Oscillating line source in a shear flow with a free surface: critical layer-like contributions

TL;DR

This paper analytically investigates the wave radiation from an oscillating line source in a shear flow with a free surface, revealing a new non-dispersive wave linked to a critical layer-like flow and resonance phenomena.

Contribution

It introduces a novel wave component caused by shear and mass flux, differing from potential flow solutions, and analyzes the resonance effects at a specific frequency.

Findings

Discovery of a non-dispersive downstream wave due to shear effects

Identification of a critical layer-like flow structure at the surface

Resonance at a specific frequency amplifies downstream waves

Abstract

The linearized water-wave radiation problem for an oscillating submerged line source in an inviscid shear flow with a free surface is investigated analytically at finite, constant depth in the presence of a shear flow varying linearly with depth. The surface velocity is taken to be zero relative to the oscillating source, so that Doppler effects are absent. The radiated wave out from the source is calculated based on Euler's equation of motion with the appropriate boundary and radiation conditions, and differs substantially from the solution obtained by assuming potential flow. To wit, an additional wave is found in the downstream direction in addition to the previously known dispersive wave solutions; this wave is non-dispersive and we show how it is the surface manifestation of a critical layer-like flow generated by the combination of shear and mass flux at the source, passively advected with the flow. As seen from a system moving at the fluid velocity at the source's depth, streamlines form closed curves in a manner similar to Kelvin's cat's eye vortices. A resonant frequency exists at which the critical wave resonates with the downstream propagating wave, resulting in a downstream wave pattern diverging linearly in amplitude away from the source.