waves - Why is the angle of the wake of a duck constant?

Why is the angle of a wake of a duck constant? And why are some conditions on the water depth necessary?

I realize that this question turns up in google searches, but I did not see a good discussion. I will be quite happy with a link.

Edited to add:

Could anyone tell me how the two up-voted answers are related?

Answer

The ideal Kelvin boat wake ignores surface tension, and it assumes deep water waves with an (in general) broad spectrum of frequencies $\omega$ with dispersion relation $\omega^2=gk$, where $g\approx 9.8 \frac{m}{s^2}$. The ideal Kelvin wake furthermore assumes that the ship sails with a constant velocity, and that the wave amplitudes of the partial waves are so small that they obey a linear superposition principle. The Kelvin wake does not describe the narrow turbulent band behind a ship, nor shock waves. The Kelvin wake consists of two types of waves: transverse and divergent waves. There are two characteristic angles

$$\alpha\approx 19^{\circ} \qquad \mathrm{and} \qquad \beta\approx 35^{\circ},$$

corresponding to

$$\tan(\alpha)= \frac{1}{2\sqrt{2}} \qquad \mathrm{and} \qquad \tan(\beta) = \frac{1}{\sqrt{2}},$$

or equivalently,

$$\sin(\alpha)= \frac{1}{3} \qquad \mathrm{and} \qquad \sin(\beta) = \frac{1}{\sqrt{3}}.$$

In polar coordinates $(r,\theta)$ of a co-moving coordinate system, where the position of the boat is at the origin, the transverse waves are in the region $|\theta|\leq \beta$, and divergent waves are in the region $\alpha\leq |\theta|\leq \beta$.

The angles $\alpha$ and $\beta$ are constant in at least two ways: Firstly, they don't depend on the distance $r$ to the ship. This is because the speed of each partial wave (with frequency $\omega$) is independent of the position $(x,y)$. Secondly, $\alpha$ and $\beta$ are, evidently, universal angles, independent of, for instance, $g$. This is explained in the references below.

_{(source: wikiwaves.org)}

References:

1) Howard Georgi, "The Physics of Waves", Chapter 14. (Hat tip:user1631.)

2) MIT on-line open course ware, mechanical engineering, wave propagation, lecture notes, fall 2006, Chapter 4.7.

3) Wikiwaves.