A path integral formulation of acoustic wave propagation

There exists a strong need in seismology for fast, accurate calculation of broad-band synthetic seismograms for waves propagating in three-dimensionally heterogeneous media. We present a promising tool for such calculations in the form of a path integral solution to the acoustic wave equation. Unlike other work in this area, we consider the effects of variable density as well as variable wave velocity. An additional parameter is introduced to transform the hyperbolic wave equation into a parabolic one similar in form to the Schrodinger equation, allowing the solution by path integration. Two cases are considered. First, we develop the solution for a constant-density medium in substantial detail for the benefit of readers unfamiliar with path integrals. In the process, we indicate how geometric ray theory may be extracted from our solution in the high-frequency limit. Next, we solve the general problem which includes the effects of variable density. Here the smoothness of the paths that contribute the most to the integral is contrasted to the extreme roughness of those dominant in the quantum mechanical analogue. In both cases, the final solutions have had both frequency and the additional parameter integrated out analytically, leaving a 'configuration space' path integral. Finally, an outline for numerical implementation is presented as well as the prospects for extension to isotropic elastic wave propagation.