Modeling of one-dimensional weakly nonlinear waves that propagate in media with arbitrary dissipation and dispersion mechanisms.

Abstract

In his paper [J. Acoust. Soc. Am. 77, 2050 (1985)] Blackstock presented a generalized Burgers equation for the propagation of one-dimensional weakly nonlinear waves in various media. His results, and the approach he employed there, however, are limited to harmonic waves. In this paper, we present a general approach to model nonlinear waves of more general wave forms that propagate in media with arbitrary absorption and dispersion relations. The resulting equation is again called the generalized Burgers equation (to follow the terminology of the literature). It is found that steady shock solutions for various media can be described by the corresponding simplified version of the equation. An efficient numerical method by means of spectral analysis is developed for solving the generalized Burgers equation. Typical results exemplified by the case of a sinusoidal wave source are also reported in this paper.