  The phonon Boltzmann transport equation (BTE) is widely utilized to study
non-diffusive thermal transport. We find a solution of the BTE in the thin film
transient thermal grating (TTG) experimental geometry by using a recently
developed variational approach with a trial solution supplied by the Fourier
heat conduction equation. We obtain an analytical expression for the thermal
decay rate that shows excellent agreement with Monte Carlo simulations. We also
obtain a closed form expression for the effective thermal conductivity that
demonstrates the full material property and heat transfer geometry dependence,
and recovers the limits of the one-dimensional TTG expression for very thick
films and the Fuchs-Sondheimer expression for very large grating spacings. The
results demonstrate the utility of the variational technique for analyzing
non-diffusive phonon-mediated heat transport for nanostructures in
multi-dimensional transport geometries, and will assist the probing of the mean
free path (MFP) distribution of materials via transient grating experiments.
