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

## TL;DR

This paper introduces a novel path integral method for solving the one-dimensional wave equation with smoothly varying coefficients, expressing solutions as convergent series of integrals involving initial data and PDE coefficients.

## Contribution

It develops a new series-based approach for wave equations with variable coefficients, linking classical reflection/transmission concepts to Green's coefficients in smooth regions.

## Key findings

- Series converges with proven bounds on truncation error
- Method effectively approximates solutions with examples
- Establishes combinatorial connection between coefficients

## Abstract

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.