Linear Kelvin Wave Predictions in the $z\to 0$ Limit

Gabriel D Weymouth

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arXiv · 2603.14945

Linear Kelvin Wave Predictions in the $z\to 0$ Limit

Gabriel D Weymouth

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Abstract

Linear wave theory captures the essential physics of free-surface flows at a fraction of the computational cost of nonlinear and viscous methods, making it attractive for design, real-time control, and surrogate modeling applications. However, the Kelvin Green's function for a translating point-source generates unbounded wave energy in the $z\to 0$ limit, causing both numerical difficulties and physical inconsistencies. This paper develops a modified kernel for the Kelvin potential incorporating an elliptic spanwise line integration that naturally resolves this ill-posedness, yielding finite wave energy over the entire free surface. We then present a fast evaluator for both point and line kernels using contour deformation adapted to the non-analytic Kelvin phase, achieving $10^4$-$10^5$ speedup over direct quadrature while preserving the wake asymptotics. Predictions on the most challenging $z=0$ limit demonstrate physically consistent wave patterns and wave resistance trends. An open-source Julia implementation is provided.

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Gabriel D Weymouth. 2026-03-16. Linear Kelvin Wave Predictions in the $z\to 0$ Limit. https://arxiv.org/abs/2603.14945

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