digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Surfing on shoaling and breaking waves.

In fluid dynamics, wave shoaling is the effect by which 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.[1] Shoaling waves will also exhibit a reduction in wavelength while the frequency remains constant.

In shallow water and parallel depth contours, non-breaking waves will increase in wave height as the wave packet enters shallower water.[2] This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

## Mathematics

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 waves, the energy flux associated with the wave motion, which is the product of the wave energy density with the group velocity, between two wave rays is a 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,

$\frac{d}{ds}(c_g E) = 0,$

where s is the co-ordinate along the wave ray and $c_g E$ is the energy flux per unit crest length. A decrease in group speed $c_g$ 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.[3][4]

Let us follow Phillips (1977)[5] and Mei (1989)[6] and denote the phase of a wave ray as

$S = S(\mathbf{x},t), 0\leq S<2\pi$.

The local wave number vector is the gradient of the phase function,

$\mathbf{k} = \nabla S$,

and the angular frequency is proportional to its local rate of change,

$\omega = -\partial S/\partial t$.

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;

$\frac{\partial k}{\partial t} + \frac{\partial \omega}{\partial x} = 0$.

Assuming stationary conditions ($\partial/\partial t = 0$), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as $\partial \omega / \partial x = 0$. As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length $\lambda = 2\pi/k$ because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

$\omega/k \equiv c = \sqrt{gh}$

dictates that

$k = \omega/\sqrt{gh}$,

i.e., a steady increase in k (decrease in $\lambda$) as the phase speed decreases under constant $\omega$.

## Notes

1. ^ Longuet-Higgins, M S; Stewart, RW (1964). "Radiation stresses in water waves; a physical discussion, with applications". Deep Sea Research and Oceanographic Abstracts 11 (4): 529–562.
2. ^ WMO (1998). Guide to Wave Analysis and Forecasting 702 (2 ed.). World Meteorological Organization. ISBN 92-63-12702-6.
3. ^ Dean, R.G.; Dalrymple, R.A. (1991). Water wave mechanics for engineers and scientists. Advanced Series on Ocean Engineering 2. Singapore: World Scientific. ISBN 978-981-02-0420-4.
4. ^ Goda, Y. (2000). Random Seas and Design of Maritime Structures. Advanced Series on Ocean Engineering 15 (2 ed.). Singapore: World Scientific. ISBN 978-981-02-3256-6.
5. ^ Phillips, Owen M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0-521-29801-6.
6. ^ Mei, Chiang C. (1989). The Applied Dynamics of Ocean Surface Waves. Singapore: World Scientific. ISBN 9971-5-0773-0.
 131 videos foundNext >
 Coastal Systems: Waves 2In the field with Simon Haslett, Professor of Physical Geography at the University of Wales, Newport, and author of Coastal Systems (2008, Routledge). Topics... Wave shoaling demonstrationThis is an experimental demonstration of wave shoaling. Here we have a water body of varying depth. Waves are generated in the deeper region of the tank. As ... coastal wavesHigh fidelity water wave simulation of the coastal process for video games. Based on the original Boussinesq equation modified to approximate non breaking wa... Lower Trestles Simulation of Wave Shoaling and Breaking using funwaveCThis is a simulation of a set of waves breaking at Lower Trestles using the funwaveC model. Silver color is the wave face and the wave goes blue-ish when wav... Standing Waves in Shoaling WaterWritten from the perspective of an American in the US attempting to get detailed accounts of the Tsunami, Standing Waves in Shoaling Water was written in mem... FM3 water wave introduction.mpgLooking at the waves in Coldingham Bay. Sea Life Center Berlin - Shoaling RingSea Life Center Berlin - Shoaling Ring. Three-time shoaling cyclist gets defensive, then passive-aggressive when confrontedHis language about 'speaking out of turn' and 'his space' was a bit strange. Was he a teacher who has had training in conflict management or what? Shoaling Shoaling Roadie Santa CruzSoquel Drive Santa Cruz California.
 131 videos foundNext >

We're sorry, but there's no news about "Wave shoaling" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter