Harmonics dispersion relation: A new fundamental theory of strongly nonlinear 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. While the theory of linear waves is fairly established, nonlinear wave motion remains a complex, often mysterious, object—particularly when the nonlinearity is strong. For example, an unbalanced nonlinear wave distorts acutely as it travels and appears to ultimately fully lose its original shape, and in many instances the final outcome is onset of a form of instability. Inherent to this distortion is an intricate mechanism of harmonic generation manifesting in the form of intensive time-varying exchange of energy between the harmonics in a manner that matches the wave’s ongoing nonlinear evolution in space and time.

In this work, a general theory is presented for the dispersion of these generated harmonics as they emerge and develop in a traveling nonlinear wave. The harmonics dispersion relation−derived by the theory−provides direct and exact analytical prediction of the collective harmonics spectrum in the frequency-wavenumber domain, and does so without prior knowledge of the spatial-temporal solution. Despite its time-independence, the new relation is shown to be applicable at any temporal state of evolution of the nonlinear wave as long as the wave is balanced or has not yet reached its breaking point. The theory is applied to nonlinear elastic waves in a homogeneous rod and an extension is demonstrated to rods with a periodic array of property modulation (phononic crystal) or intrinsic resonators (elastic metamaterial). Finally, the theory is shown to provide a rigorous foundation for the analytical synthesis of solitons.