The Kelvin Green's function describes the wave pattern generated by a point source translating beneath a free surface — the mathematical foundation for computing ship waves. The classical theory works at finite depth below the surface. As the source approaches the surface (z → 0), the wave energy diverges. The mathematics says a surface-skimming source generates infinite wave energy, which is physically absurd.
The paper resolves the singularity by replacing the point source with a spanwise line integration using an elliptic kernel. The modification is minimal — it changes the source's cross-sectional profile from a delta function to a smooth distribution — but the effect is decisive: the wave energy becomes finite at z = 0. The corrected Green's function produces physically consistent wave patterns and wave resistance values in the exact regime where the classical theory fails.
The resolution reveals what the singularity meant. A mathematical point source at the free surface concentrates wave generation at a single location with infinite intensity. The physical source — a ship hull — has finite width, and that width is exactly what the elliptic integration provides. The singularity was not a failure of the physics but a failure of the idealization: the point-source approximation that works at depth breaks at the surface because the surface is where the approximation's error becomes load-bearing. Finite width was always necessary; it just didn't matter until z → 0 made it matter.
A 10⁴–10⁵ speedup over direct quadrature makes the correction practical. The physics was there; the computation caught up.