The Limited Aperture Problem of Inverse Acoustic Scattering: Dirichlet Boundary Conditions

Abstract

Let D be a bounded, simply connected domain in the plane and let $F( {\theta ;k,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l } )$ be the far field pattern arising from the scattering of a time harmonic, acoustic plane wave $u_l ( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} } ) = \exp ( {ik\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{x} \cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l } )$, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l $ is a unit vector and k is the wave number and the time harmonic factor $e^{ - i\omega t} $ has been factored out. It is assumed, in addition, that the total field satisfies homogeneous Dirichlet boundary conditions on $\partial D$. In this paper, a method is presented for recovering $\partial D$ given the far field patterns $F( {\theta ;k,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\alpha } _l } )$, $l = 1, \cdots ,N$, for all $\theta $ in some interval $[ a,a + \delta ]$ strictly contained in $[ 0,2\pi ]$. The method used is a generalization of the orthogonal projection approach of Colton and Monk for solving the full aperture problem. In addition, the above method is numerically implemented. These computations show that one can recover by this method the shape of $\partial D$ if the length of the interval $[ a,a + \delta ]$ is as small as $180^ \circ $.