A comparison of three perturbation methods for nonlinear hyperbolic waves
Abstract
Simple wave solutions of nonlinear 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 noncumulative term. This firstorder 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.
 Publication:

Journal of Sound Vibration
 Pub Date:
 September 1976
 DOI:
 10.1016/0022460X(76)904673
 Bibcode:
 1976JSV....48..293N
 Keywords:

 Acoustic Propagation;
 Hyperbolic Differential Equations;
 Perturbation Theory;
 Wave Equations;
 Inviscid Flow;
 Method Of Characteristics;
 Plane Waves;
 Potential Flow;
 Wave Propagation;
 Acoustics