An exact shock wave solution to Burgers' equation for parametric excitation of the boundary

Abstract

It is well known that Burgers' equation for plane waves can be transformed into the linear heat conduction equation yielding a general solution for any driving waveform on the boundary. Of especial current interest is the extent to which the difference-frequency signal can be “pumped” when the primary waves are “saturated”: i.e., weak shock waves occur. A direct harmonic analysis of the general solution for parametric excitation at the boundary would yield information on the conversion efficiency even in the presence of weak shock waves. By past experience, it is difficult to maintain good numerical accuracy for acoustic Reynolds numbers above 50. It is at the higher Reynolds numbers that “saturation” effects become more pronounced and so an alternative method for calculating the conversion efficiency is desired. The saddle point method of integration is utilized to develop from the general plane wave solution of Burgers an exact weak shock solution for boundary excitation by either two signals or by one amplitude modulated signal. Approximate solutions are also given for spherical waves which are applicable to the Fraunhofer zone. Criteria for predicting the range at which weak shock waves appear are also given.