-y)(n(y)-1)u_{\rm inc}(y)dy.
\]
What is the far field pattern of $u_{\rm B}$?

\item Calculate the far field pattern of the Born approximation
explicitly, when the refractive index is given as
\[
 n(x)=\left\{\begin{array}{ll} 1+h, & |x|<R \\
                               1  , & |x|\geq R
             \end{array}
      \right.
\]
where $h>0$ is a constant.

How does the approximate
far field pattern behave in the (a) forward scattering direction,
$\widehat x=\widehat\alpha$, and (b) the Backscattering direction,
$\widehat x= -\widehat\alpha$?


\item Let $D\subset\R^3$ be a bounded domain containing the scatterer,
so that outside $D$ the field $u$ satisfies the Helmholtz equation. 
As in the previous problems, let
\[
 u= u_{\rm inc} + u_{\rm sc},
\]
where the incoming field satisfies the Helmholtz equation in the
whole $\R^3$ and the scattered field satisifes the radiation condition.
Show that then
\[
 u_{\rm sc}(x)=\int_{\partial D}\bigg(u(y)\frac{\partial\Phi}{\partial n(y)}
(x-y) - \Phi(x-y)\frac{\partial u}{\partial n(y)}
(y)\bigg)dS,
\]
i.e., in the Helmholtz representation formula, we may replace
$u_{\rm sc}$ by $u$.


%Using the result of the
%previous exercise in $\R^n$, $n=2,3$, let the incoming field be a plane wave
%propagating in the direction $\widehat\alpha$
%\[
% u_{\rm inc}(x)= e^{ik\widehat\alpha\cdot x},\quad \widehat\alpha\in S^{n-1}.
%\]
%Assume that the scatterer $D$ is sound soft ball $B_R$. In the physical
%optics approximation, we assume that the incoming wave reflects specularly
%from the scatterer, i.e,  niear the scatterer the scattered wave is
%\[
% u_{\rm sc}(x)\approx - e^{ik\widehat\beta\cdot x}
%\]
%where the new direction vector makes the same angle
%Write a formula for the
%far field pattern using the {\em Kirchhoff  approximation}

\end{enumerate}

\end{document}
