Abstract

A new method of solution is proposed for the 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.

Keywords

  1. wave equation
  2. characteristics
  3. hyperbolic PDE

MSC codes

  1. 35L05
  2. 35L45
  3. 35C15

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Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 2615 - 2638
ISSN (online): 1095-712X

History

Submitted: 14 January 2019
Accepted: 7 October 2019
Published online: 17 December 2019

Keywords

  1. wave equation
  2. characteristics
  3. hyperbolic PDE

MSC codes

  1. 35L05
  2. 35L45
  3. 35C15

Authors

Affiliations

Funding Information

King Abdullah University of Science and Technology https://doi.org/10.13039/501100004052

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