Wake

小咪咪 2022-07-13 01:27 161阅读 0赞

https://en.wikipedia.org/wiki/Wake

From Wikipedia, the free encyclopedia

For other uses, see [Wake (disambiguation)][Wake _disambiguation].

[![280px-Fjordn_surface_wave_boat.jpg][]][280px-Fjordn_surface_wave_boat.jpg 1]

Kelvin wake pattern generated by a small boat.

In [fluid dynamics][], a **wake** may either be:

*  the region of recirculating flow immediately behind a moving or stationary blunt body, caused by [viscosity][], which may be accompanied by [flow separation][] and [turbulence][], or
 *  the wave pattern on the water surface downstream of an object in a flow, or produced by a moving object (e.g. a ship), caused by [pressure][] differences of the fluids above and below the [free surface][] and [gravity][] (or [surface tension][]), or both.

## Contents ##

\[hide\]

*  [1Wake effects caused by viscosity][]
 *  [2Waves by density differences, like a water surface][2Waves by density differences_ like a water surface]
    
     *  [2.1Kelvin wake pattern][]
 *  [3Gallery][]
 *  [4Other effects][]
 *  [5Recreation][]
    
     *  [5.1Image gallery][]
 *  [6See also][]
 *  [7References][]
 *  [8External links][]

## Wake effects caused by viscosity\[[edit][]\] ##

[![400px-Karmansche_Wirbelstr_kleine_Re.JPG][]][400px-Karmansche_Wirbelstr_kleine_Re.JPG 1]

[Visualisation][] of the [Kármán vortex street][K_rm_n vortex street] in the wake behind a circular cylinder in air; the flow is made visible through release of oil vapour in the air near the cylinder.

The wake is the region of disturbed flow (often [turbulent][]) downstream of a solid body moving through a fluid, caused by the flow of the [fluid][] around the body.

For a blunt body in [subsonic][] external flow, for example the [Apollo][] or [Orion][] capsules during descent and landing, the wake is massively [separated][flow separation] 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][] testing of aircraft, and is especially important when [parachute][] 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][] forces. High-fidelity [computational fluid dynamics][] simulations are often undertaken to model wake flows, although such modeling has uncertainties associated with [turbulence modeling][] (for example [RANS][] versus [LES][] implementations), in addition to unsteady flow effects. Example applications include rocket stage separation and aircraft store separation.

## Waves by density differences, like a water surface\[[edit][edit 1]\] ##

In incompressible fluids (liquids) such as water, a [bow][] 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][] is overcome or lost, usually by [friction][] or [dispersion][].

The non-dimensional parameter of interest is the [Froude number][].

[![344px-Wave_cloud.jpg][]][344px-Wave_cloud.jpg 1]

Wave cloud pattern in the wake of the [Île Amsterdam][le Amsterdam] (lower left, at the "tip" of the triangular formation of clouds) in the southern [Indian Ocean][]

[![299px-Cloud_Wakes_from_Juan_Fernandez_Islands.jpg][]][299px-Cloud_Wakes_from_Juan_Fernandez_Islands.jpg 1]

Cloud wakes from the [Juan Fernández Islands][Juan Fern_ndez Islands]

[![327px-Atmospheric_bow_waves01.jpg][]][327px-Atmospheric_bow_waves01.jpg 1]

Wake patterns in cloud cover over [Possession Island][], [East Island][], [Ile aux Cochons][], [Île des Pingouins][le des Pingouins]

### Kelvin wake pattern\[[edit][edit 2]\] ###

Waterfowl 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**.[\[1\]][1]

This pattern consists of two wake lines that form the arms of a chevron, V, with the source of the wake at the vertex of the V. For sufficiently slow motion, each wake line is offset from the path of the wake source by around arcsin(1/3) = 19.47° and is made up of feathery wavelets angled at roughly 53° to the path.

The inside of the V (of total opening 39° as indicated above) 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.[\[2\]][2]

However, the pattern changes *at high speeds* (only), viz., above a hull [Froude number][] of approximately 0.5. Then, as the source's speed increases, the transverse waves diminish and the points of maximum amplitude on the wavelets form a second V within the wake pattern, which grows narrower with the increased speed of the source.[\[3\]][3]

The angles in this pattern are not intrinsic properties of merely water: Any [isentropic][] and incompressible liquid with low viscosity will exhibit the *same* phenomenon. Furthermore, this phenomenon has nothing to do with turbulence. Everything discussed here is based on the linear theory of an ideal fluid, cf. [Airy wave theory][].

Parts of the pattern may be obscured by the effects of propeller wash, and tail eddies behind the boat's stern, and by the boat being a large object and not a point source. The water need not be stationary, but may be moving as in a large river, and the important consideration then is the velocity of the water relative to a boat or other object causing a wake.

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

\{\\displaystyle \\omega =\{\\sqrt \{gk\}\},\} ![\\omega = \\sqrt\{g k\},][omega _ _sqrt_g k]

where

*g* = the strength of the [gravity][gravity 1] field

*ω* is the [angular frequency][] in [radians][] per second

*k* = [angular wavenumber][] in radians per metre

"Deep" means that the depth is greater than half of the wavelength. This formula implies that *the [group velocity][] of a deep water wave is half of its [phase velocity][]*, which, in turn, goes as the square root of the wavelength. Two velocity parameters of importance for the wake pattern are:

*v* is the relative velocity of the water and the surface object that causes the wake.

*c* is the [phase velocity][] of a wave, varying with wave frequency.

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 *v* and dissipate into the surrounding water and are not easily observed. Other waves with phase speeds at or below *v*, however, are amplified through [constructive interference][] and form visible [shock waves][], stationary in position w.r.t. the boat.

[![300px-Bodensee_at_Lindau_-_DSC06962.JPG][]][300px-Bodensee_at_Lindau_-_DSC06962.JPG 1]

Classic duck wake

The angle *θ* between the phase [shock wave][] front and the path of the object is *θ* = arcsin(*c/v*). 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][].

All shock waves, that each by itself would have had an angle between 33° and 72°, are compressed into a narrow band of wake with angles between 15° and 19°, *with the strongest constructive interference at the outer edge* (angle arcsin(1/3) = 19.47°), placing the two arms of the V in the celebrated **Kelvin wake pattern**.

A concise geometric construction[\[4\]][4] demonstrates that, strikingly, this group shock angle w.r.t. the path of the boat, 19.47°, *for any and all of the above* *θ*, is actually *independent of* *v*, *c*, and *g*; it merely relies on the fact that the [group velocity][] is half of the [phase velocity][] *c*. On any planet, slow-swimming objects have "effective [Mach number][]" 3!

Concise argument for the universal opening of 39° of the slow-swimmers' Kelvin wake [\[show\]][show]

The [wavefronts][] of the wavelets in the wake are at 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. They 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 seen as small ripples on top of the interior transverse waves.

## Gallery\[[edit][edit 3]\] ##

*  [![120px-070107-Konigssee-dramaticwake.jpg][]][120px-070107-Konigssee-dramaticwake.jpg 1]
    
    Wake of a boat crossing an alpine lake (Koenigsee)
 *  [![120px-Bugwinkel39.jpg][]][120px-Bugwinkel39.jpg 1]
    
    39° ship wake
 *  [![120px-Wake.avon.gorge.2boats.arp.750pix.jpg][]][120px-Wake.avon.gorge.2boats.arp.750pix.jpg 1]
    
    The wakes of two slow-moving boats. The nearer boat has made a striking series of ruler-straight waves
 *  [![120px-Powwave1908s.png][]][120px-Powwave1908s.png 1]
    
    John Harvard Biles: "The Design and Construction of Ships, Vol. II: Stability, Resistance, Propulsion and Oscillation of Ships" (1908)
 *  [![120px-Aktersvall1.jpg][]][120px-Aktersvall1.jpg 1]
    
    Wake from a fast, small motorboat with an [outboard motor][]
 *  [![120px-CGI_Wake_in_Pond.jpg][]][120px-CGI_Wake_in_Pond.jpg 1]
    
    Wake simulation
 *  [![120px-Maple_density_plot_of_Kelvin_Wake.png][]][120px-Maple_density_plot_of_Kelvin_Wake.png 1]
    
    Density plot of Kelvin Wake integral
 *  ![File:Bergeshoevede GMS Silja 2 Decin CZ ENI 08451019 01.webmhd.webm][File_Bergeshoevede GMS Silja 2 Decin CZ ENI 08451019 01.webmhd.webm] 
    
    Video of barge wake
 *  ![File:Wake behind boat.gk.webm][File_Wake behind boat.gk.webm] 
    
    Wake behind a large motorboat

## Other effects\[[edit][edit 4]\] ##

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 wavefronts is usually mixed with the effects of propeller backwash and eddying behind the boat's (usually square-ended) stern.

## Recreation\[[edit][edit 5]\] ##

"No wake zones" may prohibit wakes in [marinas][], near moorings and within some distance of shore[\[5\]][5] in order to facilitate recreation by other boats and reduce the damage wakes cause. Powered [narrowboats][] on British canals are not permitted to create a breaking wash (a wake large enough to create a breaking wave) along the banks, as this erodes them. This rule normally restricts these vessels to 4 statute miles per hour or less.

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][], 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][].

### Image gallery\[[edit][edit 6]\] ###

*  [![120px-Surfer_about_to_catch_the_wake_from_a_ship.png][]][120px-Surfer_about_to_catch_the_wake_from_a_ship.png 1]
    
    Germany's only consistent surf spot is a giant wake from a ship
 *  [![120px-Sunseeker_Wake.JPG][]][120px-Sunseeker_Wake.JPG 1]
    
    Wake from a fast motor yacht on the [Indian River][] looking at the [17th Street Bridge][]
 *  [![120px-Wake_28Kilwater_29_behind_a_ferry.jpg][]][120px-Wake_28Kilwater_29_behind_a_ferry.jpg 1]
    
    Wake behind a [ferry][] in the [Baltic Sea][]
 *  [![120px-Starr_021108-0070_returning_from_Molokini_-_wake_28November_2002_29.jpg][]][120px-Starr_021108-0070_returning_from_Molokini_-_wake_28November_2002_29.jpg 1]
    
    Wake of a boat in the [Hawaiian Islands][]
 *  [![120px-Large_engine_boat_wake.jpg][]][120px-Large_engine_boat_wake.jpg 1]
    
    Wake of a ferryboat just off British Columbia, Canada.

## See also\[[edit][edit 7]\] ##

*  [Shock wave][shock wave]
 *  [Wake turbulence][]
 *  [Wakeboarding][]
 *  [Slipstream][]
 *  [Wakesurfing][]
 *  [Bow shock (aerodynamics)][Bow shock _aerodynamics]

## References\[[edit][edit 8]\] ##

1.  **[Jump up^][Jump up]** William Thomson (1887) ["On ship waves,"][On ship waves] *Institution of Mechanical Engineers, Proceedings*, **38** : 409–434 ; illustrations, [pp. 641-649.][]
2.  **[Jump up^][Jump up 1]** The corresponding theory for the transient Kelvin wakes has further been derived for the Marangoni (Shu, Jian-Jun (2004). "Transient Marangoni waves due to impulsive motion of a submerged body". *International Applied Mechanics*. **40** (6): 709–714. [arXiv][]:[1402.4474][]![Freely accessible][]. [Bibcode][]:[2004IAM....40..709S][]. [doi][]:[10.1023/B:INAM.0000041400.70961.1b][10.1023_B_INAM.0000041400.70961.1b].) and free-surface (Shu, Jian-Jun (2006). "Transient free-surface waves due to impulsive motion of a submerged source". *Underwater Technology*. **26** (4): 133–137. [doi][]:[10.3723/175605406782725023][10.3723_175605406782725023].) waves.
3.  **[Jump up^][Jump up 2]** The "hull Froude number" (*Fr*) of a ship is *Fr* = *U* / √*gL*, where *U* is the ship's speed, *g* is the acceleration of gravity at the earth's surface, and *L* is the length of the ship's hull, a characteristic wavelength. See Marc Rabaud and Frédéric Moisy (2013) "Ship wakes: Kelvin or Mach angle?," *Physical Review Letters*, **110** (21) : 214503. Available on-line at: [University of Paris, Sud][University of Paris_ Sud]; Alexandre Darmon, Michael Benzaquen, and Elie Raphaël (2014) "Kelvin wake pattern at large Froude numbers," *Journal of Fluid Mechanics*, **738** : R3-1 - R3-8. Available on-line at: [ESPCI ParisTech][]
4.  **[Jump up^][Jump up 3]** G.B. Whitham (1974). *Linear and Nonlinear Waves* (John Wiley & Sons Inc., 1974) pp 409-410 [Online scan][]
5.  **[Jump up^][Jump up 4]** BoatWakes.org, [Table of distances][]

## External links\[[edit][edit 9]\] ##

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*  [Erosion caused by boat wakes][]
 *  [NIST detailed derivation][]

[Categories][]:

*  [Fluid dynamics][]
 *  [Water wave][]

[Wake _disambiguation]: https://en.wikipedia.org/wiki/Wake_%28disambiguation%29
[280px-Fjordn_surface_wave_boat.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Fjordn_surface_wave_boat.jpg/280px-Fjordn_surface_wave_boat.jpg
[280px-Fjordn_surface_wave_boat.jpg 1]: https://en.wikipedia.org/wiki/File:Fjordn_surface_wave_boat.jpg
[fluid dynamics]: https://en.wikipedia.org/wiki/Fluid_dynamics
[viscosity]: https://en.wikipedia.org/wiki/Viscosity
[flow separation]: https://en.wikipedia.org/wiki/Flow_separation
[turbulence]: https://en.wikipedia.org/wiki/Turbulence
[pressure]: https://en.wikipedia.org/wiki/Pressure
[free surface]: https://en.wikipedia.org/wiki/Free_surface
[gravity]: https://en.wikipedia.org/wiki/Gravity_of_Earth
[surface tension]: https://en.wikipedia.org/wiki/Surface_tension
[1Wake effects caused by viscosity]: https://en.wikipedia.org/wiki/Wake#Wake_effects_caused_by_viscosity
[2Waves by density differences_ like a water surface]: https://en.wikipedia.org/wiki/Wake#Waves_by_density_differences.2C_like_a_water_surface
[2.1Kelvin wake pattern]: https://en.wikipedia.org/wiki/Wake#Kelvin_wake_pattern
[3Gallery]: https://en.wikipedia.org/wiki/Wake#Gallery
[4Other effects]: https://en.wikipedia.org/wiki/Wake#Other_effects
[5Recreation]: https://en.wikipedia.org/wiki/Wake#Recreation
[5.1Image gallery]: https://en.wikipedia.org/wiki/Wake#Image_gallery
[6See also]: https://en.wikipedia.org/wiki/Wake#See_also
[7References]: https://en.wikipedia.org/wiki/Wake#References
[8External links]: https://en.wikipedia.org/wiki/Wake#External_links
[edit]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=1
[400px-Karmansche_Wirbelstr_kleine_Re.JPG]: https://upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Karmansche_Wirbelstr_kleine_Re.JPG/400px-Karmansche_Wirbelstr_kleine_Re.JPG
[400px-Karmansche_Wirbelstr_kleine_Re.JPG 1]: https://en.wikipedia.org/wiki/File:Karmansche_Wirbelstr_kleine_Re.JPG
[Visualisation]: https://en.wikipedia.org/wiki/Flow_visualisation
[K_rm_n vortex street]: https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street
[turbulent]: https://en.wikipedia.org/wiki/Turbulent
[fluid]: https://en.wikipedia.org/wiki/Fluid
[subsonic]: https://en.wikipedia.org/wiki/Subsonic_flight
[Apollo]: https://en.wikipedia.org/wiki/Apollo_program
[Orion]: https://en.wikipedia.org/wiki/Orion_Multi-Purpose_Crew_Vehicle
[wind tunnel]: https://en.wikipedia.org/wiki/Wind_tunnel
[parachute]: https://en.wikipedia.org/wiki/Parachute
[dynamic pressure]: https://en.wikipedia.org/wiki/Dynamic_pressure
[drag]: https://en.wikipedia.org/wiki/Drag_%28physics%29
[computational fluid dynamics]: https://en.wikipedia.org/wiki/Computational_fluid_dynamics
[turbulence modeling]: https://en.wikipedia.org/wiki/Turbulence_modeling
[RANS]: https://en.wikipedia.org/wiki/Reynolds-averaged_Navier-Stokes_equations
[LES]: https://en.wikipedia.org/wiki/Large_eddy_simulation
[edit 1]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=2
[bow]: https://en.wikipedia.org/wiki/Bow_wave
[wave forms]: https://en.wikipedia.org/wiki/Wave_form
[energy]: https://en.wikipedia.org/wiki/Energy
[friction]: https://en.wikipedia.org/wiki/Friction
[dispersion]: https://en.wikipedia.org/wiki/Dispersion_%28water_waves%29
[Froude number]: https://en.wikipedia.org/wiki/Froude_number
[344px-Wave_cloud.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Wave_cloud.jpg/344px-Wave_cloud.jpg
[344px-Wave_cloud.jpg 1]: https://en.wikipedia.org/wiki/File:Wave_cloud.jpg
[le Amsterdam]: https://en.wikipedia.org/wiki/%C3%8Ele_Amsterdam
[Indian Ocean]: https://en.wikipedia.org/wiki/Indian_Ocean
[299px-Cloud_Wakes_from_Juan_Fernandez_Islands.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Cloud_Wakes_from_Juan_Fernandez_Islands.jpg/299px-Cloud_Wakes_from_Juan_Fernandez_Islands.jpg
[299px-Cloud_Wakes_from_Juan_Fernandez_Islands.jpg 1]: https://en.wikipedia.org/wiki/File:Cloud_Wakes_from_Juan_Fernandez_Islands.jpg
[Juan Fern_ndez Islands]: https://en.wikipedia.org/wiki/Juan_Fern%C3%A1ndez_Islands
[327px-Atmospheric_bow_waves01.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Atmospheric_bow_waves01.jpg/327px-Atmospheric_bow_waves01.jpg
[327px-Atmospheric_bow_waves01.jpg 1]: https://en.wikipedia.org/wiki/File:Atmospheric_bow_waves01.jpg
[Possession Island]: https://en.wikipedia.org/wiki/%C3%8Ele_de_la_Possession
[East Island]: https://en.wikipedia.org/wiki/%C3%8Ele_de_l%27Est
[Ile aux Cochons]: https://en.wikipedia.org/wiki/Ile_aux_Cochons
[le des Pingouins]: https://en.wikipedia.org/wiki/%C3%8Ele_des_Pingouins
[edit 2]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=3
[Lord Kelvin]: https://en.wikipedia.org/wiki/Lord_Kelvin
[1]: https://en.wikipedia.org/wiki/Wake#cite_note-1
[2]: https://en.wikipedia.org/wiki/Wake#cite_note-2
[3]: https://en.wikipedia.org/wiki/Wake#cite_note-3
[isentropic]: https://en.wikipedia.org/wiki/Isentropic
[Airy wave theory]: https://en.wikipedia.org/wiki/Airy_wave_theory
[dispersion relation of deep water waves]: https://en.wikipedia.org/wiki/Airy_wave_theory#Dispersion_relation
[omega _ _sqrt_g k]: https://wikimedia.org/api/rest_v1/media/math/render/svg/25f3f29c35a7aca477d0b9d71757102fbd1c6a09
[gravity 1]: https://en.wikipedia.org/wiki/Gravity
[angular frequency]: https://en.wikipedia.org/wiki/Angular_frequency
[radians]: https://en.wikipedia.org/wiki/Radian
[angular wavenumber]: https://en.wikipedia.org/wiki/Angular_wavenumber
[group velocity]: https://en.wikipedia.org/wiki/Group_velocity
[phase velocity]: https://en.wikipedia.org/wiki/Phase_velocity
[phase speeds]: https://en.wikipedia.org/wiki/Phase_speed
[constructive interference]: https://en.wikipedia.org/wiki/Constructive_interference
[shock waves]: https://en.wikipedia.org/wiki/Shock_waves
[300px-Bodensee_at_Lindau_-_DSC06962.JPG]: https://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Bodensee_at_Lindau_-_DSC06962.JPG/300px-Bodensee_at_Lindau_-_DSC06962.JPG
[300px-Bodensee_at_Lindau_-_DSC06962.JPG 1]: https://en.wikipedia.org/wiki/File:Bodensee_at_Lindau_-_DSC06962.JPG
[shock wave]: https://en.wikipedia.org/wiki/Shock_wave
[4]: https://en.wikipedia.org/wiki/Wake#cite_note-4
[Mach number]: https://en.wikipedia.org/wiki/Mach_number
[show]: https://en.wikipedia.org/wiki/Wake#
[wavefronts]: https://en.wikipedia.org/wiki/Wavefronts
[edit 3]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=4
[120px-070107-Konigssee-dramaticwake.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/070107-Konigssee-dramaticwake.jpg/120px-070107-Konigssee-dramaticwake.jpg
[120px-070107-Konigssee-dramaticwake.jpg 1]: https://en.wikipedia.org/wiki/File:070107-Konigssee-dramaticwake.jpg
[120px-Bugwinkel39.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Bugwinkel39.jpg/120px-Bugwinkel39.jpg
[120px-Bugwinkel39.jpg 1]: https://en.wikipedia.org/wiki/File:Bugwinkel39.jpg
[120px-Wake.avon.gorge.2boats.arp.750pix.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Wake.avon.gorge.2boats.arp.750pix.jpg/120px-Wake.avon.gorge.2boats.arp.750pix.jpg
[120px-Wake.avon.gorge.2boats.arp.750pix.jpg 1]: https://en.wikipedia.org/wiki/File:Wake.avon.gorge.2boats.arp.750pix.jpg
[120px-Powwave1908s.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/Powwave1908s.png/120px-Powwave1908s.png
[120px-Powwave1908s.png 1]: https://en.wikipedia.org/wiki/File:Powwave1908s.png
[120px-Aktersvall1.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Aktersvall1.jpg/120px-Aktersvall1.jpg
[120px-Aktersvall1.jpg 1]: https://en.wikipedia.org/wiki/File:Aktersvall1.jpg
[outboard motor]: https://en.wikipedia.org/wiki/Outboard_motor
[120px-CGI_Wake_in_Pond.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/CGI_Wake_in_Pond.jpg/120px-CGI_Wake_in_Pond.jpg
[120px-CGI_Wake_in_Pond.jpg 1]: https://en.wikipedia.org/wiki/File:CGI_Wake_in_Pond.jpg
[120px-Maple_density_plot_of_Kelvin_Wake.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/0/06/Maple_density_plot_of_Kelvin_Wake.png/120px-Maple_density_plot_of_Kelvin_Wake.png
[120px-Maple_density_plot_of_Kelvin_Wake.png 1]: https://en.wikipedia.org/wiki/File:Maple_density_plot_of_Kelvin_Wake.png
[File_Bergeshoevede GMS Silja 2 Decin CZ ENI 08451019 01.webmhd.webm]: https://upload.wikimedia.org/wikipedia/commons/thumb/5/56/Bergeshoevede_GMS_Silja_2_Decin_CZ_ENI_08451019_01.webmhd.webm/120px--Bergeshoevede_GMS_Silja_2_Decin_CZ_ENI_08451019_01.webmhd.webm.jpg
[File_Wake behind boat.gk.webm]: https://upload.wikimedia.org/wikipedia/commons/thumb/0/04/Wake_behind_boat.gk.webm/120px--Wake_behind_boat.gk.webm.jpg
[edit 4]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=5
[edit 5]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=6
[marinas]: https://en.wikipedia.org/wiki/Marinas
[5]: https://en.wikipedia.org/wiki/Wake#cite_note-5
[narrowboats]: https://en.wikipedia.org/wiki/Narrowboat
[water polo]: https://en.wikipedia.org/wiki/Water_polo
[crawl stroke]: https://en.wikipedia.org/wiki/Crawl_stroke
[dribbling]: https://en.wikipedia.org/wiki/Dribbling
[edit 6]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=7
[120px-Surfer_about_to_catch_the_wake_from_a_ship.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Surfer_about_to_catch_the_wake_from_a_ship.png/120px-Surfer_about_to_catch_the_wake_from_a_ship.png
[120px-Surfer_about_to_catch_the_wake_from_a_ship.png 1]: https://en.wikipedia.org/wiki/File:Surfer_about_to_catch_the_wake_from_a_ship.png
[120px-Sunseeker_Wake.JPG]: https://upload.wikimedia.org/wikipedia/en/thumb/b/bc/Sunseeker_Wake.JPG/120px-Sunseeker_Wake.JPG
[120px-Sunseeker_Wake.JPG 1]: https://en.wikipedia.org/wiki/File:Sunseeker_Wake.JPG
[Indian River]: https://en.wikipedia.org/wiki/Indian_River_%28Florida%29
[17th Street Bridge]: https://en.wikipedia.org/wiki/17th_Street_Bridge
[120px-Wake_28Kilwater_29_behind_a_ferry.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Wake_%28Kilwater%29_behind_a_ferry.jpg/120px-Wake_%28Kilwater%29_behind_a_ferry.jpg
[120px-Wake_28Kilwater_29_behind_a_ferry.jpg 1]: https://en.wikipedia.org/wiki/File:Wake_%28Kilwater%29_behind_a_ferry.jpg
[ferry]: https://en.wikipedia.org/wiki/Ferry
[Baltic Sea]: https://en.wikipedia.org/wiki/Baltic_Sea
[120px-Starr_021108-0070_returning_from_Molokini_-_wake_28November_2002_29.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Starr_021108-0070_returning_from_Molokini_-_wake_%28November_2002%29.jpg/120px-Starr_021108-0070_returning_from_Molokini_-_wake_%28November_2002%29.jpg
[120px-Starr_021108-0070_returning_from_Molokini_-_wake_28November_2002_29.jpg 1]: https://en.wikipedia.org/wiki/File:Starr_021108-0070_returning_from_Molokini_-_wake_%28November_2002%29.jpg
[Hawaiian Islands]: https://en.wikipedia.org/wiki/Hawaii
[120px-Large_engine_boat_wake.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Large_engine_boat_wake.jpg/120px-Large_engine_boat_wake.jpg
[120px-Large_engine_boat_wake.jpg 1]: https://en.wikipedia.org/wiki/File:Large_engine_boat_wake.jpg
[edit 7]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=8
[Wake turbulence]: https://en.wikipedia.org/wiki/Wake_turbulence
[Wakeboarding]: https://en.wikipedia.org/wiki/Wakeboarding
[Slipstream]: https://en.wikipedia.org/wiki/Slipstream
[Wakesurfing]: https://en.wikipedia.org/wiki/Wakesurfing
[Bow shock _aerodynamics]: https://en.wikipedia.org/wiki/Bow_shock_%28aerodynamics%29
[edit 8]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=9
[Jump up]: https://en.wikipedia.org/wiki/Wake#cite_ref-1
[On ship waves]: https://books.google.com/books?id=Q84-AQAAMAAJ&pg=PA409#v=onepage&q&f=false
[pp. 641-649.]: https://books.google.com/books?id=Q84-AQAAMAAJ&pg=PA641#v=onepage&q&f=false
[Jump up 1]: https://en.wikipedia.org/wiki/Wake#cite_ref-2
[arXiv]: https://en.wikipedia.org/wiki/ArXiv
[1402.4474]: https://arxiv.org/abs/1402.4474
[Freely accessible]: https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png
[Bibcode]: https://en.wikipedia.org/wiki/Bibcode
[2004IAM....40..709S]: http://adsabs.harvard.edu/abs/2004IAM....40..709S
[doi]: https://en.wikipedia.org/wiki/Digital_object_identifier
[10.1023_B_INAM.0000041400.70961.1b]: https://dx.doi.org/10.1023%2FB%3AINAM.0000041400.70961.1b
[10.3723_175605406782725023]: https://dx.doi.org/10.3723%2F175605406782725023
[Jump up 2]: https://en.wikipedia.org/wiki/Wake#cite_ref-3
[University of Paris_ Sud]: http://www.fast.u-psud.fr/~moisy/papers/2013_rabaud_prl.pdf
[ESPCI ParisTech]: http://www.pct.espci.fr/~elie/Publications/Kelvin_wake_pattern.pdf
[Jump up 3]: https://en.wikipedia.org/wiki/Wake#cite_ref-4
[Online scan]: https://archive.org/details/LinearAndNonlinearWaves
[Jump up 4]: https://en.wikipedia.org/wiki/Wake#cite_ref-5
[Table of distances]: http://boatwakes.homestead.com/files/wakesc.htm#distances
[edit 9]: https://en.wikipedia.org/w/index.php?title=Wake&action=edit&section=10
[Erosion caused by boat wakes]: http://boatwakes.info/
[NIST detailed derivation]: http://dlmf.nist.gov/36.13#F1
[Categories]: https://en.wikipedia.org/wiki/Help:Category
[Fluid dynamics]: https://en.wikipedia.org/wiki/Category:Fluid_dynamics
[Water wave]: https://en.wikipedia.org/wiki/Category:Water_waves