Geometrical Acoustic Propagation with Frequency Dependence

Abstract

One characteristic of geometrical-acoustic solutions is their frequency independence. Geometrical acoustics stems from the solution of the eikonal equation, which is a form of the wave equation in the limit of infinite frequency. Since the effects of the medium and the boundary conditions on the acoustic fields are definitely frequency dependent, it is desirable to obtain approximate solutions that reflect this dependence on frequency. Although data are available on propagation losses for signals of different frequencies, geometrical acoustics does not offer a means of comparison. An asymptotic expansion in terms of inverse powers of the frequency is presented. The first term in the expansion obeys the eikonal equation; all the ray-path solutions obtained from geometrical acoustics are applicable. Here, the first term of the expansion, which is the geometrical-acoustic field, is obtained in closed form using a simplified profile in which the first derivative is continuous everywhere. The calculation of the higher-order terms then involve operators that have known variable coefficients that are functions of the lower-order solutions.