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:
where:
- 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:
where
- 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 .
Read more about this topic: Mild-slope Equation
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