Asymptotics of Long Waves Generated by Time-Harmonic Spatially Localized Sources in Basins with Gently Sloping Shores

For a nonlinear and linearized system of shallow water equations in a basin with an uneven bottom and gently sloping shores, the problem of short-wavelength asymptotic solutions describing waves excited by a time-harmonic spatially localized source is considered. In the linear approximation, such asymptotic solutions are essentially expressed via solutions of the Helmholtz equation, and the problem of constructing them is close to the problem of the asymptotics of the Green’s function. We use a recently developed approach based on the Maslov canonical operator and allowing one to find a global asymptotic solution of the linearized problem in any predetermined region, taking into account caustics and focal points, as well as Fermat’s variational principle, which, in combination with the canonical operator, makes it possible to construct such an asymptotic solution locally, i.e., in the neighborhood of a given observation point. The linearized problem is considered in a fixed domain, which is bounded by a shoreline corresponding to the fluid at rest. On this line, the equations degenerate; accordingly, a correct statement of the problem does not require (and does not admit) classical boundary conditions; instead, the condition of finiteness of the energy integral is used. From the viewpoint of asymptotic theory, the shoreline is a nonstandard caustic, in the neighborhood of which the asymptotic solution of the linearized problem is expressed via a modified canonical operator. For the original nonlinear system, a free-boundary problem is considered: the position of the shoreline depends on the elevation of the free surface. According to a recently developed approach based on the modified Carrier–Greenspan transform, the asymptotic solution of the nonlinear system is expressed via the solution of the linearized system in the form of parametrically specified functions. The resulting formulas, in particular, describe the effects of wave inrush on the shore.