  We study the hydrodynamics of a freely-standing smectic-A film in the
isothermal, incompressible limit theoretically by analyzing the linearized
hydrodynamic equations of motion with proper boundary conditions. The dynamic
properties for the system can be obtained from the response functions for the
free surfaces. Permeation is included and its importance near the free surfaces
is discussed. The hydrodynamic mode structure for the dynamics of the system is
compared with that of bulk systems. We show that to describe the dynamic
correlation functions for the system, in general, it is necessary to consider
the smectic layer displacement $u$ and the velocity normal to the layers,
$v_z$, together. Finally, our analysis also provides a basis for the
theoretical study of the off-equilibrium dynamics of freely-standing smectic-A
films.
