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

TL;DR

This paper introduces a modified kernel for linear Kelvin wave predictions that resolves unbounded wave energy issues at the free surface, enabling fast, physically consistent simulations with an open-source Julia implementation.

Contribution

It develops a new elliptic spanwise line integration kernel for flat-ship theory that ensures finite wave energy and provides a fast, accurate evaluator for wave predictions.

Findings

The modified kernel yields finite wave energy at the free surface.

The fast evaluator achieves 10,000 to 100,000 times speedup over direct methods.

Predictions show physically consistent wave patterns and resistance trends.

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 flat-ship theory 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.