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 ζ ( x , y , t ) = ℜ { η ( x , y ) e − i ω t } {\displaystyle \zeta (x,y,t)=\Re \left\{\eta (x,y)\,e^{-i\omega t}\right\}} and the waves propagating on a fluid layer of mean water depth h ( x , y ) {\displaystyle h(x,y)} —the mild-slope equation is:
∇ ⋅ ( c p c g ∇ η ) + k 2 c p c g η = 0 , {\displaystyle \nabla \cdot \left(c_{p}\,c_{g}\,\nabla \eta \right)\,+\,k^{2}\,c_{p}\,c_{g}\,\eta \,=\,0,}
where:
η ( x , y ) {\displaystyle \eta (x,y)} is the complex-valued amplitude of the free-surface elevation ζ ( x , y , t ) ; {\displaystyle \zeta (x,y,t);}
( x , y ) {\displaystyle (x,y)} is the horizontal position;
ω {\displaystyle \omega } is the angular frequency of the monochromatic wave motion;
i {\displaystyle i} is the imaginary unit;
ℜ { ⋅ } {\displaystyle \Re \{\cdot \}} means taking the real part of the quantity between braces;
∇ {\displaystyle \nabla } is the horizontal gradient operator;
∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator;
k {\displaystyle k} is the wavenumber;
c p {\displaystyle c_{p}} is the phase speed of the waves and
c g {\displaystyle c_{g}} 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:
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