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An asymptotic method is developed for the study of gravity-capillary waves in a compressible viscous fluid with edge constraints in an inclined, straight channel. The Navier-Stokes equations subject to free surface and rigid bottom conditions are reduced to a sequence of elliptic boundary problems over a cross section of the channel. Their solutions are used to determine the wave speed and to construct the Burgers equation for the evolution of the gravity-capillary waves. The Burgers equation may become ill-posed when the Reynolds number exceeds some critical value. A criterion for the stability of the flow is then defined in terms of the critical Reynolds number. (Author)