On resonances of nonlinear elastic waves in a cubic crystal

Abstract

Using the method of weakly nonlinear geometric optics, we obtain asymptotic transport evolution equations for high-frequency, small amplitude nonlinear elastic waves in a cubic crystal. Both geometrical and physical nonlinearities are included in our model. We expand strain energy up to the third order terms with respect to the strain matrix components. The nonlinear resonant asymptotic equations obtained are of integro-differential type. The coefficients of these equations are called resonant interaction coefficients (RIC). They determine whether and between which waves the nonlinear resonant interactions occur. We have calculated all the RIC in the explicit analytical form for three different crystalline directions of a one-dimensional wave motion. Comparison of the results shows that the direction of propagation influences the resonant interactions in an essential way. Moreover, our analytical formulas for RIC can be used to determine the material constants of a crystal.