Mild-slope equation

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting geomorphology changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.

A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972.[1][2][3] Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.

In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.

CGWAVE Tedious Creek MD
Simulation of wave penetration—involving diffraction and refraction—into Tedious Creek, Maryland, using CGWAVE (which solves the mild-slope equation).

Formulation for monochromatic wave motion

For monochromatic waves according to linear theory—with the free surface elevation given as and the waves propagating on a fluid layer of mean water depth —the mild-slope equation is:[4]


  • is the complex-valued amplitude of the free-surface elevation
  • is the horizontal position;
  • is the angular frequency of the monochromatic wave motion;
  • is the imaginary unit;
  • means taking the real part of the quantity between braces;
  • is the horizontal gradient operator;
  • is the divergence operator;
  • is the wavenumber;
  • is the phase speed of the waves and
  • is the group speed of the waves.

The phase and group speed depend on the dispersion relation, and are derived from Airy wave theory as:[5]


For a given angular frequency , the wavenumber has to be solved from the dispersion equation, which relates these two quantities to the water depth .

Transformation to an inhomogeneous Helmholtz equation

Through the transformation

the mild slope equation can be cast in the form of an inhomogeneous Helmholtz equation:[4][6]

where is the Laplace operator.

Propagating waves

In spatially coherent fields of propagating waves, it is useful to split the complex amplitude in its amplitude and phase, both real valued:[7]


  • is the amplitude or absolute value of and
  • is the wave phase, which is the argument of

This transforms the mild-slope equation in the following set of equations (apart from locations for which is singular):[7]


  • is the average wave-energy density per unit horizontal area (the sum of the kinetic and potential energy densities),
  • is the effective wavenumber vector, with components
  • is the effective group velocity vector,
  • is the fluid density, and
  • is the acceleration by the Earth's gravity.

The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy is transported in the -direction normal to the wave crests (in this case of pure wave motion without mean currents).[7] The effective group speed is different from the group speed

The first equation states that the effective wavenumber is irrotational, a direct consequence of the fact it is the derivative of the wave phase , a scalar field. The second equation is the eikonal equation. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with the splitting into amplitude and phase leads to consistent-varying and meaningful fields of and . Otherwise, κ2 can even become negative. When the diffraction effects are totally neglected, the effective wavenumber κ is equal to , and the geometric optics approximation for wave refraction can be used.[7]

Derivation of the mild-slope equation

The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach.[4][8] The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. These assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using potential flow theory.

The following time-dependent equations give the evolution of the free-surface elevation and free-surface potential [4]

From the two evolution equations, one of the variables or can be eliminated, to obtain the time-dependent form of the mild-slope equation:[4]

and the corresponding equation for the free-surface potential is identical, with replaced by The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around

Monochromatic waves

Consider monochromatic waves with complex amplitude and angular frequency

with and chosen equal to each other, Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:[4]

Applicability and validity of the mild-slope equation

The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3.[11] However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.


  1. ^ Eckart, C. (1952), "The propagation of gravity waves from deep to shallow water", Circular 20, National Bureau of Standards: 165–173
  2. ^ Berkhoff, J. C. W. (1972), "Computation of combined refraction–diffraction", Proceedings 13th International Conference on Coastal Engineering, Vancouver, pp. 471–490
  3. ^ Berkhoff, J. C. W. (1976), Mathematical models for simple harmonic linear water wave models; wave refraction and diffraction (PDF) (PhD. Thesis), Delft University of Technology
  4. ^ a b c d e f g h i j Dingemans (1997, pp. 248–256 & 378–379)
  5. ^ Dingemans (1997, p. 49)
  6. ^ Mei (1994, pp. 86–89)
  7. ^ a b c d Dingemans (1997, pp. 259–262)
  8. ^ Booij, N. (1981), Gravity waves on water with non-uniform depth and current (PDF) (PhD. Thesis), Delft University of Technology
  9. ^ Luke, J. C. (1967), "A variational principle for a fluid with a free surface", Journal of Fluid Mechanics, 27 (2): 395–397, Bibcode:1967JFM....27..395L, doi:10.1017/S0022112067000412
  10. ^ Miles, J. W. (1977), "On Hamilton's principle for surface waves", Journal of Fluid Mechanics, 83 (1): 153–158, Bibcode:1977JFM....83..153M, doi:10.1017/S0022112077001104
  11. ^ Booij, N. (1983), "A note on the accuracy of the mild-slope equation", Coastal Engineering, 7 (1): 191–203, doi:10.1016/0378-3839(83)90017-0


  • Dingemans, M. W. (1997), Water wave propagation over uneven bottoms, Advanced Series on Ocean Engineering, 13, World Scientific, Singapore, ISBN 981-02-0427-2, OCLC 36126836, 2 Parts, 967 pages.
  • Liu, P. L.-F. (1990), "Wave transformation", in B. Le Méhauté and D. M. Hanes (ed.), Ocean Engineering Science, The Sea, 9A, Wiley Interscience, pp. 27–63, ISBN 0-471-52856-0
  • Mei, Chiang C. (1994), The applied dynamics of ocean surface waves, Advanced Series on Ocean Engineering, 1, World Scientific, ISBN 9971-5-0789-7, 740 pages.
  • Porter, D.; Chamberlain, P. G. (1997), "Linear wave scattering by two-dimensional topography", in J. N. Hunt (ed.), Gravity waves in water of finite depth, Advances in Fluid Mechanics, 10, Computational Mechanics Publications, pp. 13–53, ISBN 1-85312-351-X
  • Porter, D. (2003), "The mild-slope equations", Journal of Fluid Mechanics, 494: 51–63, Bibcode:2003JFM...494...51P, doi:10.1017/S0022112003005846
Airy wave theory

In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states – giving a description of the wave kinematics and dynamics of high-enough accuracy for many purposes. Further, several second-order nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. Airy wave theory is also a good approximation for tsunami waves in the ocean, before they steepen near the coast.

This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the wave height to water depth (for waves in shallow water), and wave height to wavelength (for waves in deep water).

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.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.

Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.

Index of physics articles (M)

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.

List of submarine volcanoes

A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

Luke's variational principle

In fluid dynamics, Luke's variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the so-called mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. This is often used when modelling the spectral density evolution of the free-surface in a sea state, sometimes called wave turbulence.

Both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity.

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.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

Physical oceanography

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.


In computational fluid dynamics, TELEMAC is short for the open TELEMAC-MASCARET system, or a suite of finite element computer program owned by the Laboratoire National d'Hydraulique et Environnement (LNHE), part of the R&D group of Électricité de France. After many years of commercial distribution, a Consortium (the TELEMAC-MASCARET Consortium) was officially created in January 2010 to organize the open source distribution of the open TELEMAC-MASCARET system now available under GPLv3.

Torricelli's law

Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The law states that the speed v of efflux of a fluid through a sharp-edged hole at the bottom of a tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. , where g is the acceleration due to gravity (9.81 m/s2 near the surface of the Earth). This expression comes from equating the kinetic energy gained, , with the potential energy lost, mgh, and solving for v. The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

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.

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.

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies (like lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.

Ocean zones
Sea level

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