M. F. Hamilton
Yu. A. Il
E. A. Zabolotskaya
Dept. of Mech. Eng., Univ. of Texas at Austin, Austin, TX 78712-1063
Time domain evolution equations are derived for nonlinear Rayleigh wave propagation along the free surface of an isotropic solid. The evolution equations are expressed in different forms: one for the horizontal displacement component, and one for a complex displacement variable. Both equations are derived from spectral equations published earlier [Zabolotskaya, J. Acoust. Soc. Am. 91, 2569 (1992)]. To simplify the derivation of equations for the displacement variables, not all components of the nonlinearity coefficient matrix are retained. However, the terms which are retained possess the same mathematical properties (symmetries in the frequency domain, singularities in the time domain, and others) as the terms that are not taken into account. Moreover, with appropriate definition of the shock formation distance, it is shown that numerical solutions based on the simplified equations yield close agreement with numerical solutions based on the complete nonlinearity matrix. The nonlinear terms in the new evolution equations are expressed as time derivatives and Hilbert transforms of the displacement variables. Waveforms calculated with the time domain evolution equations are shown to agree with results based on the original frequency domain equations. [Work supported by the Packard Foundation, ONR, and NSF.]