# Environment: Data for random waves

Random waves are specified by their energy spectrum. The wave type gives the particular type of spectrum, and the spectral parameters then define the actual spectrum for that type. The spectral data for the available spectra are described as follows:

Data common to all random waves are described below.

For a given spectrum, sea state and simulation time origin, different wave conditions can be obtained by shifting the wave time origin. For more information, see setting up a random sea.

### Number of wave directions, spreading exponent, direction spectrum discretisation method

These data allow you to model a directional spread spectrum. You specify the number of discrete wave directions to be used, and the exponent for the spreading function. If the number of wave directions is 1 (the default), then no spreading is applied.

OrcaFlex uses a $\cos^n$ spreading function, where $n$ is the spreading exponent.

For directional spread spectra, technical reasons dictate that the spectral discretisation algorithm depends upon whether dynamics is to be solved in the time domain or the frequency domain. This choice has an effect on the static analysis too, even though no waves are active, because the wave components arising from the discretisation determine the mean wave drift load on vessels, which is also applied during the static analysis. If dynamics is disabled, then this becomes a problem because it is unclear whether to apply the time domain or frequency domain discretisation methods for purposes of the static analysis. In this scenario, it is therefore necessary to choose between the time domain and frequency domain methodologies.

### Number of components, seed

Random wave trains are represented by a user-defined number of component waves whose amplitudes and periods are selected by OrcaFlex to give a sea state having the required spectrum. Note that, if a value for the maximum component frequency range (see below) is given (i.e. it is not '~'), then the actual number of components used can be greater than the number of components specified here.

If you are using a directional spreading spectrum then the number of wave components you give is per direction. Otherwise, it is the total number of wave components for the wave train.

The phases associated with each wave component are pseudo-random. OrcaFlex uses a random number generator and the user-defined seed to assign phases. The sequence is repeatable, so the same seed will always give the same phases and consequently precisely the same wavetrain.

If user specified seeds is checked then you must give a seed for each wave train; if it is not checked, seeds are chosen automatically.

### Frequency spectrum discretisation method

This gives you control over the spectral discretisation algorithm used to choose the wave components. The following options are available:

• Arithmetic progression, wave components equally-spaced in frequency
• Geometric progression, the ratio of successive wave component frequencies is constant
• Equal energy, each component represents an equal amount of spectral energy.
 Note: There are three distinct equal-energy discretisation methods named equal energy, equal energy, 9.3a, deprecated and equal energy, legacy, deprecated. The deprecated variants are not available in normal usage, and are only retained to maintain backwards compatibility with simulation files produced by older versions of OrcaFlex. These methods are only shown in the list when a model is loaded which uses one of them. The deprecated methods should not be used for new analysis.
 Warning: The arithmetic progression method leads to a realisation with a repeat period which is the reciprocal of the frequency spacing. This can be shorter than the simulation duration, in which case the results of the simulation are likely to be invalid and OrcaFlex will display a warning to this effect. The problem can be alleviated by increasing the number of components, thereby reducing the frequency spacing. This may, however, require so many wave components that the simulation run time becomes impractical. Furthermore, the arithmetic progression often results in a poor discretisation of the spectral peak. Again, using more components improves the discretisation but at the cost of increased run time. For these reasons we recommend using equal energy (the default method) or geometric progression. The arithmetic progression method has been included largely for completeness. The equal energy and geometric progression methods have much longer repeat periods than arithmetic progression and, for practical purposes, the realisations do not repeat. You should, however, still take care to avoid unrealistic autocorrelation. The autocorrelation of the wave process can be inspected by viewing the autocorrelation graph.

### Relative frequency range

The minimum $(r\!_\mathrm{min})$ and maximum $(r\!_\mathrm{max})$ relative frequency determine the range of frequencies considered by the spectral discretisation algorithm. The default values of $r\!_\mathrm{min}{=}0.5$ and $r\!_\mathrm{max}{=}10$ are usually sufficient to produce a good representation of the spectrum.

### Maximum component frequency range

Available only when the equal energy frequency spectrum discretisation method is selected. The maximum component frequency range $\delta\!f_\mathrm{max}$ places an upper limit on the width of the frequency range represented by each wave component: each is constrained to cover a frequency range no greater than $\delta\!f_\mathrm{max}$. This data item is provided because the equal energy discretisation algorithm used by OrcaFlex can result in some wave components (e.g. those in the low and high frequency tails of the spectrum, where the spectral density is low) covering a large frequency range. Such wave components can result in poor modelling of system responses, since the spectral energy represented by the component is concentrated at a single frequency. This might result in over-exciting a system response, if the component's frequency happens to match a system response frequency; conversely, it might result in failure to excite an important system response, even though the response frequency falls within the (over-wide) frequency band represented by the component.

The maximum component frequency range can be used to avoid such problems, and we recommend that you do limit the frequency range. If any of the wave components generated by the equal energy discretisation algorithm covers a frequency range wider than the specified maximum component frequency range, then that component is subdivided into multiple components (which now each have lower energy) repeatedly until all the components satisfy the specified maximum frequency range. A value of '~' means that the component frequency ranges are not limited in this way.

## Spectrum graphs and tables

The view frequency spectrum button displays a graph of the energy spectrum.

The view direction spectrum button similarly displays a graph of the direction spectrum. This button will only be available if the selected wave train has more than one wave direction.

The view wave components button displays a spreadsheet giving details of the wave components that OrcaFlex has chosen to represent a random or time history wave train.

For a random wave train the spreadsheet also reports the following properties of the spectrum:

• Spectral moments $m_0, m_1, \ldots, m_4$
• Spectral bandwidth parameter $\epsilon = \left[ 1 - m_2^2/\left( m_0 m_4 \right) \right]^{1/2}$
• Mean period $T_1 = m_0/m_1$. Note that $T_1$ is sometimes denoted by $T_m$.
• Peak period and frequency $\Tp$ and $\fm$. These are the period and frequency at which the spectrum has the greatest spectral density.

### Spectral moments

The ith spectral moment, $m_i$, of a process with spectrum $S(f)$ is defined to be $$m_i = \int_0^\infty f^i\ S(f)\,\ud f$$ where $f$ denotes temporal frequency.