Abstract. The method of characteristics is extended to solve the Cauchy problem for linear hyperbolic PDEs in one space dimension with arbitrary variation of coefficients. In the presence of continuous variation of coefficients, the number of characteristics that must be dealt with is uncountable. This difficulty is overcome by writing the solution as an infinite series in terms of the number of reflections involved in each characteristic path. We illustrate an interesting combinatorial connection between the traditional reflection and transmission coefficients for a sharp interface to Green's coefficient for transmission through a smoothly-varying region. We prove that the series converges and provide bounds for the truncation error. The effectiveness of the approximation is illustrated with examples.
Preprint: https://arxiv.org/abs/1901.04158
NOTE: This paper was substantially revised and will be published in SIAM J. Appl. Math under a new title, "A path integral method for solution of the wave equation with continuously-varying coefficients"
bibtex entry:
@misc{GeorgeKetchesonLeVeque2019b,
Author = {J. D. George and D. I. Ketcheson and R. J. LeVeque},
Title = {A characteristics-based approximation for wave scattering from an
arbitrary obstacle in one dimension},
Howpublished = "arXiv:1901.04158",
Year = {2019}
}