The nonlinear dispersion of Rayleigh waves

Rayleigh waves in linear elasticity are non-dispersive-all profiles propagate without change of form, at the speed c_R Previously, the author has determined periodic non-distorting waveforms for nonlinear elastic surface waves. They are far from sinusoidal. For each waveform, the difference between the phase speed c and cR is proportional to the wave steepness (the ratio amplitude/wavelength). The present paper shows, using Whitham's methods for analysing modulations of wavetrains, that gradual changes of amplitude and wavelength of these nonlinear Rayleigh waves propagate in a particularly simple manner. The loci of constant phase speed always propagate as a simple wave, with group velocityc_G = G(c). The phase curves also are characteristic curves of the modulation equations.It is shown that these two properties are general properties of the modulation of waveforms having phase speed depending only on wave steepness. Such waveforms arise from physical systems with no intrinsic scales of length or time.