§36.13 Kelvin’s Ship-Wave Pattern

A ship moving with constant speed $V$ on deep water generates a surface gravity wave. In a reference frame where the ship is at rest we use polar coordinates $r$ and $\varphi$ with $\varphi =0$ in the direction of the velocity of the water relative to the ship. Then with $g$ denoting the acceleration due to gravity, the wave height is approximately given by

where

The integral is of the form of the real part of (36.12.1) with $y=\varphi$, $u=\theta$, $g=1$, $k=\rho$, and

When $\rho >1$, that is, everywhere except close to the ship, the integrand oscillates rapidly. There are two stationary points, given by

These coalesce when

This is the angle of the familiar V-shaped wake. The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency $\omega$ as a function of wavevector $𝐤$:

Here $k=|𝐤|$, and $𝐕$ is the ship velocity (so that $\mathrm{V}=|𝐕|$).

The disturbance $z(\rho ,\varphi )$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that ${\theta }_{\pm }(\varphi )$ are real for and complex for $|\varphi |>{\varphi }_c$. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

the disturbance is

See Figure 36.13.1.

For further information see Lord Kelvin (1891, 1905) and Ursell (1960, 1994).