Dispersion relation for harmonic generation in nonlinear elastic waves

Abstract

We present a theory for the dispersion of generated harmonics in a traveling nonlinear wave. The harmonics dispersion relation—derived by the theory—provides direct and exact prediction of the collective harmonics spectrum in the frequency-wavenumber domain and does so without prior knowledge of the spatial-temporal solution. The new relation is applicable to a family of initial wave functions characterized by an initial amplitude and wavenumber. We demonstrate the theory on nonlinear elastic waves in a homogeneous rod and demonstrate its extension to periodic rods. We investigate a thick elastic rod admitting longitudinal motion. In the linear limit, this rod is dispersive due to the effect of lateral inertia. The nonlinearity is introduced through either the stress–strain relation and/or the strain–displacementgradient relation. Using a theory we have developed earlier, we derive an exact general nonlinear dispersion relation for the thick rod. We then derive a special case of this relation and show that it provides an exact prediction of the generated harmonics spectrum, in frequency versus wavenumbers. Both relations are validated by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) for the general nonlinear dispersion relation and short-term, pre-breaking dispersion (by Fourier transformations) for both the general and specialized relation.