The discrete-analytical solution method is proposed for the solution to problems related to the dynamics of the hydro-elastic system consisting of an axially-moving pre-stressed plate, compressible viscous fluid and rigid wall. The fluid flow caused by the axial movement of the plate and the pre-stresses in the plate are taken into consideration as the initial state of the system under consideration. It is assumed that the additional lineally-located time-harmonic forces act on the plate and these forces cause additional flow field in the fluid and an additional stress-strain state in the plate. The additional stress-strain state in the plate is described by utilizing the equations and relations of the three-dimensional linearized theory of elastic waves in initially stressed elastic bodies. The additional fluid flow field is described with linearized Navier-Stokes equations for compressible viscous fluid. As the fluid flow velocities in the initial state are non-homogeneous, the linearized Navier-Stokes equations have variable coefficients and this situation causes difficulties in obtaining an analytical solution to these equations. The proposed discrete-analytical solution method allows this difficulty to be overcome and for approximate analytical solutions for these types of problems to be obtained. The proposed solution method is examined with respect to concrete problems. Numerical results obtained with the proposed approach are presented and discussed.