The mathieu functions applied to some problems in underwater acoustics

The purpose of this paper is to present examples of the application of the Mathieu functions to solving problems in the field of acoustics, mainly underwater acoustics. The Mathieu functions have application to aspects of waves related to ellipses and elliptical cylinders: elliptical membrane vibration, vibration of water in elliptical containers, etc. The reason why theses functions have attracted little attention so far is mainly the complexity of the issues they involve. Among all the other functions occurring during the separation of the variables in a wave equation (with partial derivatives) in various co-ordinate systems, the Mathieu functions were the first non-hypergeometric ones. Hence, there are difficulties in the theory of these functions, as well as the calculations involved. The reflection of a sound-wave is studied at an inhomogeneous layer with parallel surfaces separating two homogeneous semi-infinite media having different indices.