Abstract

Asymptotic Frequency-Domain Methods in Modeling Nonlinear Waves with Shocks

O.A.Sapozhnikov, V.A.Khokhlova, A.A.Slavnov (Department of Acoustics, Faculty of Physics, Moscow State University, Moscow, Russia)

e-mail: oleg@acs366b.phys.msu.su

In an earlier work a modified spectral approach was proposed for the description of nonlinear waves containing shocks. An abrupt shock has an analytical high frequency asymptote inversely proportional to the frequency. This asymptotic result was used in the numerical algorithm to model strongly distorted plane waves and acoustic beams with a relatively few number of harmonics N ~ 30. The method was generalized later on to include the next asymptotic term responsible for the discontinuity in the first derivative at the shock front; and to account for the finite thickness of the shock using the Fay solution of the Burgers equation for high frequencies. A general asymptotic approach, that includes all previously developed ones as the limiting cases, is presented here. The approach is based on the high frequency asymptote of the periodically repetitive pulse with hyperbolic tangent shock profile and exponential tail. The approach accounts for the finite thickness of the shock front and asymmetric waveform before and behind the shock. Several model problems are considered. The accuracy and stability of the method, a possibility to include the existence of two shocks in one period, their relative movement, and interaction are discussed. Work was supported by CRDF and RFBR.

Section : 1