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A new method of solution is proposed for solution of the wave equation in onespace dimension with continuously-varying coefficients. By considering allpaths along which information arrives at a given point, the solution isexpressed as an infinite series of integrals, where the integrand involves onlythe initial data and the PDE coefficients. Each term in the series representsthe influence of paths with a fixed number of turning points. We prove that theseries converges and provide bounds for the truncation error. The effectivenessof the approximation is illustrated with examples. We illustrate an interestingcombinatorial connection between the traditional reflection and transmissioncoefficients for a sharp interface, and Green's coefficient for transmissionthrough a smoothly-varying region.

A path integral method for solution of the wave equation with continuously-varying coefficients