Angles

All shock waves, that each by itself would have had an angle between 33° and 72°, are compressed into a narrow band of wake with angles between 15° and 19°, with the strongest constructive interference at the outer edge (angle arcsin(1/3) = 19.47°), placing the two arms of the V in the celebrated Kelvin wake pattern.

A concise geometric construction^[3] demonstrates that, strikingly, this group shock angle w.r.t. the path of the boat, 19.47°, for any and all of the above θ, is actually independent of v, c, and g; it merely relies on the fact that the group velocity is half of the phase velocity c. On any planet, slow-swimming objects have "effective Mach number" 3.

For slow swimmers, low Froude number, the Lighthill−Whitham geometric argument that the opening of the Kelvin chevron (wedge, V pattern) is universal goes as follows. Consider a boat moving from right to left with constant speed v, emitting waves of varying wavelength, and thus wavenumber k and phase velocity c(k), of interest when < v for a shock wave (cf., e.g., Sonic boom or Cherenkov radiation). Equivalently, and more intuitively, fix the position of the boat and have the water flow in the opposite direction, like a piling in a river.

Focus first on a given k, emitting (phase) wavefronts whose stationary position w.r.t. the boat assemble to the standard shock wedge tangent to all of them, cf. Fig.12.3.

As indicated above, the openings of these chevrons vary with wavenumber, the angle θ between the phase shock wavefront and the path of the boat (the water) being θ = arcsin(c/v) ≡ π/2 − ψ. Evidently, ψ increases with k. However, these phase chevrons are not visible: it is their corresponding group wave manifestations which are observed.

Consider one of the phase circles of Fig.12.3 for a particular k, corresponding to the time t in the past, Fig.12.2. Its radius is QS, and the phase chevron side is the tangent PS to it. Evidently, PQ= vt and SQ = ct = vt cosψ, as the right angle PSQ places S on the semicircle of diameter PQ.

Since the group velocity is half the phase velocity for any and all k, however, the visible (group) disturbance point corresponding to S will be T, the midpoint of SQ. Similarly, it lies on a semicircle now centered on R, where, manifestly, RQ=PQ/4, an effective group wavefront emitted from R, with radius vt/4 now.

Significantly, the resulting wavefront angle with the boat's path, the angle of the tangent from P to this smaller circle, obviously has a sine of TR/PR=1/3, for any and all k, c, ψ, g, etc.: Strikingly, virtually all parameters of the problem have dropped out, except for the deep-water group-to-phase-velocity relation! Note the (highly notional) effective group disturbance emitter moves slower, at 3v/4.

Thus, summing over all relevant k and ts to flesh out an effective Fig.12.3 shock pattern, the universal Kelvin wake pattern arises: the full visible chevron angle is twice that, 2arcsin(1/3) ≈ 39°.

The wavefronts of the wavelets in the wake are at 53°, which is roughly the average of 33° and 72°. The wave components with would-be shock wave angles between 73° and 90° dominate the interior of the V. They end up half-way between the point of generation and the current location of the wake source. This explains the curvature of the arcs.

Those very short waves with would-be shock wave angles below 33° lack a mechanism to reinforce their amplitudes through constructive interference and are usually seen as small ripples on top of the interior transverse waves.

The nature of two types of crests, longitudinal and transverse, is graphically illustrated by the pattern of wavefronts of a moving point source in proper frame. The radii of wavefronts are proportional, due to dispersion, to the square of time (measured from the moment of emission), and the envelope of the wavefronts represents the Kelvin wake pattern.