Exact dispersion relation for linear surface waves on arbitrary vertical shear
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911626480844800 |
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| author | Heinrich, Kjell S. Ellingsen, Simen Å. |
| author_facet | Heinrich, Kjell S. Ellingsen, Simen Å. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24484 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact dispersion relation for linear surface waves on arbitrary vertical shear Heinrich, Kjell S. Ellingsen, Simen Å. Fluid Dynamics 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. |
| title | Exact dispersion relation for linear surface waves on arbitrary vertical shear |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2604.24484 |