Duh... everyone knows that this pattern follows from the dispersion relation of deep water waves, which is often written as,
where g is the strength of the gravity field and "deep" means that the depth is greater than half of the wavelength. This formula has two implications: first, the speed of the wave scales with the wavelength and second, the group velocity of a deep water wave is half of its phase velocity.
As a surface object moves along its path at a constant velocity v, it continuously generates a series of small disturbances corresponding to waves with a wide spectrum of wavelengths. Those waves with the longest wavelengths have phase speeds above v and simply dissipate into the surrounding water without being easily observed. Only the waves with phase speeds at or below v get amplified through the process of constructive interference and form visible shock waves.
In a medium like air, where the dispersion relation is linear, i.e.
the phase velocity c is the same for all wavelengths and the group velocity has the same value as well. The angle θ of the shock wave thus follows from simple trigonometry and can be written as,
This angle is dependent on v, and the shock wave only forms when v > c.
In deep water, however, shock waves always form even from slowmoving sources because waves with short enough wavelengths move still more slowly. These shock waves also manifest themselves at sharper angles than one would naively expect because it is group velocity that dictates the area of constructive interference and, in deep water, the group velocity is only half of the phase velocity.
