The author considers the linearized Navier-Stokes equations governing the motion of a compressible viscous fluid with a free surface. The approach used in the paper is based upon the ray method expansion originally developed by J. B. Keller [Bull. Am. Math. Soc. 84, 727-750 (1978; Zbl 0393.35002)]. By assuming a wave-like solution with an amplitude function and a phase function and taking into account the boundary conditions as an integral part of the formulation of the method, the three-dimensional problem is reduced to the problem of constructing rays and wave fronts on the two-dimensional equilibrium free surface, and the solution of partial differential equations is reduced to the integration of ordinary differential equations along rays. A general uniform asymptotic expansion is also constructed to remove anomalies where an amplitude function in the ray method expansion becomes infinite.