  The reduced phonon specularity $p$ from boundary roughness scattering plays a
major role in the lower thermal conductivity in semiconducting and insulating
nanowires and films. Although the well-known Ziman formula
$p=\exp(-4\sigma^{2}q_{x}^{2})$, where $\sigma$ and $q_{x}$ denote the
root-mean-square boundary roughness and the normal component of the incident
phonon wave vector, respectively, and its variants are commonly used in the
literature to estimate how roughness attenuates $p$, their validity and
accuracy remain poorly understood, especially when the effects of mode
conversion cannot be ignored. In this paper, we investigate the accuracy and
validity of the more general Ogilvy formula, from which the Ziman formula is
derived, by comparing its predictions to the $p$ values computed from Atomistic
Green's Function (AGF) simulations for an ensemble of rough boundaries in
single-layer graphene. The effects of phonon dispersion, incident angle,
polarization, mode conversion, and correlation length are analyzed. Our results
suggest that the Ogilvy formula is remarkably accurate for
$0<q_{x}<\frac{\pi}{4\sigma}$ when the lateral correlation length $L$ is large
or the phonon is at normal incidence. At large $q_{x}$ in the short-wavelength
limit, the $q_{x}$-dependence of $p$ becomes significantly weaker. In the
large-$L$ limit, the numerical results suggest the existence of a minimum $p$
for short-wavelength phonons, given by $p\sim p_{0}\exp(-\pi^{2}/4)$, where
$p_{0}$ is the baseline specularity for the ideal boundary.
