Interaction of elastic waves in solids with quadratic and cubic nonlinearity

This article investigates the interactions of two-plane waves in weakly nonlinear elastic solids containing quadratic and cubic nonlinearity. The analytical solutions for generated combined harmonic waves are derived using the Green's function approach applied to a generated system of quasi-linear equations of motion. Wave mixing solutions are obtained and include shape functions that permit closed-form solutions for a variety of interaction geometries. An explicit example is highlighted for a spherical interaction volume assuming isotropic elastic constants. Several parameters of the generated field after mixing are analyzed including resonant and nonresonant mixing, the role of interaction angle, and the frequencies of the two incident waves. Wave mixing offers the potential for sensing localized elastic nonlinearity and the present model can be used to help design experimental configurations.