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. 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.
- 1 Formulation for monochromatic wave motion
- 2 Transformation to an inhomogeneous Helmholtz equation
- 3 Propagating waves
- 4 Derivation of the mild-slope equation
- 5 Applicability and validity of the mild-slope equation
- 6 Notes
- 7 References
Formulation for monochromatic wave motion
- η(x,y) is the complex-valued amplitude of the free-surface elevation ζ(x,y,t);
- (x,y) is the horizontal position;
- ω is the angular frequency of the monochromatic wave motion;
- i is the imaginary unit;
- means taking the real part of the quantity between braces;
- is the horizontal gradient operator;
- is the divergence operator;
- k is the wavenumber;
- cp is the phase speed of the waves and
- cg is the group speed of the waves.
- g is Earth's gravity and
- tanh is the hyperbolic tangent.
For a given angular frequency ω, the wavenumber k has to be solved from the dispersion equation, which relates these two quantities to the water depth h.
Transformation to an inhomogeneous Helmholtz equation
Through the transformation
where Δ is the Laplace operator.
This transforms the mild-slope equation in the following set of equations (apart from locations for which is singular):
- E 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,
- ρ is the fluid density, and
- g 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 E is transported in the -direction normal to the wave crests (in this case of pure wave motion without mean currents). The effective group speed is different from the group speed cg.
The first equation states that the effective wavenumber is irrotational, a direct consequence of the fact it is the derivative of the wave phase θ, 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 a and phase θ leads to consistent-varying and meaningful fields of a and . Otherwise, κ2 can even become negative. When the diffraction effects are totally neglected, the effective wavenumber κ is equal to k, and the geometric optics approximation for wave refraction can be used.
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. 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 ζ(x,y,t) and free-surface potential ϕ(x,y,t):
From the two evolution equations, one of the variables φ or ζ can be eliminated, to obtain the time-dependent form of the mild-slope equation:
and the corresponding equation for the free-surface potential is identical, with ζ replaced by φ. The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around ω0.
Consider monochromatic waves with complex amplitude η(x,y) and angular frequency ω:
with ω and ω0 chosen equal to each other, ω = ω0. Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:
Applicability and validity of the mild-slope equation
The standard mild slope equation, without extra terms for bottom curvature, provides accurate results for the wave field over bottom slopes ranging from 0 to about 1/3. 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 bottom slopes.
- ^ Eckart, C. (1952), "The propagation of gravity waves from deep to shallow water", Circular 20 (National Bureau of Standards): 165–173
- ^ Berkhoff, J. C. W. (1972), "Computation of combined refraction–diffraction", Proceedings 13th International Conference on Coastal Engineering, Vancouver, pp. 471–490
- ^ Berkhoff, J. C. W. (1976), Mathematical models for simple harmonic linear water wave models; wave refraction and diffraction (PhD. Thesis), Delft University of Technology, http://repository.tudelft.nl/assets/uuid:381c691b-eea8-4f67-be8f-d471a7da1d58/261254.pdf
- ^ a b c d e f g h i j See Dingemans (1997), pp. 248–256 & 378–379.
- ^ See Dingemans (1997), p. 49.
- ^ See Mei (1994), pp. 86–89.
- ^ a b c d See Dingemans (1997), pp. 259–262.
- ^ Booij, N. (1981), Gravity waves on water with non-uniform depth and current (PhD. Thesis), Delft University of Technology, http://repository.tudelft.nl/assets/uuid:05f9b2b1-b237-491f-927a-2a470e0808f3/Booij1981.pdf
- ^ 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
- ^ 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
- ^ 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, Ocean Engineering Science, The Sea, 9A, Wiley Interscience, pp. 27–63, ISBN 0471528560
- Mei, Chiang C. (1994), The applied dynamics of ocean surface waves, Advanced Series on Ocean Engineering, 1, World Scientific, ISBN 997 15 0789 7 , 740 pages.
- Porter, D.; Chamberlain, P. G. (1997), "Linear wave scattering by two-dimensional topography", in J. N. Hunt, Gravity waves in water of finite depth, Advances in Fluid Mechanics, 10, Computational Mechanics Publications, pp. 13–53, ISBN 185312351X
Wikimedia Foundation. 2010.
Look at other dictionaries:
Cnoidal wave — US Army bombers flying over near periodic swell in shallow water, close to the Panama coast (1933). The sharp crests and very flat troughs are characteristic for cnoidal waves. In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic… … Wikipedia
Wind wave — Ocean wave redirects here. For the film, see Ocean Waves (film). North Pacific storm waves as seen from the NOAA M/V Noble Star, Winter 1989 … Wikipedia
Oceanic trench — Oceanic crust is formed at an oceanic ridge, while the lithosphere is subducted back into the asthenosphere at trenches. The oceanic trenches are hemispheric scale long but narrow topographic depressions of the sea floor. They are also the… … Wikipedia
Continental shelf — Marine habitats Anatomy of a continental shelf off the south eastern coast of the United States Littoral zone Intertidal zone … Wikipedia
Cold seep — Marine habitats Tube worms are among the dominant species in one of four cold seep community types in the Gulf of Mexico. Littoral zone … Wikipedia
Contourite — A contourite is a sedimentary deposit produced by thermohaline induced deepwater bottom currents and may be influenced by wind or tidal forces. Most contourites are formed in continental rise to lower slope settings, although they may occur … Wikipedia
El Niño-Southern Oscillation — El Niño redirects here. For other uses, see El Niño (disambiguation). ENSO redirects here. For other uses, see Enso (disambiguation). The 1997 El Niño observed by TOPEX/Poseidon. The white areas off the tropical coasts of South and North America… … Wikipedia
Dispersion (water waves) — This article is about dispersion of waves on a water surface. For other forms of dispersion, see Dispersion (disambiguation). In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of… … Wikipedia
Ocean — For other uses, see Ocean (disambiguation). Maps exhibiting the world s oceanic waters. A continuou … Wikipedia
Fluid dynamics — Continuum mechanics … Wikipedia