A Comparison of Perturbation Methods for Nonlinear Hyperbolic Waves

Abstract

This chapter discusses the comparison of perturbation methods for nonlinear hyperbolic waves. It discusses five numbers of perturbation methods including the method of renormalization, the method of strained coordinates, the analytic method of characteristics, the method of multiple scales, and the Krylov–Bogoliubov–Mitropolsky method. These techniques are applied to nonlinear acoustic waves propagating in thermoviscous fluids. For lossless, oppositely traveling, one-dimensional waves, a first-order uniformly valid expansion can be obtained by using any of these techniques provided the waves do not mutually interact in the body of the medium. This is so if the waves are periodic or pulses. If the mutual-interaction terms are not negligible, only the method of renormalization, strained coordinates, and characteristics can be used. For dissipative media, it is not clear how one can use the method of renormalization, strained coordinates, and characteristics to determine an approximate solution when the dissipation term is the same order as the nonlinear term. For conservative multidimensional waves, a combination of the methods of renormalization and matched asymptotic expansions appear to be the most powerful. In applying the analytic method of characteristics to multi-dimensional waves, one can obtain uniformly valid expansions only if the characteristic surfaces are chosen appropriately. For waves that are short compared with the radii of curvature of the wave fronts, choosing the appropriate characteristics does not present any difficulty because the linear solution displays the role of the characteristic and geometric rays.