Abstract:

In an earlier work [V. A. Khokhlova and O. A. Sapozhnikov, J. Acoust. Soc. Am. 96, 3321 (1994)], a modified spectral approach was proposed for the description of nonlinear waves containing shocks. An abrupt shock has an analytical high-frequency (omega)[sup -1] asymptote. This asymptotic result was used in the numerical algorithm to model strongly nonlinear waves with a relatively few number of harmonics N~20 [Pischal'nikov et al., Acoust. Phys. 42, 362--367 (1996)]. However, in real dissipative medium the shock front is not a discontinuity, but a transition region of finite thickness. This region can be adequately described by a hyperbolic tangent profile, so that the correspondent wave spectrum at high frequencies is governed by the Fay solution. Here, the Fay spectrum asymptote of the finite thickness shock is used, instead of the (omega)[sup -1] asymptote of the abrupt shock, to derive a set of coupled differential equations for the harmonic amplitudes. Several model problems are considered. It is shown that this method permits increasing the accuracy and stability of the modified spectral approach, and still leads to a reduction in the number of equations by a factor of 10--100 in comparison with direct frequency domain schemes. [Work supported by FIRCA and RFBR.]