Evolution equation for nonlinear Lucassen waves

Abstract

A nonlinear, fractional, surface-wave equation was developed recently by Kappler et al. [Phys. Rev. Fluids 2, 114804 (2017)] for propagation along an elastic interface coupled to a viscous incompressible medium. Linear theory for attenuation and dispersion of such a wave was developed originally by Lucassen [Trans. Faraday Soc. 64, 2221 (1968)]. Kappler et al. employ a fractional derivative to account for the Lucassen attenuation and dispersion, and they include quadratic and cubic nonlinearity of the elastic interface. Presented here is a simplified form of their model equation for plane progressive waves. The resulting nonlinear evolution equation has the form of a Burgers equation but with a fractional derivative in place of the second derivative for viscosity, and with cubic as well as quadratic nonlinearity. In addition to facilitating analytical and numerical calculations, the evolution equation enables interpretation of a threshold phenomenon, revealed in numerical simulations presented by Kappler et al., as competition between quadratic and cubic nonlinearity. It is also suitable for determining critical source amplitudes above which Lucassen attenuation and dispersion alone cannot preclude formation of unphysical multivalued waveforms [Cormack and Hamilton, Wave Motion 85, 18 (2019)]. [BES and JMC were supported by the ARL:UT McKinney Fellowship in Acoustics.]