Waves Propagating on Fluids with Piecewise Constant Vorticity

Abstract

The purpose of this study was to find the dispersion relation for waves on an arbitrary current velocity profile approximated by N linear layers.

Using a Stokes expansion to first order in the steepness parameter ka the dispersion relation for systems with one, two and three linear layers was derived, a process revealing a general pattern for an increase in N. The pattern was then used to derive the dispersion relation for the N-layer system, which is shown valid even to second order theory. The dispersion relation was significantly simplified by a recursion formula which made it easy to solve numerically. To test the formula two possible current velocity profiles of water flowing down a river was investigated; a parabolic and a logarithmic profile.

The numerical analysis revealed that even a linear current velocity profile serve as a good approximation. The results also showed that we get N+1 solutions. However, data supported a theory that only two of them are physical possible. The extra solutions probably originates from the unphysical bends in the velocity profile, and are not waves but layers of vortices drifting with the current.