Harmonic dispersion relation for strongly nonlinear elastic waves

Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat, or fluid flow are all likely to involve wave dynamics at some level. 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. It is valid throughout the evolution of a distorting unbalanced wave or the steady-steady propagation of a balanced wave with waveform invariance. 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 traveling in a homogeneous rod with and without linear dispersion, showing that the theory is not limited by the strength of the nonlinearity or wave amplitude. Finally, we use the underlying formulation to present an analysis on the condition required for synthesis of solitary waves.