On the Connection Between the Asymptotic Waveform and the Fading Tail of an Initial N-Wave in Nonlinear Acoustics

The propagation and decay of plane and cylindrical acoustic N-waves are studied using the nonlinear Burgers' and generalized Burgers' equations. The asymptotic waveform, which satisfies a wave equation with the nonlinear term neglected, is determined in these two cases. The asymptotic waveform is given by an integral, whose parameters are determined by identification with that part of the shockwave profile, which fades out and therefore is also determined by a linear wave equation. The method is tested in the plane wave case and gives the same asymptotic waveform as is obtained by the exact Cole-Hopf solution of Burgers' equation. In the cylindrical case the integral representation of the asymptotic waveform (also called the old-age waveform) is evaluated approximately to give a simple expression which can be compared with numerical results obtained by other authors. The agreement is satisfactory.