# §36.13 Kelvin’s Ship-Wave Pattern

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

where

36.13.2

The integral is of the form of the real part of (36.12.1) with , , , , and

36.13.3

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

36.13.4

These coalesce when

36.13.5

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 as a function of wavevector :

36.13.6

Here , and is the ship velocity (so that ).

The disturbance 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 are real for and complex for . (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

36.13.7

the disturbance is

See Figure 36.13.1.

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of , .

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