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.
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.
[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.
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
where: The phase and group speed depend on the dispersion relation, and are derived from Airy wave theory as:[5]
has to be solved from the dispersion equation, which relates these two quantities to the water depth
In spatially coherent fields of propagating waves, it is useful to split the complex amplitude
is irrotational, a direct consequence of the fact it is the derivative of the wave phase
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).
[9] For the case of a horizontally unbounded domain with a constant density
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.
in the fluid interior, as well as all the boundary conditions both on the free surface
In case of linear wave theory, the vertical integral in the Lagrangian density
Using a Taylor series expansion for the second integral around the mean free-surface elevation
Next, the mild-slope assumption is made, in that the vertical shape function
with ∇ the horizontal gradient operator: ∇ ≡ (∂/∂x, ∂/∂y)T where superscript T denotes the transpose.
Since the objective is the description of waves over mildly sloping beds, the shape function
This is the linear theory of waves propagating in constant depth
a constant angular frequency, chosen in accordance with the characteristics of the wave field under study.
The following time-dependent equations give the evolution of the free-surface elevation
can be eliminated, to obtain the time-dependent form of the mild-slope equation:[4]
The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around
[11] However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero.
