# 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:

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