Elastic pulsed wave propagation in media with second- or higher-order nonlinearity. Part I. Theoretical framework

A theoretical model is presented that describes the interaction of frequency components in arbitrary pulsed elastic waves during one-dimensional propagation in an infinite medium with extreme nonlinear response. The model is based on one-dimensional Green’s function theory in combination with a perturbation method, as has been developed for a general source function by McCall [J. Geophys. Res. 99 (B2), 2591–2600 (Feb. 1994)]. A polynomial expansion of the equation of state is used in which four orders of nonlinearity in the moduli are accounted for. The nonlinear wave equation is solved for the displacement field at distance x from a symmetric ‘‘breathing’’ source with arbitrary Fourier spectrum imbedded in an infinite medium. The perturbation expression corresponds to a higher-order equivalent of the Burgers’ equation solution for velocity fields in solids. The solution is implemented numerically in an iterative procedure which allows one to include an arbitrary attenuation function. Energy conservation is investigated in the absence of (linear) attenuation, and the notion of a hybrid (linear and nonlinear) dissipation is illustrated. Examples are provided showing the effect of each term in the perturbation solution on the spectral content of the waveform. Finally, the possibility of creating a parametric array for seismic exploration is briefly considered from a theoretical point of view.