Theory of non-collinear interactions of acoustic waves in an isotropic material with hysteretic quadratic nonlinearity.

Abstract

A particular form of the energy potential (cubic in strains) is proposed, which leads to the bow-tie behavior of the nonlinear modulus in an isotropic material with hysteresis of quadratic nonlinearity. The nonlinear scattering of a weak probe wave in the field of a strong pump wave is analyzed. It is demonstrated that collinear interactions of the shear waves are allowed in materials with nonlinearity hysteresis. Both in collinear and non-collinear frequency-mixing processes the combination frequency is composed of the probe wave frequency and one of the even harmonics of the pump wave. In general, the developed theory predicts that in the presence of the hysteretic nonlinearity the number of possible resonant scattering processes increases. In particular, if frequency-mixing processes are forbidden in the material with the elastic quadratic nonlinearity (for a fixed ratio of primary frequencies), they may be allowed in the materials with hysteretic quadratic nonlinearity. Moreover, in materials with hysteresis of the nonlinearity the resonant frequency mixing for a fixed ratio of primary frequencies may be allowed for multiple mutual orientations of the primary wave vectors.