A comparison of three perturbation methods for non-linear hyperbolic waves

Abstract

Simple wave solutions of non-linear hyperbolic equations are studied by using the method of renormalization, the analytic method of characteristics, and the method of multiple scales. It is shown that the results of the method of renormalization depend on whether the potential function or the velocity is normalized. This arbitrariness does not occur when using either the analytic method of characteristics or the method of multiple scales. However, special consideration must be given in determining the potential from the velocity obtained by the analytic method of characteristics. No such consideration is needed when the method of multiple scales is applied. The first term obtained for the potential by the method of multiple scales contains a cumulative term in addition to a non-cumulative term. This first-order term is shown to yield the equal area rule for shock waves, and the slope of an equipotential line is the arithmetic mean of the slope of the characteristic in the unperturbed medium and the slope of the characteristic at the point under consideration.