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[[File:Surfer 2.jpg|upright=1.3|thumb|[[Surfing]] on shoaling and [[breaking wave]]s.]]
[[File:Phase and group velocity as a function of depth.svg|upright=1.3|thumb|The [[phase velocity]] ''c''<sub>p</sub> (blue) and [[group velocity]] ''c''<sub>g</sub> (red) as a function of water depth ''h'' for [[surface gravity wave]]s of constant [[frequency]], according to [[Airy wave theory]]. <br>Quantities have been made [[dimensionless]] using the [[Earth's gravity|gravitational acceleration]] ''g'' and [[frequency|period]] ''T'', with the deep-water [[wavelength]] given by ''L''<sub>0</sub> = ''gT''<sup>2</sup>/(2π) and the deep-water phase speed ''c''<sub>0</sub> = ''L''<sub>0</sub>/''T''. The grey line corresponds with the shallow-water limit ''c''<sub>p</sub> =''c''<sub>g</sub> = √(''gh''). <br>The phase speed – and thus also the wavelength ''L'' = ''c''<sub>p</sub>''T'' – decreases [[monotonic function|monotonically]] with decreasing depth. However, the group velocity first increases by 20% with respect to its deep-water value (of ''c''<sub>g</sub> = {{sfrac|1|2}}''c''<sub>0</sub> = ''gT''/(4π)) before decreasing in shallower depths.<ref>{{cite book |title=Oceanographical Engineering |last=Wiegel |first=R.L. |publisher=Dover Publications |year=2013 |isbn=978-0-486-16019-1 |page=17, Figure 2.4 }}</ref>]]
In [[fluid dynamics]], '''wave shoaling''' is the effect by which [[ocean surface waves|surface waves]] entering shallower water change in [[wave height]]. It is caused by the fact that the [[group velocity]], which is also the wave-energy transport velocity, changes with water depth. Under stationary conditions, a decrease in transport speed must be compensated by an increase in [[energy density]] in order to maintain a constant energy flux.<ref name=lon64>{{cite journal | last1=Longuet-Higgins |first1=M.S. |last2=Stewart |first2=R.W. | title = Radiation stresses in water waves; a physical discussion, with applications | journal = Deep-Sea Research and Oceanographic Abstracts | volume = 11 | number = 4 | pages = 529–562 | year = 1964 | url = http://chinacat.coastal.udel.edu/cieg682/protect/longuet-higgins-stewart-dsr64.pdf |doi=10.1016/0011-7471(64)90001-4 |bibcode=1964DSRA...11..529L }}</ref> Shoaling waves will also exhibit a reduction in [[wavelength]] while the [[frequency]] remains constant.
In [[Waves and shallow water|shallow water]] and parallel [[depth contour]]s, non-breaking waves will increase in wave height as the [[wave packet]] enters shallower water.<ref name=wmo98>{{cite book | last=WMO | year=1998 | title=Guide to Wave Analysis and Forecasting | volume=702 | publisher=World Meteorological Organization | edition=2 | isbn=92-63-12702-6 |
url=http://www.wmo.int/pages/prog/amp/mmop/documents/WMO%20No%20702/WMO702.pdf}}</ref> This is particularly evident for [[tsunami]]s as they wax in height when approaching a [[coast]]line, with devastating results.
==
Waves nearing the coast change wave height through different effects. Some of the important wave processes are [[refraction]], [[diffraction]], [[reflection (physics)|reflection]], [[wave breaking]], [[wave–current interaction]], friction, wave growth due to the wind, and ''wave shoaling''. In the absence of the other effects, wave shoaling is the change of wave height that occurs solely due to changes in mean water depth – without changes in wave propagation direction and [[dissipation]]. Pure wave shoaling occurs for [[wave crest|long-crested]] waves propagating [[perpendicular]] to the parallel depth [[contour line]]s of a mildly sloping sea-bed. Then the wave height <math>H</math> at a certain location can be expressed as:<ref name=god00/><ref name=dal91/>
:<math>H = K_S\; H_0,</math>
with <math>K_S</math> the shoaling coefficient and <math>H_0</math> the wave height in deep water. The shoaling coefficient <math>K_S</math> depends on the local water depth <math>h</math> and the wave [[frequency]] <math>f</math> (or equivalently on <math>h</math> and the wave period <math>T=1/f</math>). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth <math>h</math> is larger than about half the deep-water [[wavelength]] <math>L_0=gT^2/(2\pi).</math>
==Physics==
[[File:Propagation du tsunami en profondeur variable.gif|right|thumb|When waves enter shallow water they slow down. Under stationary conditions, the wave length is reduced. The energy flux must remain constant and the reduction in group (transport) speed is compensated by an increase in wave height (and thus wave energy density).]]
[[File:Mavericks wave diagram.gif|thumb|right|Convergence of wave rays (reduction of width <math>b</math>) at [[Mavericks, California]], producing high [[surfing]] waves. The red lines are the wave rays; the blue lines are the [[wavefront]]s. The distances between neighboring wave rays vary towards the coast because of [[refraction]] by [[bathymetry]] (depth variations). The distance between wavefronts (i.e. the wavelength) reduces towards the coast because of the decreasing [[phase speed]].]]
[[File:Shoaling coefficient as a function of depth.svg|thumb|right|Shoaling coefficient <math>K_S</math> as a function of relative water depth <math>h/L_0,</math> describing the effect of wave shoaling on the [[wave height]] – based on [[conservation of energy]] and results from [[Airy wave theory]]. The local wave height <math>H</math> at a certain mean water depth <math>h</math> is equal to <math>H=K_S\;H_0,</math> with <math>H_0</math> the wave height in deep water (i.e. when the water depth is greater than about half the [[wavelength]]). The shoaling coefficient <math>K_S</math> depends on <math>h/L_0,</math> where <math>L_0</math> is the wavelength in deep water: <math>L_0=gT^2/(2\pi),</math> with <math>T</math> the [[frequency|wave period]] and <math>g</math> the [[gravity of Earth]]. The blue line is the shoaling coefficient according to [[Green's law]] for waves in shallow water, i.e. valid when the water depth is less than 1/20 times the local wavelength <math>L=T\,\sqrt{gh}.</math><ref name=dal91/>]]
For non-[[breaking wave]]s, the [[energy flux]] associated with the wave motion, which is the product of the [[wave energy]] density with the [[group velocity]], between two [[ray tracing (physics)|wave rays]] is a [[conservation of energy|conserved quantity]] (i.e. a constant when following the energy of a [[wave packet]] from one location to another). Under stationary conditions the total energy transport must be constant along the wave ray – as first shown by [[William Burnside]] in 1915.<ref>{{cite journal | title = On the modification of a train of waves as it advances into shallow water | first = W. | last = Burnside |author-link = William Burnside | year = 1915 | journal = Proceedings of the London Mathematical Society | series = Series 2 | volume = 14 | pages = 131–133 | doi = 10.1112/plms/s2_14.1.131 | url = https://zenodo.org/record/1447774 }}</ref>
For waves affected by refraction and shoaling (i.e. within the [[geometric optics]] approximation), the [[rate of change (mathematics)|rate of change]] of the wave energy transport is:<ref name=dal91/>
:<math>\frac{d}{ds}(b c_g E) = 0,</math>
where <math>s</math> is the co-ordinate along the wave ray and <math>b c_g E</math> is the energy flux per unit crest length. A decrease in group speed <math>c_g</math> and distance between the wave rays <math>b</math> must be compensated by an increase in energy density <math>E</math>. This can be formulated as a shoaling coefficient relative to the wave height in deep water.<ref name=dal91>{{cite book | title=Water wave mechanics for engineers and scientists | author=Dean, R.G. |author2=Dalrymple, R.A. | year=1991 | series=Advanced Series on Ocean Engineering | volume=2 | publisher=World Scientific | location=Singapore | url = https://books.google.com/books?id=9-M4U_sfin8C&q=Water%20wave%20mechanics%20for%20engineers%20and%20scientists&pg=PP1 | isbn=978-981-02-0420-4 }}</ref><ref name=god00>{{cite book | first=Y. | last=Goda | title=Random Seas and Design of Maritime Structures | year=2010 | series=Advanced Series on Ocean Engineering | volume=33 | publisher=World Scientific | location=Singapore | edition=3 | url = https://books.google.com/books?id=kneahaZ-2UQC&q=Random%20Seas%20and%20Design%20of%20Maritime%20Structures.%20Advanced%20Series%20on%20Ocean%20Engineering&pg=PP1 | isbn=978-981-4282-39-0 |pages=10–13 & 99–102 }}</ref>
For shallow water, when the [[wavelength]] is much larger than the water depth – in case of a constant ray distance <math>b</math> (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies [[Green's law]]:
:<math>H\, \sqrt[4]{h} = \text{constant},</math>
with <math>h</math> the mean water depth, <math>H</math> the wave height and <math>\sqrt[4]{h}</math> the [[fourth root]] of <math>h.</math>
==Water wave refraction{{anchor|Refraction}}==
Following [[Owen Martin Phillips|Phillips]] (1977) and [[Chiang C. Mei|Mei]] (1989),<ref name=phi77>{{cite book | first=Owen M. | last=Phillips | author-link=Owen Martin Phillips |year=1977 | title=The dynamics of the upper ocean (2nd ed.) | isbn=0-521-29801-6 | publisher=Cambridge University Press | url=https://www.google.com/books?id=fYk6AAAAIAAJ&lpg=PP9&ots=htfM77pfaz&dq=phillips%20dynamics%20of%20the%20upper%20ocean&lr=&pg=PA23#v=onepage&q=phillips%20dynamics%20of%20the%20upper%20ocean&f=false}}</ref><ref name=mei89>{{cite book | first=Chiang C. | last=Mei | author-link=Chiang C. Mei | year=1989 | title=The Applied Dynamics of Ocean Surface Waves | publisher=World Scientific | location = Singapore | url=https://www.google.com/books?id=LKCQorj3XZwC&lpg=PA1&ots=qk-kn82-z2&dq=mei%201989&lr=&pg=PA62#v=onepage&q=mei%201989%20page%2063&f=false | isbn=9971-5-0773-0}}</ref> denote the [[Phase (waves)|phase]] of a [[Ray (optics)|wave ray]] as
:<math>S = S(\mathbf{x},t), \qquad 0\leq S<2\pi</math>.
The local [[wave vector|wave number vector]] is the gradient of the phase function,
:<math>\mathbf{k} = \nabla S</math>,
and the [[angular frequency]] is proportional to its local rate of change,
:<math>\omega = -\partial S/\partial t</math>.
Simplifying to one dimension and cross-differentiating it is now easily seen that the above definitions indicate simply that the rate of change of wavenumber is balanced by the convergence of the frequency along a ray;
:<math>\frac{\partial k}{\partial t} + \frac{\partial \omega}{\partial x} = 0</math>.
Assuming stationary conditions (<math>\partial/\partial t = 0</math>), this implies that wave crests are conserved and the [[frequency]] must remain constant along a wave ray as <math>\partial \omega / \partial x = 0</math>.
As waves enter shallower waters, the decrease in [[group velocity]] caused by the reduction in water depth leads to a reduction in [[wave length]] <math>\lambda = 2\pi/k</math> because the nondispersive [[Waves and shallow water|shallow water limit]] of the [[Dispersion (water waves)|dispersion relation]] for the wave [[phase speed]],
:<math>\omega/k \equiv c = \sqrt{gh}</math>
dictates that
:<math>k = \omega/\sqrt{gh}</math>,
i.e., a steady increase in ''k'' (decrease in <math>\lambda</math>) as the [[phase speed]] decreases under constant <math>\omega</math>.
==See also==
{{refbegin|2}}
*{{annotated link|Airy wave theory}}
*{{annotated link|Breaking wave}}
*{{annotated link|Dispersion (water waves)}}
*{{annotated link|Ocean surface waves}}
*{{annotated link|Shallow water equations}}
*{{annotated link|Shoal}}
*{{annotated link|Waves and shallow water}}
*{{annotated link|Wave height}}
*{{annotated link|Ursell number}}
{{refend}}
==Notes==
{{reflist|2}}
==External links==
{{Commons category|Wave shoaling}}
*[http://www.encora.eu/coastalwiki/Wave_transformation#Shoaling Wave transformation at Coastal Wiki]
{{coastal geography}}
{{physical oceanography}}
{{Underwater diving|scidiv}}
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[[Category:Coastal geography]]
[[Category:Physical oceanography]]
[[Category:Water waves]]
[[Category:Oceanographical terminology]]
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