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arXiv · 2604.24484

Exact dispersion relation for linear surface waves on arbitrary vertical shear

Kjell S. Heinrich·Simen Å. Ellingsen
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Abstract

We derive the formal solution to the dispersion relation for linear surface waves on a horizontal mean current with arbitrary vertical dependence. The problem is cast in a Green's function framework for the Rayleigh equation, neglecting viscosity but making no further approximations about the mean velocity profile. The solution is the dispersion relation in the form of a single, implicit equation relating -- and containing only -- the velocity profile, wave frequency, and wavenumber. By isolating curvature effects in a path-ordered exponential, we obtain a solution that serves as a natural starting point for systematic approximations. We demonstrate that our solution reduces to the expression found by Shrira (1993, J. Fluid Mech. 252, 565--584) in the deep-water limit, yields known asymptotic approximations, and recovers known analytical solutions in special cases.

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Kjell S. Heinrich, Simen Å. Ellingsen. 2026-04-27. Exact dispersion relation for linear surface waves on arbitrary vertical shear. https://arxiv.org/abs/2604.24484

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