Generalized dispersion relation predicts harmonic generation in strongly nonlinear systems
In recent work, we have derived an exact nonlinear dispersion relation for elastic wave propagation in a thin rod (linearly nondispersive) and a thick rod (linearly dispersive). The derivation is generally applicable to any type of nonlinearity, geometric (related to the kinematics) or material (related to the stress-strain constitution). The derived relation has been verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations), as well as by perturbation theory. In this talk, I will present new results showing that our derived nonlinear dispersion relation provides direct and exact prediction of harmonic generation, thus merging two key tenets of wave propagation–dispersion (usually associated with linear response) and harmonic generation (a fundamentally nonlinear mechanism). This fundamental unification of nonlinear dispersion and harmonic generation is applicable to any arbitrary wave profile, irrespective of the strength of the nonlinearity, and regardless of whether the medium is dispersive or nondispersive in the linear limit.