  Understanding ballistic phonon transport effects in transient
thermoreflectance experiments and explaining the observed deviations from
classical theory remains a challenge. Diffusion equations are simple and
computationally efficient but are widely believed to break down when the
characteristic length scale is similar or less than the phonon mean-free-path.
Building on our prior work, we demonstrate how well-known diffusion equations,
namely the hyperbolic heat equation and the Cattaneo equation, can be used to
model ballistic phonon effects in frequency-dependent periodic steady-state
thermal transport. Our analytical solutions are found to compare excellently to
rigorous numerical results of the phonon Boltzmann transport equation. The
correct physical boundary conditions can be different from those traditionally
used and are paramount for accurately capturing ballistic effects. To
illustrate the technique, we consider a simple model problem using two
different, commonly-used heating conditions. We demonstrate how this framework
can easily handle detailed material properties, by considering the case of bulk
silicon using a full phonon dispersion and mean-free-path distribution. This
physically transparent approach provides clear insights into the nonequilibrium
physics of quasi-ballistic phonon transport and its impact on thermal transport
properties.
