5.9 Ship Waves

The surface wave pattern of a moving source, such as that made by a ship, lacks the isotropy of many of the problems discussed previously. Let us consider a time harmonic surface pressure distribution translating at uniform speed V in the negative x direction over water of depth h, so that applied surface pressure (5.8.1) becomes

By the usual procedure, the resulting surface displacement is given by

The phase ? = k x ? ( ? _o ? Vk ₁) t, and the frequency of the surface displacement of wavenumber k is

This is the frequency with respect to an observer fixed relative to the undisturbed water and evidently depends on the direction of propagation. Relative to the moving source, the frequency is

It is therefore convenient to adopt a moving reference frame, by setting = x + Vt, = y, whereupon

5.9.1 Moving line pressure source

Let the moving pressure source be uniform in the y direction and be given by

Then p _o( k) = ( P _o ?/2)e ^{? k ?}, and conditions are steady in a frame moving with the source. On deep water, Equation (5.9.3) becomes

and the dispersion function

These factors respectively vanish at when ? _o = 0. In either case ? ? / ? k = V. Therefore the poles at k = g/V