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 bathymetric 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.
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:
- 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:
- is Earth's gravity and
- is the hyperbolic tangent.
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
In spatially coherent fields of propagating waves, it is useful to split the complex amplitude in its amplitude and phase, both real valued:
- 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):
- 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). 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.
When is used in the mild-slope equation, the result is, apart from a factor :
Now both the real part and the imaginary part of this equation have to be equal to zero:
The effective wavenumber vector is defined as the gradient of the wave phase:
Note that is an irrotational field, since the curl of the gradient is zero:
Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by :
The first equation directly leads to the eikonal equation above for , while the second gives:
which—by noting that in which the angular frequency is a constant for time-harmonic motion—leads to the wave-energy conservation equation.
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.
Luke's variational principle
Luke's Lagrangian formulation gives a variational formulation for non-linear surface gravity waves. For the case of a horizontally unbounded domain with a constant density , a free fluid surface at and a fixed sea bed at Luke's variational principle uses the Lagrangian
where is the velocity potential, with the flow velocity components being and in the , and directions, respectively. 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. Taking the variations of with respect to the potential and surface elevation leads to the Laplace equation for in the fluid interior, as well as all the boundary conditions both on the free surface as at the bed at
Linear wave theory
In case of linear wave theory, the vertical integral in the Lagrangian density is split into a part from the bed to the mean surface at and a second part from to the free surface . Using a Taylor series expansion for the second integral around the mean free-surface elevation and only retaining quadratic terms in and the Lagrangian density for linear wave motion becomes
The term in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the Euler–Lagrange equations, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to in the potential energy.
The waves propagate in the horizontal plane, while the structure of the potential is not wave-like in the vertical -direction. This suggests the use of the following assumption on the form of the potential
Here is the velocity potential at the mean free-surface level Next, the mild-slope assumption is made, in that the vertical shape function changes slowly in the -plane, and horizontal derivatives of can be neglected in the flow velocity. So:
As a result:
The Euler–Lagrange equations for this Lagrangian density are, with representing either or
Now is first taken equal to and then to As a result, the evolution equations for the wave motion become:
The next step is to choose the shape function and to determine and
Vertical shape function from Airy wave theory
Since the objective is the description of waves over mildly sloping beds, the shape function is chosen according to Airy wave theory. This is the linear theory of waves propagating in constant depth The form of the shape function is:
Here a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals and become:
The following time-dependent equations give the evolution of the free-surface elevation and free-surface potential 
From the two evolution equations, one of the variables or can be eliminated, to obtain the time-dependent form of the mild-slope equation:
Consider monochromatic waves with complex amplitude and angular frequency :
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. 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.
- ^ Eckart, C. (1952), "The propagation of gravity waves from deep to shallow water", Circular 20, National Bureau of Standards: 165–173, Bibcode:1952grwa.conf..165E
- ^ 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 (PDF) (PhD. Thesis), Delft University of Technology
- ^ a b c d e f g h i j Dingemans (1997, pp. 248–256 & 378–379)
- ^ Dingemans (1997, p. 49)
- ^ Mei (1994, pp. 86–89)
- ^ a b c d Dingemans (1997, pp. 259–262)
- ^ Booij, N. (1981), Gravity waves on water with non-uniform depth and current (PDF) (PhD. Thesis), Delft University of Technology, Bibcode:1981PhDT........37B
- ^ 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, S2CID 123409273
- ^ 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, S2CID 121777750
- ^ 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, vol. 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, vol. 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, vol. 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, vol. 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, S2CID 121112316