text: As the surface object moves, it continuously generates small disturbances which are the sum of sinusoidal waves with a wide spectrum of wavelengths. Those waves with the longest wavelengths have phase speeds above  and dissipate into the surrounding water and are not easily observed. Other waves with phase speeds at or below , however,  are amplified through constructive interference and form visible shock waves, stationary in position w.r.t. the boat.
  The angle  between the phase shock wave front and the path of the object is  = . If c/v > 1 or < −1, no later waves can catch up with earlier waves and no shockwave forms.
  In deep water, shock waves form even from slow-moving sources, because waves with short enough wavelengths move slower. These shock waves are 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 half of the phase velocity.