Mild-slope Equation - Formulation For Monochromatic Wave Motion

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 .