The linearized hydrodynamic equations of motion for a thin, flat, superfluid helium film are derived in some detail from standard two-fluid hydrodynamics. Interactions of the film with both the He vapor and the substrate which are in contract with it are included and discussed in detail. Boundary conditions for both the film-substrate and film-gas interfaces are derived. It is indicated how one may construct the equations of motion for the entire coupled system (gas-film-substrate). The equations are actually constructed and solved for the case when a certain parameter is small, which includes all the third-sound experiments on unsaturated films. A dispersion equation is found which is exact in the limit of vanishing frequency, and which is eminently suited to describe both the velocity and the attenuation of third sound in the regime of unsaturated films. No hydrodynamic instability is found. Results for the attenuation are shown to be in good agreement with preliminary experiments on unsaturated He films.