Let D be a bounded, simply connected domain in the plane, and let F((theta);k,(alpha)(,L)) be the far field pattern arising from the scattering of an incoming, time harmonic, acoustic plane wave u(,L)(x) = exp (ikx(.)(alpha)(,L)), where (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 addi- tion, that the total field satisfies homogeneous Dirichlet or Neumann boundary conditions on (PAR-DIFF)D. In this dissertation, a method is pre- sented for recovering (PAR-DIFF)D given the far field patterns F((theta);k,(alpha)(,L)), L = 1,...,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 (SIAM J. Appl. Math., 45, 1039-1053 (1985)) for solving the full aperture problem.

In addition, the above method is numerically implemented for Dirichlet boundary data on (PAR-DIFF)D. These computations show that one can recover by this method the shape of (PAR-DIFF)D if the length of the inter- val a,a + (delta) is as small as 180(DEGREES). While numerical experiments have only been performed for Dirichlet boundary data, it is made clear that with appropriate modifications the same method will work for Neumann boundary data on (PAR-DIFF)D.