Model equations for nonlinear surface waves

Abstract

The theoretical framework developed originally for modeling nonlinear Rayleigh waves in isotropic elastic half-spaces [Zabolotskaya, J. Acoust. Soc. Am. 91, 2569 (1992); see also Knight et al., ibid. 102, 1402 (1997)] has been extended to encompass a broad class of nonlinear surface wave problems. The theory is based on Hamiltonian mechanics, in which spatial Fourier amplitudes are used as generalized coordinates, and coupled spectral equations are obtained for the harmonic amplitudes in the wave. The nonlinearity coefficient matrices are expressed explicitly in terms of the second- and third-order elastic moduli of the materials. Solutions of the equations describe harmonic generation, waveform distortion, and shock formation not only at the interface but also in the interior of the solid. Cylindrical spreading and beam diffraction may be taken into account, as can transient effects in pulses. Model equations, and simulations of nonlinear surface waves in real materials, shall be presented for Rayleigh, Stoneley, Scholte, and Lamb waves, and surface waves in crystals and piezoelectric materials. The validity of the theory has been demonstrated via quantitative agreement with measurements of laser-generated Rayleigh wave pulses with shocks in fused quartz [Lomonosov et al., J. Acoust. Soc. Am. 101, 3080(A) (1997)]. [Work supported by NSF and ONR.]