# 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:

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 :

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

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)

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

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

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

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

## 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 \(\Phi_{mn}\).

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)

The velocity potential can thus be given by

where

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

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

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

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