The long time asymptotics of solutions to the generalized Burgers equation

Abstract

The long time (i.e. range following a wavefront) behaviour of finiteamplitude sound waves in ducts (‘horns’) of variable cross section is considered, the wave evolution being governed by a generalized form of Burgers equation. The type of initial data far behind and far ahead of the wavefront is restricted to a transition from one constant state to another. It is proved that the form of the wave for large range is dependent on the limiting value of a certain function of the duct parameters. We have termed the various cases that arise ‘supercylindrical’, ‘cylindrical’ and ‘subcylindrical’, for reasons that should be clear from the details to follow. The analytic form of the limiting profile is determined in all but one case. In particular, it is shown that the similarity solution for cylindrical waves possesses very strong global stability properties.