Environment
During dynamic simulation, wind velocity and wave elevation may be read from an ascii file.
1. Wind
1.1. Computation of wind at multiple locations
In some situations it is necessary to compute wind velocity at multiple locations during time domain simulation. A typical example of this is when computing wind force on slender elements. For wind spectra where the spectral density does not depend on the vertical position this is simply achieved by scaling the pregenerated wind time series with the wind profile. This applies for the following wind spectra:

Davenport

Harris

Wills
The spectral density of the following wind spectra depends on the vertical position, which means that the approach described above will not give correct results:

Sletringen

ISO 199011 (NPD)

API

ESDU

Simiu normal gust
To account for the vertical variation in spectral density, wind velocity is pregenerated at multiple levels in an evenly distributed grid from \(z=z_{min}\) to \(z=z_{max}\):
During timedomain simulation, the wind velocity at an arbitrary vertical position \(\mathrm {z}\) is found using linear interpolation between the pregenerated time series before the wind profile is applied:
Computation of wind velocity outside the specified domain can be enabled by the user. If so, the nearest pregenerated wind velocity is used when computing wind velocity outside the domain:
2. Waves
2.1. Irregular waves
The sea state is described as a sum of two spectra: a wind sea contribution and a swell contribution.
in which
\(S_{\zeta,1}^+, S_{\zeta,2}^+\) 
describe the onesided wave spectra (power spectra of wave elevation). 
\(\theta_1, \theta_2\) 
describe the directionality of the waves. Unidirectional waves and cosine spreading functions are included. 
\(\beta\) 
direction of wave propagation. 
For more information on the power spectra, see Spectral analysis.
The spreading function satisfies the relations
in which \(\mathrm {\sigma _{\zeta}^2}\) is the variance of the surface elevation.
2.2. Linear potential model
Linear wave potential theory is used throughout the study. The incoming undisturbed wave field is determined by the wave potential \(\mathrm {\Phi_0}\). \(\mathrm {\Phi_0}\) defines a longcrested sinusoidal wave. Unidirectional wave spectra are thought of as a sum of a large number of regular waves at different frequencies. Shortcrested waves are constructed by introducing a directional distribution in addition to the frequency distribution. Further details of the method used are outlined below.
The wave potential \(\mathrm {\Phi_0}\) for a regular wave is, according to Airy’s theory, expressed by
where
\(\zeta_a\) 
wave amplitude 
\(g\) 
acceleration due to gravity 
\(k\) 
wave number, \(~ \omega^2 = gk \tanh(k d)\) 
\(\beta\) 
direction of wave propagation. (\(\beta = 0\) corresponds to wave propagation along the positive xaxis.) 
\(\phi_\zeta\) 
wave component phase angle 
\(\mathrm {C_1}\) is given by
where \(\mathrm {d}\) is the water depth. In deep water \(\mathrm {C_1}\) tends to
We then obtain the following relations for the particle velocities and accelerations in the undisturbed wave field
where \(\mathrm {\alpha =\omega tkx\cos(\beta )ky\sin(\beta )+\phi _\zeta}\).
Using the deep water approximation we have
Taking into account finite water depth we have
The surface elevation is given by
Similarly, the linearized dynamic pressure is given by
In Figure 1 examples of the wave elevation and particle velocities and accelerations are shown.
2.3. Phase angles of wave particle motions
The surface elevation, \(\mathrm {\zeta}\), is selected as the reference when describing waves and waveinduced responses.
where
\(\mathrm {\phi _p}\) is a positiondependent phase angle.
Table 1 and Table 2 present summaries of phase angles and amplitudes of wave particle velocities and accelerations in x, y and z directions: \(\mathrm {C_2}\) and \(\mathrm {C_3}\) are depth and frequencydependent functions, compare with Linear potential model.
Direction  Velocity  Acceleration 

X 
0 
\(\mathrm {\pi /2}\) 
Y 
0 
\(\mathrm {\pi /2}\) 
Z 
\(\mathrm {\pi /2}\) 
\(\mathrm {\pi }\) 
Direction  Displacement  Velocity  Acceleration 

X 
\(\mathrm {\cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega \cos(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega ^2\cos(\beta )C_2\zeta_a}\) 
Y 
\(\mathrm {\sin(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega \sin(\beta )C_2\zeta_a}\) 
\(\mathrm {\omega ^2\sin(\beta )C_2\zeta_a}\) 
Z 
\(\mathrm {C_3\zeta_a}\) 
\(\mathrm {\omega C_3\zeta_a}\) 
\(\mathrm {\omega ^2C_3\zeta_a}\) 
2.4. Complex notation
The complex harmonic wave component is defined by
The surface elevation is
All other responses, \(\mathrm {r}\), are related to the surface elevation by complex transfer functions, \(\mathrm {H_r}\), and can be derived from a complex harmonic function, \(\mathrm {\tilde{r}}\).