Half space scattering problems at low frequencies

Low frequency scattering by isolated targets in free space hasbeen well studied and there exists a general theory as wellas explicit results for special target shapes. In the presentpaper we develop a comparable theory for low frequency scatteringof targets above a flat plane. The presence of the ground planehas a considerable effect on the way in which the target scattersan incident field and this effect is highly dependent on theboundary condition used to model the ground. To gain an understandingof how the target-ground interaction affects the scatteringamplitude at low frequencies a number of different models aretreated. Attention is directed to scalar scattering by smallthree-dimensional objects on which either Dirichlet or Neumannboundary conditions are imposed. The object is located abovea ground plane on which again either Dirichlet or Neumann conditionsare imposed, resulting in four different combined boundary-valueproblems. The incident wave originates in the half-space containingthe object. The full low frequency expansion of the scatteredfield is obtained in terms of solutions of arbitrarily shapedscatterers. The first non-trivial term is found explicitly fora spherical target using separation of variables in bisphericalcoordinates. This is compared with the exact result for thetranslated sphere in the absence of the ground plane, also foundin terms of bispherical coordinates. The presence of the groundplane is demonstrated to have a profound effect on the scatteringamplitude and this effect is shown to change drastically withthe boundary condition on the plane. Amazingly, the presenceof an acoustically soft plane changes the signature of a softsphere so that it more closely resembles the signature of ahard sphere. These results provide some essential benchmarksfor making a reasonable extrapolation from the free space targetsignature of a general object to its signature in the presenceof a ground plane.