What are the wavevectors contributing to the Kelvin wake pattern?

The V-shaped wake behind a boat, a duck, or a cargo ship always ¹ opens at the same angle: $\arcsin(1/3)\approx 19.5^\circ$. This is the Kelvin wake pattern, named after Lord Kelvin who worked out the mathematics in 1887 ². Curiously, the angle does not depend on the size of the object, nor on its speed (as long as the approximation that they are gravity waves holds).

I wanted to understand this phenomenon, and quickly encountered the problem is most of the derivations are not convincing. They typically invoke point disturbances created by the moving object (this is the approach of the original derivation and some other ones ³), but these disturbances somehow have well-defined wavelength and are initially localized at the same time (plot twist: this is impossible, it's basically Heisenberg uncertainty relation).

My approach is less handwavy – I hope so anyway. This is not to discredit the people who first think it through, I simply have access to more polished mathematics. The critical ideas are: (a) the assumption that the object moving through the water interacts substantially only with a subset of monochromatic waves and (b) the realization that the family of waves is described by a curve in the wavevector space, which (c) can be split into wave groups, and (d) each of the wave groups has a limited opening angle; the wake edge itself is sort of a caustic.

The (c,d) point I reserve for a later note. Writing this one already takes too long. But here come (a,b):

The resonance condition

Consider a point-like emitter (say, a boat) moving with constant velocity $\vec v$ through still water. Its position is $\vec x(t) = \vec v t$. The emitter interacts with water waves; it pushes water around, creating a broad spectrum of disturbances.

Here we will describe the disturbances with the usage of waves composed from sums and integrals of 2D plane waves. This is an idealized approach (such waves are characterized by vertical displacement only, and in our model can not break), but captures what happens in the world surprisingly good.

A plane wave with wavevector $\vec k$ and frequency $\omega$ has the form

such that $\psi(x,y,t)$ can be thought of as vertical displacement of the water table at 2D horizontal position $(x,y)$ at time $t$.

What does the emitter "see" as it moves through this wave? At the emitter's position, the phase of the wave is

If $\omega \neq \vec k \cdot \vec v$, this phase changes in time—the emitter moves through crests and troughs, and any effect it has on the wave averages out over time. But if

the emitter always sits at the same phase of the wave. There is a resonance: the emitter can continuously feed energy into (or extract energy from) this particular wave mode. My assumption is that while I do not understand the exact mechanism, these are the waves that survive in the wake pattern.

Figure 1: A wave satisfying the resonance condition $\omega = \vec k \cdot \vec v$. The emitter (marked point) always sees the same local wave pattern.

Water waves in deep water obey the dispersion relation

where $g$ is gravitational acceleration. This tells us that longer waves (smaller $\lvert\vec k\rvert$) travel faster. Combining this with the resonance condition $\omega = \vec k \cdot \vec v$, we get

This is the principal equation: it selects which wavevectors can participate in the wake.

The constraint curve

Let's take $\vec v = (v, 0)$—the boat moves in the positive $x$-direction. Writing $\vec k = (k_x, k_y)$, the constraint becomes

Rearranging the equation we get:

For real solutions, we need $k_x \geq g/v^2$. The constraint defines a curve in the $(k_x, k_y)$ plane.

Figure 2: The constraint curve in wavevector space. Only wavevectors on this curve can contribute to the wake. The curve starts at $(k_x, k_y) = (g/v^2, 0)$ and extends toward larger $|k_x|$ and $|k_y|$.

The amplitude $A(\vec k)$ for wavevectors on this curve is not something we can determine from first principles without a detailed model of the source. But we can argue that only these wavevectors matter: all others average to zero. The wake is built from waves with