Half space scattering problems at low frequencies

Abstract

Low frequency scattering by isolated targets in free space has been well studied and there exists a general theory as well as explicit results for special target shapes. In the present paper we develop a comparable theory for low frequency scattering of targets above a flat plane. The presence of the ground plane has a considerable effect on the way in which the target scatters 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 target-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 ground 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 full low frequency expansion of the scattered field is obtained in terms of solutions of arbitrarily shaped scatterers. The first non-trivial term is found explicitly for a spherical target using separation of variables in bispherical coordinates. This is compared with the exact result for the translated sphere in the absence of the ground plane, also found in terms of bispherical coordinates. The presence of the ground plane is demonstrated to have a profound effect on the scattering amplitude and this effect is shown to change drastically 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 provide some essential benchmarks for making a reasonable extrapolation from the free space target signature of a general object to its signature in the presence of a ground plane.