Kelvin wake pattern

Remarks

Check: work in the frame of the water, and let $P$ be the point that the body passed a time $T$ ago. Take a wave component sent out from $P$ in a direction at angle $\theta$ to the track. It is steady in the frame of the body only if its phase speed is $c = U\cos\theta$. Its energy moves at the group velocity, which by the deep-water dispersion relation is $c_g = c/2 = \tfrac{1}{2}U\cos\theta$. So after the time $T$ this energy sits at $P + \tfrac{1}{2}UT\cos\theta\,(\cos\theta, \sin\theta)$, and these points fill the circle of diameter $\tfrac{1}{2}UT$ through $P$, centred at distance $\tfrac{1}{4}UT$ from $P$ towards the body. The body is at distance $UT$ from $P$, hence at distance $\tfrac{3}{4}UT$ from the centre of a circle of radius $\tfrac{1}{4}UT$. The tangent lines from the body to all these circles make the angle $\arcsin\left(\tfrac{1}{4} / \tfrac{3}{4}\right) = \arcsin(1/3)$ with the track. Observations of fast ships show wakes narrower than this angle; the point-source model ignores the finite size of the hull, which matters at high speed.