  A new method of solution is proposed for solution of the wave equation in one
space dimension with continuously-varying coefficients. By considering all
paths along which information arrives at a given point, the solution is
expressed as an infinite series of integrals, where the integrand involves only
the initial data and the PDE coefficients. Each term in the series represents
the influence of paths with a fixed number of turning points. We prove that the
series converges and provide bounds for the truncation error. The effectiveness
of the approximation is illustrated with examples. We illustrate an interesting
combinatorial connection between the traditional reflection and transmission
coefficients for a sharp interface, and Green's coefficient for transmission
through a smoothly-varying region.
