Half space scattering problems at low frequencies

Low frequency scattering by isolated targets in free space has been well st udied and there exists a general theory as well as explicit results for spe cial target shapes. In the present paper we develop a comparable theory for low frequency scattering of targets above a fiat plane. The presence of th e ground plane has a considerable effect on the way in which the target sca tters an incident field and this effect is highly dependent on the boundary condition used to model the ground. To gain an understanding of how the ta rget-ground interaction affects the scattering amplitude at low frequencies a number of different models are treated. Attention is directed to scalar scattering by small three-dimensional objects on which either Dirichlet or Neumann boundary conditions are imposed. The object is located above a grou nd plane on which again either Dirichlet or Neumann conditions are imposed, resulting in four different combined boundary-value problems. The incident wave originates in the half-space containing the object. The fun low frequ ency expansion of the scattered field is obtained in terms of solutions of arbitrarily shaped scatterers. The first non-trivial term is found explicit ly for a spherical target using separation of variables in bispherical coor dinates. This is compared with the exact result for the translated sphere i n the absence of the ground plane, also found in terms of bispherical coord inates. The presence of the ground plane is demonstrated to have a profound effect on the scattering amplitude and this effect is shown to change dras tically with the boundry condition on the plane. Amazingly, the presence of an acoustically soft plane changes the signature of a soft sphere so that it more closely resembles the signature of a hard sphere. These results pro vide some essential benchmarks for making a reasonable extrapolation from t he free space target signature of a general object to its signature in the presence of a ground plane.