# Regular waves

The wave kinematics of regular waves in Ashes are computed used linear theory, also called Airy wave theory. In this theory, the water is considered inviscid, incompressible and irrotational. By assuming that water is inviscid, incompressible and irrotational, the water can then be described by a velocity potential as follows (see Faltinsen (1990)):

$$\Phi = \frac{gA}{\omega}\frac{\cosh\left[k(z+h)\right]}{\cosh(kh)}\cos(\omega t -k_xx-k_yy)$$

where

$$x, y, z$$
are spatial coordinates in a Cartesian reference system [m]

$$t$$
is the time coordinate [s]

$$A$$
is the wave amplitude [m]

$$g$$
is the acceleration due to gravity [m.s-2]

$$\omega$$
is the wave circular frequency, [rd.s-1]

$$k$$
is the magnitude of the wave number vector [rd.m-1]

$$h$$
is the water depth (for a flat bottom) [m]

$$k_x,k_y$$
are the horizontal components of the wave number vector [rd.m-1]

Note: it is common to use the wave height
$$H=2A$$
and the wave period
$$T=2\pi/\omega$$
when describing a regular wave.

The wave number
$$k$$
and the wave frequency
$$\omega$$
are linked through the linear dispersion relationship
$$\omega^2=gk\cdot tanh(kh)$$

The wave elevation is then given by
$$\zeta = A\cdot cos(\omega t - k_xx - k_yy)$$

The water particle velocity in the x, y and z direction are respectively
$$u=\frac{\partial \Phi}{\partial x}$$

$$v=\frac{\partial \Phi}{\partial y}$$
$$w=\frac{\partial \Psi}{\partial z}$$

The water particle accelerations are the time derivatives of the water particle velocities.

## 1 Deep water approximation

It is common to distinguish between Deep or Infinite water and Finite water depending on the wave characteristics and water depth. In deep waters, it is assumed that the sea bed does not affect the wave kinematics. By defining the wave length
$$\lambda$$
as
$$\lambda = 2\pi/k$$
, deep water is assumed when
$$\lambda/h>0.5$$
. Under these conditions, the approximation
$$kh\gg 1$$
is made. The expression for the velocity potential then becomes
$$\Phi=\frac{gA}{\omega}e^{kz}\cos(\omega t -k_xx-k_yy)$$

and the dispersion relationship becomes
$$\omega^2=gk$$

In Ashes, the parameter Depth approximation in the Sea part enables you to select whether deep water waves kinematics are computed based on this approximation or based on the finite water equations.

## 2 Regular wave characteristics

When modeling regular waves, It is common to give as input the wave height
$$H$$
, the wave period
$$T$$
and the water depth
$$h$$
. Several characteristics of the linear wave can be derived from these input, and are available in the Information pane of the Regular wave part. The following section shows how these characteristics are derived:

The wave length is
$$\lambda = 2\pi/k$$
The wave number is obtained by solving the dispersion relationship
$$\omega^2=gk\cdot tanh(kh)$$
The circular frequency is
$$\omega = 2\pi/T$$
The wave steepness is
$$H/\lambda$$
The critical wave height
$$H_c$$
corresponds to the theoretical maximum height that a wave can have before it breaks. For deep water,
$$H_c = \lambda/7$$
and for finite water
$$H_c = 0.78h$$
The wave (phase) velocity is
$$c=\lambda/T$$
The wave group velocity is
$$c_g= \frac{c}{2}\left(1+\frac{2kh}{\sinh(2kh)}\right)$$