On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface

We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a $\delta $-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that $x$ is a real variable and to allow time $t$ to take complex values.