The acoustic field of a near-surface source at intermediate distances: A simple result using matched asymptotic expansions

The method of matched asymptotic expansions is used to determine simple expressions for the acoustic field due to a source near a rigid boundary in an inhomogeneous medium. The observation distance is assumed to be large compared to the wavelength but short compared to the length scale over which the medium varies. Three asymptotic regions are identified: a local or source region, an outer or geometric-acoustics region, and a boundary layer containing an outer scattered field. The approach is illustrated in detail for the case of a simple source above a rigid surface in an environment for which the square of the index of refraction varies linearly with distance from the surface. Numerical comparisons show that, over a wide parameter range, the asymptotic results agrees well with the Hankel-transform solution for the linearly varying medium. The theory is then extended to account for more general media variations and sources with complex directivities.