On Kelvin's ship-wave pattern

Abstract

When a concentrated pressure travels with constant velocity over the free surface of water, it carries with it a familiar pattern of ship waves. Let viscosity and surface tension be neglected, let the free-surface condition be linearized, let the depth of water be assumed infinite, and let initial transient effects be ignored. Then, as is well known, the wave motion everywhere can be found by standard methods in the form of a double integral. The wave pattern at a great distance behind the disturbance can be found by an application of the ordinary method of stationary phase, which shows that the wave amplitude is considerable inside an angle bounded by the two horizontal rays θ = ± θ_c from the disturbance, where $\theta_c = \rm {sin}^{-1} {\frac{1}{3} \eDot 19{\frac{1}{2}\deg$. But the method fails in two regions, near the track θ = 0 of the pressure point, and near the critical lines θ = ±θ_c.

These two regions are treated in the present paper. It is shown that near θ = 0 the linearized surface elevation oscillates with indefinitely increasing amplitude and indefinitely decreasing wavelength. (This result holds only when the pressure is concentrated at a point and applied at the free surface.) Near the critical lines the surface elevation at a greater distance behind the pressure point can be expressed in terms of Airy functions, and this expression goes over into the known wave pattern inside the critical angle. It is shown that near the critical lines the crest length increases as the cube root of the distance, and that the separation between crests remains constant. Contour maps of the wave surface are given for three distances behind the moving pressure point.