Frequency shift and attenuation length of a Rayleigh wave due to surface roughness

Abstract

On the basis of Rayleigh's method we study the effect of surface roughness on the dispersion relation of a Rayleigh wave on an isotropic medium. The stress-free boundary conditions (applied at the rough interface) lead us to an integral eigenvalue equation for the elastic displacement field. That equation reduces to a simpler algebraic equation upon averaging the displacement field over an ensemble of rough surfaces. We obtain explicit expressions for the roughness-induced perturbation of the flat-surface dispersion relation ${\ensuremath{\omega}}_{0}({q}_{\ensuremath{\parallel}})={c}_{R}{q}_{\ensuremath{\parallel}}$. (Here ${c}_{R}$ is the speed of a Rayleigh wave and ${q}_{\ensuremath{\parallel}}$ is a two-dimensional wave vector.) The real part of this perturbation measures the shift in the frequency away from ${\ensuremath{\omega}}_{0}({q}_{\ensuremath{\parallel}})$ while its imaginary part measures the lifetime (or the inverse attenuation length) of the Rayleigh wave. We present detailed numerical results for both the shift and the lifetime. We show that the decay of a Rayleigh wave into bulk elastic waves is a much more efficient mechanism than its decay into other Rayleigh surface waves. An explanation for this disagreement with the earlier results of Maradudin and Mills is presented.