The decay of sawtooth solutions to the Burgers equation

Abstract

The paper discusses the decay of a one-dimensional periodic acoustic signal of moderate amplitude after it has developed a sawtooth profile containing one thin shock in each period. A composite asymptotic expansion is constructed by inserting Taylor (1910) shock transitions at regular intervals within a piecewise linear profile. This representation, which formally is valid to first order at distances where shocks are thin, is shown to be an exact solution of Burgers equation at all distances. It is, in fact, the profile due to Fay (1931) and is usually represented as a Fourier series. The Fay solution is thereby shown to have a simple interpretation in terms of a periodic array of spreading shocks that appear not to interact as they interpenetrate. It is also confirmed that this new representation corresponds, under the Hopf-Cole transformation, to the solution of the heat conduction equation which describes the spreading of a periodic array of point heat sources. In an appendix, two identities involving double sums of hyperbolic functions are derived.