Elastic Pulsed Wave Propagation in Media with Second or Higher Order Nonlinearity. Part I: Theoretical Framework

Koen E-A Van Den Abeele

Post-Doctoral Research Fellow, EES-4, MS D443, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.
also Post-Doctoral Fellow of the Belgian Foundation for Scientific Research, K.U.Leuven Campus Kortrijk, Interdisciplinary Research Center, E. Sabbelaan 53, B-8500 Kortrijk, Belgium.


Abstract

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. [1] A polynomial expansion of the equation of state is used in which we account for four orders of nonlinearity in the moduli. We solve the nonlinear wave equation for the displacement field at distance x from a symmetric "breathing" source with arbitrary Fourier spectra 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 us to include an arbitrary attenuation function. We investigate energy conservation in the absence of (linear) attenuation, and illustrate the notion of a hybrid (linear and nonlinear) dissipation. 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.