This paper describes mode conversion effects and subsequent waveform distortion arising when a finite amplitude dilatational (P) wave that has already experienced nonlinear distortion is obliquely incident on a stress-free boundary of an isotropic elastic half-space. A two-term perturbation expansion is first employed to identify the dominant nonlinear effects. The understanding of wave interactions obtained from the perturbation analysis is then exploited to derive a successful solution using the method of characteristics for two-dimensional wave. It is shown that the incident and reflected P waves undergo nonlinear amplitude dispersion along their ray paths. The orientation of the rays for the reflected waves are time dependent, being governed by a modified form of Snell’s law, in which the phase speed incorporates the nonlinear correction for the association particle velocity. The reflection coefficients are shown to resemble those of linear theory, except for the dependence on the variable angles of reflection. The nonlinear propagation and reflection laws are solved to determine temporal waveforms for the reflected P and SV (vertically polarized shear) waves. This requires an iterative procedure in order to trace rays arriving at a specified field point at an arbitrary instant back to their source.

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A ray theory for two-dimensional, finite-amplitude acoustic waves forming a mode within a hard-walled rectangular waveguide was described previously [K. T. Shu and J. H. Ginsberg, J. Acoust. Soc. Am. Suppl. 1 83, S1 (1988)]. The present paper extends those developments to the treatment of oblique reflection and mode conversion of a finite-amplitude dilatational wave (P wave) at the stress-free boundary of an elastic half-space. Due to nonlinear self-action, cumulative growth of second harmonics occur in the incident and reflected P waves in proportion to the square of the amplitude of the first-order incident and reflected P waves, respectively, but such growth is not encountered in the reflected vertically polarized shear wave (SV wave), nor in the many waves arising from nonlinear interaction between dilatational and shear waves. Uniformly valid expressions for strain are obtained by using renormalization techniques along the rays. The analysis indicates that the mode conversion between the nonlinear P and SV waves can be described by linear reflection theory. As a consequence of the reflection process, the nonlinear effect in the reflected P wave corresponds to a simple planar wave that originated from a weaker source at a longer range, even though the phase of that wave is governed by the actual propagation distance from the source. [Work supported by NSF and ONR.]

An earlier paper [K. T. Shu and J. H. Ginsberg, J. Acoust. Soc. Am. Suppl. 1 84, S5 (1988)] proved that the significant finite amplitude aspects of mode conversion of a plane dilatational wave obliquely incident at a plane stress-free boundary may be described in terms of Earnshaw-type solutions for the dilatational and vertically polarized shear waves. In the work described here, the method is extended to an obliquely incident finite amplitude plane dilatational wave at a plane interface between two bounded isotropic solids. It is shown that, not as a consequence of nonlinear interaction effects, but rather as a consequence of nonlinear self-action effects, cumulative growth of higher harmonics occurs in the incident, reflected, and refracted waves. A number of special circumstances of wave reflection-refraction are examined, relative to the incidence angle and the material properties of the two media. Also given are the deformed incident, reflected, and refracted waveforms near one shock formation distance. [Work supported by NSF and ONR.]