Wake
Encyclopedia
A wake is the region of recirculating flow immediately behind a moving or stationary solid body, caused by the flow of surrounding fluid around the body.

## Fluid dynamics

In fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, a wake is the region of disturbed flow (usually turbulent) downstream of a solid body moving through a fluid, caused by the flow of the fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

around the body. In incompressible fluids (liquids) such as water, a bow
Bow wave
A bow wave is the wave that forms at the bow of a ship when it moves through the water. As the bow wave spreads out, it defines the outer limits of a ship's wake. A large bow wave slows the ship down, poses a risk to smaller boats, and in a harbor can cause damage to shore facilities and moored ships...

wake is created when a watercraft moves through the medium; as the medium cannot be compressed, it must be displaced instead, resulting in a wave. As with all wave forms, it spreads outward from the source until its energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

is overcome or lost, usually by friction
Friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

or dispersion
Dispersion (water waves)
In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...

.

The formation of these waves in liquids is analogous to the generation of shockwaves in compressible flow, such as those generated by rockets and aircraft traveling supersonic through air (see also Lighthill equation). The non-dimensional parameter of interest is the Froude number
Froude number
The Froude number is a dimensionless number defined as the ratio of a characteristic velocity to a gravitational wave velocity. It may equivalently be defined as the ratio of a body's inertia to gravitational forces. In fluid mechanics, the Froude number is used to determine the resistance of an...

.

For a blunt body in subsonic external flow, for example the Apollo or Orion
Orion (spacecraft)
Orion is a spacecraft designed by Lockheed Martin for NASA, the space agency of the United States. Orion development began in 2005 as part of the Constellation program, where Orion would fulfill the function of a Crew Exploration Vehicle....

capsules during descent and landing, the wake is massively separated
Flow separation
All solid objects travelling through a fluid acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent...

and behind the body is a reverse flow region where the flow is moving toward the body. This phenomenon is often observed in wind tunnel
Wind tunnel
A wind tunnel is a research tool used in aerodynamic research to study the effects of air moving past solid objects.-Theory of operation:Wind tunnels were first proposed as a means of studying vehicles in free flight...

testing of aircraft, and is especially important when parachute
Parachute
A parachute is a device used to slow the motion of an object through an atmosphere by creating drag, or in the case of ram-air parachutes, aerodynamic lift. Parachutes are usually made out of light, strong cloth, originally silk, now most commonly nylon...

systems are involved, because unless the parachute lines extend the canopy beyond the reverse flow region, the chute can fail to inflate and thus collapse. Parachutes deployed into wakes suffer dynamic pressure deficits which reduce their expected drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

forces. High-fidelity computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

simulations are often undertaken to model wake flows, although such modeling has uncertainties associated with turbulence modeling (for example RANS
Reynolds-averaged Navier-Stokes equations
The Reynolds-averaged Navier–Stokes equations are time-averagedequations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds...

versus LES
Large eddy simulation
Large eddy simulation is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and many of the issues unique to LES were first explored by Deardorff...

implementations), in addition to unsteady flow effects. Example applications include rocket stage separation and aircraft store separation.

## 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
Airy wave theory
In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational...

, 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
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

of a deep water wave is half of its phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the 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
Shock Waves
Shock Waves, , is a 1977 horror movie written and directed by Ken Wiederhorn...

.

In a medium like air, where the dispersion relation
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

is linear, i.e.
the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

c is the same for all wavelengths and the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

has the same value as well. The angle of the shock wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

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
Shock Waves
Shock Waves, , is a 1977 horror movie written and directed by Ken Wiederhorn...

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
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

that dictates the area of constructive interference and, in deep water, the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

is only half of the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at 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
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...

at the line AB, with the angle CAB = 62° because the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

of the wave has been chosen to be = 0.883 of the boat velocity. But the group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

is only half of the phase velocity
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

, so the wake actually forms along the line AD, where D is the mid-point 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
Sonic boom
A sonic boom is the sound associated with the shock waves created by an object traveling through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding much like an explosion...

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.

## Other effects

The above describes an ideal wake, where the body's means of propulsion has no other effect on the water. In practice the wave pattern between the V-shaped wave fronts is usually mixed with the effects of propeller backwash and eddying behind the boat's (usually square-ended) stern.

## Recreation

"No wake zones" may prohibit wakes in marinas, near moorings and within some distance of shore in order to facilitate recreation by other boats, and reduce the damage wakes cause.

Wakes are occasionally used recreationally. Swimmers, people riding personal watercraft, and aquatic mammals such as dolphins can ride the leading edge of a wake. In the sport of wakeboarding the wake is used as a jump. The wake is also used to propel a surfer in the sport of wakesurfing. In the sport of water polo
Water polo
Water polo is a team water sport. The playing team consists of six field players and one goalkeeper. The winner of the game is the team that scores more goals. Game play involves swimming, treading water , players passing the ball while being defended by opponents, and scoring by throwing into a...

, the ball carrier can swim while advancing the ball, propelled ahead with the wake created by alternating armstrokes in crawl stroke, a technique known as dribbling
Dribbling
In sports, dribbling refers to the maneuvering of a ball around a defender through short skillful taps or kicks with either the legs , hands , stick or swimming strokes...

.