Higher Order Approximation of Weak Nonlinear Waves in a Compressible Fluid

Abstract

The third order approximation in the reductive method of perturbation is applied to the weak nonlinear waves in a compressible fluid governed by the Navier-Stokes equation. It is shown that secular terms appearing in the third order approximation are renormalized to the phase velocity of a weak shock, which agrees with the expression obtained from the Rankine-Hugoniot relation. The third order solution obtained analytically, in the special case, also coincides to Becker’s solution of steady shock in the approximation of the weak shock.