# Wake - Wake Pattern of A Boat

Wake Pattern of A Boat

Waterfowls and boats moving across the surface of water produce a wake pattern, first explained mathematically by Lord Kelvin and known today as the Kelvin wake pattern. This pattern consists of two wake lines that form the arms of a V, with the source of the wake at the point. Each wake line is offset from the path of the wake source by around 19° and is made up with feathery wavelets that are angled at roughly 53° to the path. The interior of the V is filled with transverse curved waves, each of which is an arc of a circle centered at a point lying on the path at a distance twice that of the arc to the wake source. This pattern is independent of the speed and size of the wake source over a significant range of values. The angles in this pattern are not intrinsic properties of water; Any isentropic and incompressible liquid with low viscosity will exhibit the same phenomenon. This phenomenon has nothing to do with turbulence. Everything discussed here is based on the linear theory of an ideal fluid.

This pattern follows from the dispersion relation of deep water waves, which is often written as,

where 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, 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 and simply dissipate into the surrounding water without being easily observed. Only the waves with phase speeds at or below 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 of the shock wave thus follows from simple trigonometry and can be written as,

This angle is dependent on, and the shock wave only forms when .

In deep water, however, shock waves always form even from slow-moving 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.

By a simple accident in geometry, all shock waves that should have had angles between 33° and 72° get compressed into a narrow band of wake with angles between 15° and 19° with the strongest constructive interference occurring at the outer edge, resulting in the two arms of the V in the Kelvin wake pattern. This can be seen easily in the diagram on the left. Here, we consider waves generated at point C by the source which has now moved to point A. These waves would have formed a shock wave at the line AB, with the angle CAB = 62° because the phase velocity of the wave has been chosen to be = 0.883 of the boat velocity. But the group velocity is only half of the phase velocity, so the wake actually forms along the line AD, where D is the midpoint on the segment BC, and the wake angle CAD turns out to be 19°. The wavefronts of the wavelets in the wake coming from the wave components in our example still maintain an angle of 62° to the AC line. In reality, all the waves with would-be-shock-wave-angles between 33° and 72° contribute to the same narrow wake band and the wavelets exhibit an angle of 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. Again, the waves that should have joined together and formed a wall similar to the phenomenon in sonic boom 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 perceived by the naked eyes as small ripples on top of the interior transverse waves.

• Wake from a small motorboat with an outboard motor.

• Wake of a boat crossing an alpine lake.

• The wakes of two slow-moving boats. The nearer boat has made a striking series of ruler-straight waves.