Wave shoaling

In fluid dynamics, wave shoaling is the effect by which 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.[2] 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.[3] This is particularly evident for tsunamis as they wax in height when approaching a coastline, with devastating results.

Surfer 2
Surfing on shoaling and breaking waves.
Phase and group velocity as a function of depth
The phase velocity cp (blue) and group velocity cg (red) as a function of water depth h for surface gravity waves of constant frequency, according to Airy wave theory.
Quantities have been made dimensionless using the gravitational acceleration g and period T, with the deep-water wavelength given by L0 = gT2/(2π) and the deep-water phase speed c0 = L0/T. The grey line corresponds with the shallow-water limit cp =cg = √(gh).
The phase speed – and thus also the wavelength L = cpT – decreases monotonically with decreasing depth. However, the group velocity first increases by 20% with respect to its deep-water value (of cg = 1/2c0 = gT/(4π)) before decreasing in shallower depths.[1]


Waves nearing the coast change wave height through different effects. Some of the important wave processes are refraction, diffraction, 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 long-crested waves propagating perpendicular to the parallel depth contour lines of a mildly sloping sea-bed. Then the wave height at a certain location can be expressed as:[4][5]

with the shoaling coefficient and the wave height in deep water. The shoaling coefficient depends on the local water depth and the wave frequency (or equivalently on and the wave period ). Deep water means that the waves are (hardly) affected by the sea bed, which occurs when the depth is larger than about half the deep-water wavelength


Propagation du tsunami en profondeur variable
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).
Mavericks wave diagram
Convergence of wave rays (reduction of width ) at Mavericks, California, producing high surfing waves. The red lines are the wave rays; the blue lines are the wavefronts. 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.
Shoaling coefficient as a function of depth
Shoaling coefficient as a function of relative water depth 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 at a certain mean water depth is equal to with the wave height in deep water (i.e. when the water depth is greater than about half the wavelength). The shoaling coefficient depends on where is the wavelength in deep water: with the wave period and 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 [5]

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 – as first shown by William Burnside in 1915.[6] For waves affected by refraction and shoaling (i.e. within the geometric optics approximation), the rate of change of the wave energy transport is:[5]

where is the co-ordinate along the wave ray and is the energy flux per unit crest length. A decrease in group speed and distance between the wave rays must be compensated by an increase in energy density . This can be formulated as a shoaling coefficient relative to the wave height in deep water.[5][4]

For shallow water, when the wavelength is much larger than the water depth – in case of a constant ray distance (i.e. perpendicular wave incidence on a coast with parallel depth contours) – wave shoaling satisfies Green's law:

with the mean water depth, the wave height and the fourth root of

Water wave refraction

Following Phillips (1977) and Mei (1989),[7][8] denote the phase of a wave ray as


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


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


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;


Assuming stationary conditions (), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as . As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length because the nondispersive shallow water limit of the dispersion relation for the wave phase speed,

dictates that


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

See also


  1. ^ Wiegel, R.L. (2013). Oceanographical Engineering. Dover Publications. p. 17, Figure 2.4. ISBN 978-0-486-16019-1.
  2. ^ Longuet-Higgins, M.S.; Stewart, R.W. (1964). "Radiation stresses in water waves; a physical discussion, with applications" (PDF). Deep-Sea Research and Oceanographic Abstracts. 11 (4): 529–562. doi:10.1016/0011-7471(64)90001-4.
  3. ^ WMO (1998). Guide to Wave Analysis and Forecasting (PDF). 702 (2 ed.). World Meteorological Organization. ISBN 92-63-12702-6.
  4. ^ a b Goda, Y. (2010). Random Seas and Design of Maritime Structures. Advanced Series on Ocean Engineering. 33 (3 ed.). Singapore: World Scientific. pp. 10–13 & 99–102. ISBN 978-981-4282-39-0.
  5. ^ a b c d 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.
  6. ^ Burnside, W. (1915). "On the modification of a train of waves as it advances into shallow water". Proceedings of the London Mathematical Society. Series 2. 14: 131–133. doi:10.1112/plms/s2_14.1.131.
  7. ^ Phillips, Owen M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0-521-29801-6.
  8. ^ Mei, Chiang C. (1989). The Applied Dynamics of Ocean Surface Waves. Singapore: World Scientific. ISBN 9971-5-0773-0.

External links

1965 Ceram Sea earthquake

The 1965 Ceram Sea earthquake occurred on January 24 at 00:11 UTC with a moment magnitude of 8.2 and its epicenter was located just off the southwestern coast of Sanana Island in eastern Indonesia. The event occurred at a depth of 28 kilometers under the Ceram Sea, and a tsunami was generated which caused damage in Sanana, Buru, and Mangole. During the tsunami three consecutive run-ups were reported in Seram Island, and a four-meter run-up was reported at Buru Island.A series of tremors were reported during the week leading up to the mainshock. The number of people reported dead was 71 and up to 3,000 buildings and a total of 14 bridges were destroyed by both the earthquake and tsunami on Sanana.

Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.


The coast, also known as the coastline or seashore, is the area where land meets the sea or ocean, or a line that forms the boundary between the land and the ocean or a lake. A precise line that can be called a coastline cannot be determined due to the coastline paradox.

The term coastal zone is a region where interaction of the sea and land processes occurs. Both the terms coast and coastal are often used to describe a geographic location or region (e.g., New Zealand's West Coast, or the East and West Coasts of the United States). Edinburgh is an example city on the coast of Great Britain.

The term pelagic coast refers to a coast that fronts the open ocean, as opposed to a more sheltered coast in a gulf or bay. A shore, on the other hand, can refer to parts of land adjoining any large body of water, including oceans (seashore) and lakes (lake shore). Similarly, the somewhat related term stream bed or stream bank refers to the land alongside or sloping down to a river (riverbank) or body of water smaller than a lake. Bank is also used in some parts of the world to refer to an artificial ridge of earth intended to retain the water of a river or pond; in other places this may be called a levee.

While many scientific experts might agree on a common definition of the term coast, the delineation of the extents of a coast differ according to jurisdiction, with many scientific and government authorities in various countries differing for economic and social policy reasons. According to the UN atlas, 44% of people live within 150 km (93 mi) of the sea.

Coastal geography

Coastal geography is the study of the constantly changing region between the ocean and the land, incorporating both the physical geography (i.e. coastal geomorphology, geology and oceanography) and the human geography (sociology and history) of the coast. It includes understanding coastal weathering processes, particularly wave action, sediment movement and weather, and the ways in which humans interact with the coast


A fajã (Portuguese pronunciation: [fa'ʒɐ̃]) is a Portuguese-language term for a supratidal talus-platform geology constructed from landslides or lava flows, that are relatively common coastal features, occurring on the toe of cliffs. Although they exist throughout the world, they are distinct features of the islands of the Azores.

Green's law

In fluid dynamics, Green's law describes the evolution of non-breaking surface gravity waves propagating in shallow water of gradually varying depth and width. The law is named after George Green. In its simplest form, for wavefronts and depth contours parallel to each other (and the coast), it states:


where and are the wave heights at two different locations – 1 and 2 respectively – where the wave passes, and and are the mean water depths at the same two locations.

Green's law is often used in coastal engineering for the modelling of long shoaling waves on a beach, with "long" meaning wavelengths in excess of about twenty times the mean water depth. Tsunamis shoal (change their height) in accordance with this law, as they propagate – governed by refraction and diffraction – through the ocean and up the continental shelf. Very close to (and running up) the coast nonlinear effects become important and Green's law no longer applies.

Index of physics articles (W)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Index of wave articles

This is a list of Wave topics.


An island or isle is any piece of sub-continental land that is surrounded by water. Very small islands such as emergent land features on atolls can be called islets, skerries, cays or keys. An island in a river or a lake island may be called an eyot or ait, and a small island off the coast may be called a holm. A grouping of geographically or geologically related islands is called an archipelago, such as the Philippines.

An island may be described as such, despite the presence of an artificial land bridge; examples are Singapore and its causeway, and the various Dutch delta islands, such as IJsselmonde. Some places may even retain "island" in their names for historical reasons after being connected to a larger landmass by a land bridge or landfill, such as Coney Island and Coronado Island, though these are, strictly speaking, tied islands. Conversely, when a piece of land is separated from the mainland by a man-made canal, for example the Peloponnese by the Corinth Canal or Marble Hill in northern Manhattan during the time between the building of the United States Ship Canal and the filling-in of the Harlem River which surrounded the area, it is generally not considered an island.

There are two main types of islands in the sea: continental and oceanic. There are also artificial islands.


Mudflats or mud flats, also known as tidal flats, are coastal wetlands that form in intertidal areas where sediments have been deposited by tides or rivers. A recent global analysis suggested they are as extensive globally as mangroves. They are found in sheltered areas such as bays, bayous, lagoons, and estuaries. Mudflats may be viewed geologically as exposed layers of bay mud, resulting from deposition of estuarine silts, clays and marine animal detritus. Most of the sediment within a mudflat is within the intertidal zone, and thus the flat is submerged and exposed approximately twice daily.

In the past tidal flats were considered unhealthy, economically unimportant areas and were often dredged and developed into agricultural land. Several especially shallow mudflat areas, such as the Wadden Sea, are now popular among those practising the sport of mudflat hiking.

On the Baltic Sea coast of Germany in places, mudflats are exposed not by tidal action, but by wind-action driving water away from the shallows into the sea. These wind-affected mudflats are called windwatts in German.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.


In oceanography, geomorphology, and earth sciences, a shoal is a natural submerged ridge, bank, or bar that consists of, or is covered by, sand or other unconsolidated material, and rises from the bed of a body of water to near the surface. Often it refers to those submerged ridges, banks, or bars that rise near enough to the surface of a body of water as to constitute a danger to navigation. Shoals are also known as sandbanks, sandbars, or gravelbars. Two or more shoals that are either separated by shared troughs or interconnected by past or present sedimentary and hydrographic processes are referred to as a shoal complex.The term shoal is also used in a number of ways that can be either similar or quite different from how it is used in the geologic, geomorphic, and oceanographic literature. Sometimes, this terms refers to either any relatively shallow place in a stream, lake, sea, or other body of water; a rocky area on the sea floor within an area mapped for navigation purposes; a growth of vegetation on the bottom of a deep lake that occurs at any depth; and as a verb for the process of proceeding from a greater to a lesser depth of water.

Surf zone

As ocean surface waves come closer to shore they break, forming the foamy, bubbly surface called surf. The region of breaking waves defines the surf zone. After breaking in the surf zone, the waves (now reduced in height) continue to move in, and they run up onto the sloping front of the beach, forming an uprush of water called swash. The water then runs back again as backswash. The nearshore zone where wave water comes onto the beach is the surf zone. The water in the surf zone, or breaker zone, is shallow, usually between 5 and 10 m (16 and 33 ft) deep; this causes the waves to be unstable.


A tsunami (from Japanese: 津波, lit. 'harbour wave';

English pronunciation: soo-NAH-mee or ) or tidal wave,, also known as a seismic sea wave, is a series of waves in a water body caused by the displacement of a large volume of water, generally in an ocean or a large lake. Earthquakes, volcanic eruptions and other underwater explosions (including detonations, landslides, glacier calvings, meteorite impacts and other disturbances) above or below water all have the potential to generate a tsunami. Unlike normal ocean waves, which are generated by wind, or tides, which are generated by the gravitational pull of the Moon and the Sun, a tsunami is generated by the displacement of water.

Tsunami waves do not resemble normal undersea currents or sea waves because their wavelength is far longer. Rather than appearing as a breaking wave, a tsunami may instead initially resemble a rapidly rising tide. For this reason, it is often referred to as a "tidal wave", although this usage is not favoured by the scientific community because it might give the false impression of a causal relationship between tides and tsunamis. Tsunamis generally consist of a series of waves, with periods ranging from minutes to hours, arriving in a so-called "internal wave train". Wave heights of tens of metres can be generated by large events. Although the impact of tsunamis is limited to coastal areas, their destructive power can be enormous, and they can affect entire ocean basins. The 2004 Indian Ocean tsunami was among the deadliest natural disasters in human history, with at least 230,000 people killed or missing in 14 countries bordering the Indian Ocean.

The Ancient Greek historian Thucydides suggested in his 5th century BC History of the Peloponnesian War that tsunamis were related to submarine earthquakes, but the understanding of tsunamis remained slim until the 20th century and much remains unknown. Major areas of current research include determining why some large earthquakes do not generate tsunamis while other smaller ones do; accurately forecasting the passage of tsunamis across the oceans; and forecasting how tsunami waves interact with shorelines.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

Wave setup

In fluid dynamics, wave setup is the increase in mean water level due to the presence of breaking waves. Similarly, wave setdown is a wave-induced decrease of the mean water level before the waves break (during the shoaling process). For short, the whole phenomenon is often denoted as wave setup, including both increase and decrease of mean elevation. This setup is primarily present in and near the coastal surf zone. Besides a spatial variation in the (mean) wave setup, also a variation in time may be present – known as surf beat – causing infragravity wave radiation.

Wave setup can be mathematically modeled by considering the variation in radiation stress (Longuet-Higgins & Stewart 1962). Radiation stress is the tensor of excess horizontal-momentum fluxes due to the presence of the waves.

Waves and shallow water

When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies.

Ocean zones
Sea level


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