  Scattering of normally incident longitudinal and transverse acoustic waves by
a randomly rough surface of an elastically isotropic solid is analyzed within
the small perturbation approach. In the limiting case of a large correlation
length $L$ compared with the acoustic wavelength, the specularity reduction is
given by $4\eta^2k^2$, where $\eta$ is the RMS roughness and $k$ is the
acoustic wavevector, which is in agreement with the well-known Kirchhoff
approximation result often referred to as Ziman's equation [J. M. Ziman,
Electrons and Phonons (Clarendon Press, Oxford, 1960)]. In the opposite
limiting case of a small correlation length, the specularity reduction is found
to be proportional to $\eta^2k^4L^2$, with the fourth power dependence on
frequency as in Rayleigh scattering. Numerical calculations for a Gaussian
autocorrelation function of surface roughness connect these limiting cases and
reveal a maximum of diffuse scattering at an intermediate value of $L$. This
maximum becomes increasingly pronounced for the incident longitudinal wave as
the Poisson's ratio of the medium approaches 1/2 as a result of increased
scattering into transverse and Rayleigh surface waves. The results indicate
that thermal transport models using Ziman's formula are likely to overestimate
the heat flux dissipation due to boundary scattering, whereas modeling
interface roughness as atomic disorder is likely to underestimate scattering.
