Exact solutions of the Helmholtz equation for plane wave propagation in a medium with variable density and sound speed

The Helmholtz equation for a variable density medium is considered and solutions are sought for the case of horizontal stratification, both sound speed and density varying continuously with depth. It is shown that the nonlinear density dependence in the Helmholtz equation can be reduced to a linear form by means of appropriate transformations. A general procedure is then outlined by which forms of density profile can be generated permitting analytical solutions for plane wave propagation in a stratified medium. The same procedure is applicable to calculate forms of sound-speed profile permitting exact solutions in the presence of a given density stratification. Some specific examples are considered, for which representative density profiles are calculated in terms of well-known special functions such as Bessel and Airy functions. Comparison with measured data shows that the analytical density and sound-speed profiles derived are capable of giving good agreement with real density and speed profiles in marine sediments.