Airy models

The Airy approximation is a linear approximation, i.e. for small amplitude waves. Two linear wave field models are available, based on the Airy approximation : the regular and irregular wave models.

Regular wave model

The regular wave is a single wave component defined by its amplitude \(A\), its circular frequency \(\omega\) and its wave direction \(\theta\) respect to the x-axis in NWU convention.

The elevation of the regular wave, at a position \((x,y)\) and a time \(t\) is given by:

\[\eta (x,y,t) = A \sin (k \bar{x} - \omega t)\]

where \(k\) is the wave number and \(\bar{x} = x \cos(\theta) + y \sin(\theta)\).

The wave height \(Hcc\) is measured from trough to crest, and is equal to twice the amplitude. The wave period is \(T = 2\pi/\omega\).

The Airy model is based on the incompressible and irrotationnal flow hypothesis, which implies that the velocity field can be derived as a gradient of a scalar function, the velocity potential :

\[\phi (x,y,z,t) = -\frac{A g}{\omega}\frac{\cosh(k(z+H))}{\cosh(kH)}\cos(k\bar{x} - \omega t)\]

The wave dispersion relation, \(\omega^2 = gk\tanh(kH)\), with H the water depth, introduced in the previous equation, yields:

\[\phi (x,y,z,t) = -\frac{A \omega}{k}\frac{\cosh(k(z+H))}{\sinh(kH)}\cos(k\bar{x} - \omega t)\]

One can note \(E(z)\) the scaling factor which can be subject to transformations to limit inaccuracy of velocity prediction principally on the crest (see kinematic stretching)

\[E(z) = \frac{\cosh(k(z+H))}{\sinh(kH)}\]

For infinite depth, the scaling factor \(E(z)\) is defined by its asymptotic formula:

\[E(z) = \exp( k z)\]

In finite depth, the orbital velocity of the flow is given by:

\[\begin{split}V_x = A \omega \cos \theta \frac{\cosh(k(z+H))}{\sinh(kH)}\sin(\bar{x} - \omega t) \\ V_y = A \omega \sin \theta \frac{\cosh(k(z+H))}{\sinh(kH)}\sin(\bar{x} - \omega t) \\ V_z = -A \omega \frac{\sinh(k(z+H))}{\sinh(kH)} \cos(\bar{x} - \omega t)\end{split}\]

The previous velocities definitions take the following form in infinite depth:

\[\begin{split}V_x = A \omega \cos(\theta) \exp(kz)\sin(\bar{x} - \omega t) \\ V_y = A \omega \sin(\theta) \exp(kz)\sin(\bar{x} - \omega t) \\ V_z = -A \omega \exp(kz)\cos(\bar{x} - \omega t)\end{split}\]

The pressure generated by a single Airy regular wave is given by:

\[\begin{split}P(x,y,z,t) = \rho gA \tanh(kH) E(z) \sin(k \bar{x} - \omega t) \\ = \rho g \tanh(kH) E(z) \eta(x,y,t)\end{split}\]

Airy irregular wave model

The Airy irregular wave is a linear superposition of Airy regular waves of different wave circular frequency \(\omega_m\) and direction \(\theta_n\), with random or specified wave phases \(\Phi_{mn}\).

\[\eta(x,y,t) = \sum_m \sum_n A_{mn} \sin(k_m\bar{x}_n - \omega_m t + \Phi_{mn})\]

where \(\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)\).

The wave frequency and direction discretizations are based on a uniform repartition, i.e the wave components are chosen to be equally spaced in frequency and direction.

The wave component amplitudes, \(A_{mn}\), are given by the directional spectrum (see wave spectra)

\[A_{mn}^2 = 2S(\omega_m,\theta_n)\delta\omega\delta\theta\]

The velocity potential can thus be given by

\[\phi(x,y,z,t) = \sum_m \sum_n -\frac{\omega_m A_{mn}}{k} E_m(z) \cos(k_m\bar{x}_n - \omega_m t + \Phi_{mn})\]

where

\[E_m(z) = \frac{\cosh(k_m(z+H))}{\sinh(k_mH)}\]

In infinite depth, the scaling factor is defined as follows:

\[E_m(z) = exp(k_m z)\]

In finite depth, the orbital velocity of the flow is given by:

\[\begin{split}V_x = \sum_m \sum_n A_{mn} \omega_m \cos \theta_n \frac{\cosh(k_m(z+H)}{\sinh(k_m H)}\sin(\bar{x_n} - \omega_m t) \\ V_y = \sum_m \sum_n A_{mn} \omega_m \sin \theta_n \frac{\cosh(k_m(z+H)}{\sinh(k_m H)}\sin(\bar{x_n} - \omega_m t) \\ V_z = \sum_m \sum_n A_{mn} \omega_m \frac{\sinh(k_m(z+H)}{\sinh(k_mH)} \cos(\bar{x_n} - \omega_m t)\end{split}\]

The previous velocities definitions take the following form in infinite depth:

\[\begin{split}V_x = \sum_m \sum_n A_{mn} \omega_m \cos(\theta_n) \exp(k_m z)\sin(\bar{x_n} - \omega_m t) \\ V_y = \sum_m \sum_n A_{mn} \omega_m \sin(\theta_n) \exp(k_m z)\sin(\bar{x_n} - \omega_m t) \\ V_z = \sum_m \sum_n A_{mn} \omega_m \exp(k_m z)\cos(\bar{x_n} - \omega_m t)\end{split}\]

The pressure generated by an Airy irregular wave field is given by:

\[P(x,y,z,t) = \rho g \sum_m \sum_n E_m(z)\tanh(k_mH) \Im(A_{mn} \exp(jk_m\bar{x}_n - j\omega_m t + j \Phi_{mn}))\]