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arxiv: 2604.24484 · v1 · submitted 2026-04-27 · ⚛️ physics.flu-dyn

Exact dispersion relation for linear surface waves on arbitrary vertical shear

Kjell S. Heinrich , Simen {\AA}. Ellingsen This is my paper

Pith reviewed 2026-05-07 17:58 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords solutionapproximationsdispersionrelationarbitraryequationknownlinear
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The pith

An exact implicit dispersion relation is derived for linear surface waves on arbitrary vertical shear using a Green's function solution to the Rayleigh equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Ocean waves often travel over currents that change speed at different depths, like faster near the surface and slower below. Earlier studies handled only simple current profiles. This work reformulates the wave problem using a Green's function approach to the Rayleigh equation, which governs the flow disturbance. The result is one exact equation linking wave frequency, wavenumber, and the full velocity profile, with no extra approximations beyond linearity and zero viscosity. It recovers known results for deep water and special cases.

Core claim

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.

Load-bearing premise

The flow is treated as inviscid with no further approximations about the mean velocity profile, relying on the Rayleigh equation framework for linear perturbations.

Figures

Figures reproduced from arXiv: 2604.24484 by Kjell S. Heinrich, Simen {\AA}. Ellingsen.

Figure 1
Figure 1. Figure 1: FIG. 1. Geometry of the problem set-up used for derivation. view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Three vertical shear profiles and their dispersion relations found with DIM and the expression derived view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of inviscid, incompressible, linear flow without introducing new free parameters or postulated entities.

axioms (2)
  • domain assumption The fluid is inviscid and the flow is incompressible.
    Standard assumption for deriving the Rayleigh equation in linear wave theory.
  • domain assumption Waves are small-amplitude linear perturbations on a horizontal mean current.
    Required to linearize the equations and obtain the dispersion relation.

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Works this paper leans on

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