Low Frequency Scattering by Local Inhomogeneities

An asymptotic expansion which is uniformly valid in space is obtained for the low frequency scattering of a plane wave incident on a localized inhomogeneity. The scattering region, which may be simply or multiply (collection of scatterers) connected, has a characteristic length which is small compared with the wave length of the incident wave. The index of refraction n is unity outside the scattering region and it is arbitrary inside the region. The method of matched asymptotic expansions is used in the analysis. The Born approximation is shown to agree with the uniform expansion in the far and the near fields. the leading term in the uniform expansion is a linear functional of 1 - n². Thus, statistics of the scattered field are easily evaluated from the statistics of n, when n is a random process. The method is then applied to the low frequency scattering of an acoustic plane wave by localized inhomogeneities in the density and the index of refraction. Finally, the scattering by a plane which is acoustically hard except for a small impedance spot, is analyzed by the same method.