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[[File:Surfer 2.jpg|right|thumb|300px|[[Surfing]] on shoaling and [[breaking wave]]s.]]
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In [[fluid dynamics]], '''wave shoaling''' is the effect by which [[ocean surface waves|surface waves]] entering shallower water increase in [[wave height]] (which is about twice the [[amplitude]]). It is caused by the fact that the [[group velocity]], which is also the wave-energy transport velocity, decreases with the reduction of water depth. Under stationary conditions, this 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 | author=Longuet-Higgins, M S | coauthors = Stewart, RW | 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 }}</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.
 
==Mathematics==
[[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).]]
 
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,
:<math>\frac{d}{ds}(c_g E) = 0,</math>
where ''s'' is the co-ordinate along the wave ray and <math>c_g E</math> is the energy flux per unit crest length. A decrease in group speed <math>c_g</math> must be compensated by an increase in energy density ''E''. 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. | coauthors=Dalrymple, R.A. | year=1991 | series=Advanced Series on Ocean Engineering | volume=2 | publisher=World Scientific | location=Singapore | url = http://books.google.com/books?id=9-M4U_sfin8C&lpg=PP1&dq=Water%20wave%20mechanics%20for%20engineers%20and%20scientists&pg=PP1#v=onepage&q=&f=false | 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=2000 | series=Advanced Series on Ocean Engineering | volume=15 | publisher=World Scientific | location=Singapore | edition=2 | url = http://books.google.com/books?id=kneahaZ-2UQC&lpg=PR2&dq=Random%20Seas%20and%20Design%20of%20Maritime%20Structures.%20Advanced%20Series%20on%20Ocean%20Engineering&pg=PP1#v=onepage&q=&f=false | isbn=978-981-02-3256-6 }}</ref>
 
Let us follow Phillips (1977)<ref name=phi77>{{cite book | first=Owen M. | last=Phillips | year=1977 | title=The dynamics of the upper ocean (2nd ed.) | isbn=0-521-29801-6
| publisher=Cambridge University Press | url=http://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>
and  
Mei (1989)<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=http://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> and denote the [[Phase (waves)|phase]] of a [[Ray (optics)|wave ray]] as
:<math>S = S(\mathbf{x},t), 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}}
*[[Airy wave theory]]
*[[Breaking wave]]
*[[Dispersion (water waves)]]
*[[Ocean surface waves]]
*[[Shallow water equations]]
*[[Shoal]]
*[[Waves and shallow water]]
*[[Wave height]]
*[[Ursell number]]
{{refend}}
 
==Notes==
{{reflist|2}}
 
==External links==
{{Commons category|Ocean surface waves}}
*[http://www.encora.eu/coastalwiki/Wave_transformation#Shoaling Wave transformation at Coastal Wiki]
 
{{coastal geography}}
{{physical oceanography}}
 
[[Category:Coastal geography]]
[[Category:Physical oceanography]]
[[Category:Water waves]]

Latest revision as of 00:58, 6 August 2014

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