The generation of beams of three-dimensional periodic internal waves in an exponentially stratified fluid

Abstract

The spectral method is used to construct an exact solution of the linearized problem of the generation of disturbances by localized sources that execute arbitrary periodic motions in a viscous exponentially stratified fluid. The expressions obtained do not contain any adjusting parameters and describe conical beams of three-dimensional periodic internal waves and two types of boundary layers, the spatial scale of which is given by the kinematic viscosity and the buoyancy frequency of the medium. The thickness of one of them, which is analogous to Stokes periodic flow in a homogeneous viscous fluid, is specified by the kinematic viscosity and the wave frequency, that is, it additionally depends on a ratio of the wave and buoyancy frequencies. The thickness of the specific internal boundary layer also depends on the geometry of the problem. In the approximation of weak stratification and low viscosity, asymptotic estimates of the expressions obtained are presented for two types of generators, namely, in the form of a plane inclined rectangle that vibrates along its surface (a frictional source) and along the normal to it (a piston source) in the non-degenerate case when the wave cone does not touch the radiating plane. In limiting cases the analytical expressions obtained agree with known exact solutions of the problem of generating axially symmetric and two-dimensional periodic internal waves.