Analytical and numerical modelling of non-collinear wave mixing at a contact interface

Abstract

An analytical model is presented for wave mixing of two incident non-collinear plane waves at a plane contact interface characterised by a nonlinear traction law that is representative of rough-surface contact, where the normal response is described by a quadratic nonlinearity whereas the tangential response is linear. The approach relies on a decomposition of the scattered field into contributions exhibiting distinctive symmetries with respect to the interface, combined with a perturbation analysis that treats the nonlinear response as a small correction relative to a linear response. The decomposition is shown to lead to uncoupled governing equations, thereby enabling explicit analytical formulae to be derived for the scattered field amplitudes for a linear interface, which in turn provides the source terms for determining the amplitudes and directions of the nonlinear mixed waves. The characteristics of ultrasound wave mixing at an interface are thus shown to differ in several respects from bulk wave mixing due to material nonlinearity. In particular, interface mixing generates mixed waves at the sum and difference frequencies that propagate in both forward and backward directions (transmitted and reflected), whereas bulk mixing only produces a forward propagating mixed wave at the sum frequency. The directions of propagation for interface mixing are determined from the sum or difference of the components of the incident wave vectors parallel to the interface, coupled with the sum or difference of the frequencies, which constitutes a generalisation of Snell's law. These directions generally differ from the sum of the wave vectors that provides the propagation direction for bulk mixing. A computational model is also presented that confirms these predictions of the analytical model. The implications of both models for the design and interpretation of experimental investigations are briefly discussed.