Watch a duck paddle across a pond, or a ferry cross a bay. Behind it spreads a neat V of ripples. Now notice something odd: a slow rowing boat and a fast launch leave a V with roughly the same opening. Speed up, and the V does not get narrower, the way a jet's shock cone does. Why does the water refuse to change its mind?
A moving boat shoves the water, and that shove sends out waves in every direction at every wavelength. Most of them wander off and fade into a mess. Only the ones that stay put relative to the boat add up into a pattern you can see, because a wake that you can photograph must look the same from the boat's point of view.
For a wave travelling at an angle θ to the boat's path to keep pace, its crests must slide along at the boat's speed projected onto that direction: phase speed = v cos θ. Deep-water waves have a fixed link between wavelength and speed, so each direction picks out exactly one wavelength. Straight behind the boat (θ = 0) that wavelength is 2πv²/g. At 5 m/s, a brisk motor launch, that is about 16 metres between the transverse crests. Go faster and it grows with the square of your speed.
The drawing below is the full pattern Lord Kelvin worked out in 1887. Each curve is one crest line. Drag the wave direction slider and a dot slides along the outer crest. Read off the dot's bearing from the boat's path.
Try it. Slide the direction from 0° upwards. The dot starts straight behind the boat, swings out to the side, and then, past a certain direction, swings back in. The bearing never gets above about 19.5°. Everything the boat makes has to stay inside that wedge, and the crests pile up along its edge, which is why you see a bright V.
Here is a short version of the argument. Deep-water waves carry their energy at only half their phase speed. So while the boat travels a distance vt, the energy of any given wave has only gone halfway along the wave's own track. Draw those halfway points for every direction and they trace a circle of radius vt/4, centred a quarter of the way back along the boat's path. The boat sits at the tip, 3vt/4 from that centre. The steepest line from the boat that still touches the circle has sin α = (1/4)/(3/4) = 1/3, so α ≈ 19.47°, and the full V is about 39°.
Nothing in that argument contains the boat's speed. It cancels out. The speed only sets the scale of the pattern, which you can see by moving the speed slider: the crests spread apart but the edges of the V stay put.
Switch the drawing to sound-like waves. Now every ripple travels at one fixed speed, however long it is, as sound does. Each ripple stays a circle that grows from the place where it was made, and the circles pile up along a straight cone whose angle depends on speed: sin α = (wave speed)/(boat speed). Go faster and the cone narrows. Water waves break that rule, because long waves outrun short ones. That is what pins the Kelvin wedge at 19.47°.
The fixed angle is a result for gentle speeds and deep water. It treats the boat as a point, and water as an ideal fluid. In 2013, Frédéric Rabaud and Frédéric Moisy measured the wakes in 37 aerial photographs of ships. At low speeds, up to a hull Froude number (speed divided by √(g × hull length)) of about 0.5, the angle sat near the Kelvin value, at roughly 18.6° ± 1.8°. Faster than that, the V narrowed, falling roughly in inverse proportion to speed, to about 7° for the fastest boats. Their explanation is that a hull much longer than the transverse wavelength cannot stir up those waves well, so the wake starts to look like a Mach cone after all.
The same wedge shows up whenever water moves past something. A stone in a stream makes it too, because only the speed relative to the water counts. Next time you stand on a bridge, look for the V. It is not about the stone or the boat. It is the water's own rule for how its waves travel.
Source notes: Kelvin's analysis is from his 1887 paper “On ship waves”. The high-speed measurements are from Rabaud and Moisy, “Ship wakes: Kelvin or Mach angle?”, Physical Review Letters (2013).