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Scattering of normally incident longitudinal and transverse acoustic waves bya randomly rough surface of an elastically isotropic solid is analyzed withinthe small perturbation approach. In the limiting case of a large correlationlength $L$ compared with the acoustic wavelength, the specularity reduction isgiven by $4\eta^2k^2$, where $\eta$ is the RMS roughness and $k$ is theacoustic wavevector, which is in agreement with the well-known Kirchhoffapproximation result often referred to as Ziman's equation [J. M. Ziman,Electrons and Phonons (Clarendon Press, Oxford, 1960)]. In the oppositelimiting case of a small correlation length, the specularity reduction is foundto be proportional to $\eta^2k^4L^2$, with the fourth power dependence onfrequency as in Rayleigh scattering. Numerical calculations for a Gaussianautocorrelation function of surface roughness connect these limiting cases andreveal a maximum of diffuse scattering at an intermediate value of $L$. Thismaximum becomes increasingly pronounced for the incident longitudinal wave asthe Poisson's ratio of the medium approaches 1/2 as a result of increasedscattering into transverse and Rayleigh surface waves. The results indicatethat thermal transport models using Ziman's formula are likely to overestimatethe heat flux dissipation due to boundary scattering, whereas modelinginterface roughness as atomic disorder is likely to underestimate scattering.

Boundary scattering of phonons: specularity of a randomly rough surface in the small perturbation limit