Mild-slope Equation - Derivation of The Mild-slope Equation - Vertical Shape Function From Airy Wave Theory

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:

with now in general not a constant, but chosen to vary with and according to the local depth and the linear dispersion relation:

Here a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals and become:

$\begin{align} F &= \int_h^0 f^2\; \text{d}z = \frac{1}{g}\, c_p\, c_g \quad \text{and} \\ G &= \int_h^0 \left( \frac{\partial{f}}{\partial{z}} \right)^2\; \text{d}z = \frac{1}{g} \left( \omega_0^2\, -\, k^2\, c_p\, c_g \right). \end{align}$

The following time-dependent equations give the evolution of the free-surface elevation and free-surface potential

$\begin{align} g\, \frac{\partial\zeta}{\partial{t}} &+ \nabla\cdot\left( c_p\, c_g\, \nabla \varphi \right) + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \varphi = 0, \\ \frac{\partial\varphi}{\partial{t}} &+ g \zeta = 0, \quad \text{with} \quad \omega_0^2\, =\, g\, k\, \tanh\, (kh). \end{align}$

From the two evolution equations, one of the variables or can be eliminated, to obtain the time-dependent form of the mild-slope equation:

$-\frac{\partial^2\zeta}{\partial{t^2}} + \nabla\cdot\left( c_p\, c_g\, \nabla \zeta \right) + \left( k^2\, c_p\, c_g\, -\, \omega_0^2 \right)\, \zeta = 0,$

and the corresponding equation for the free-surface potential is identical, with replaced by The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around

Read more about this topic: Mild-slope Equation, Derivation of The Mild-slope Equation

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