There are several pre-configured wave spectra that only require knowledge of a couple of statistical parameters to fully define the sea state. These are explained below.

 

2.1.2.1.1. JONSWAP Spectrum

The JONSWAP (Joint North Sea Wave Project) spectrum can take into account the imbalance of energy flow in the wave system (for instance, when seas are not fully developed). Energy imbalance is nearly always the case when there is a high wind speed. Parameterization of the classic form of the JONSWAP spectrum (using fetch and wind speed) was undertaken by Houmb and Overvik [17]. The peak frequency as well as empirical parameters and are used in this formulation. The spectral ordinate at a frequency is given by

(2–31)

where

is the peak frequency in rad/s, is the peak enhancement factor, is a constant that relates to the wind speed and the peak frequency of wave spectrum, and

Because is a constant, the integration of this spectrum can be expressed as

(2–32)

Therefore if , , and are known, then the variable can be determined by

(2–33)

You can define the starting and finishing frequencies of the JONSWAP spectrum used in Equation 2–31. By default, Aqwa gives the definitions as:

Starting frequency (in rad/s):

(2–34)

Finishing frequency (in rad/s):

(2–35)

where the weighting function values against are listed in Table 2.1: Weighting Function Values.

Table 2.1: Weighting Function Values

1.0	5.1101	8.0	3.3700	15.0	2.9650
2.0	4.4501	9.0	3.2900	16.0	2.9300
3.0	4.1000	10.0	3.2200	17.0	2.8950
4.0	3.8700	11.0	3.1600	18.0	2.8600
5.0	3.7000	12.0	3.1050	19.0	2.8300
6.0	3.5700	13.0	3.0550	20.0	2.8000
7.0	3.4600	14.0	3.0100

 

2.1.2.1.2. Pierson-Moskowitz Spectrum

The Pierson-Moskowitz spectrum is a special case for a fully developed long crested sea. In Aqwa, the Pierson-Moskowitz spectrum is formulated in terms of two parameters of the significant wave height and the average (mean zero-crossing) wave period. The form used in Aqwa is considered of more direct use than the classic form (in terms of the single parameter wind speed), and the form involving the peak frequency (where the spectral energy is a maximum). For more information, see [15]. The spectral ordinate at a frequency (in rad/s) is given by

(2–36)

The following relationship exists between , , and :

(2–37)

where is the mean wave period and is the peak period.

You can define the starting and finishing frequencies of the Pierson-Moskowitz spectrum used in Equation 2–36. By default, Aqwa gives the definitions as:

Starting frequency (in rad/s):

(2–38)

Finishing frequency (in rad/s):

(2–39)

 

2.1.2.1.3. Gaussian Spectrum

The standard Gaussian spectrum is:

(2–40)

where is the peak frequency (in rad/s), is the standard deviation and .

You can define the starting and finishing frequencies of the Gaussian spectrum used in Equation 2–40. By default, Aqwa gives the definitions as:

Starting frequency (in rad/s):

(2–41)

Finishing frequency (in rad/s):

(2–42)

If , they are alternatively defined as

(2–43)

 

2.1.2.1.4. Ochi-Hubble Spectrum

Ochi-Hubble bimodal spectrum represents almost all stages of the sea condition associated with storm (Ochi and Hubble, 1976 [33]). The spectrum is expressed by the sum of two sets of three-parameter spectra, to cover the lower frequency and high components of the wave energy respectively.

(2–44)

where is the significant wave height, is the spectral shape parameter and is the modal frequency in rad/s of the j-th sub-spectrum, and is the Gamma function.

The total significant wave height of the Ochi-Hubble spectrum is

(2–45)

 

2.1.2.1.5. Bretschneider Spectrum

Bretschneider spectrum is the special case of the Ochi-Hubble spectrum when and j=1. It is expressed by two parameters of significant wave height and modal frequency,

(2–46)

The zero-crossing frequency (ITTC, 2002 [20]) is

(2–47)

 

2.1.2.1.6. TMA Spectrum

Finite water depth TMA spectrum for non-breaking waves is given as the JONSAWP spectrum multiplying a water depth function (ITTC, 2002 [20]),

(2–48)

where the depth function , k is the wave number derived from the linear dispersion relationship of Equation 2–3; the approximately expression (ITTC, 2002 [20]) of is

(2–49)

where , d is the water depth and

You can input any other spectrum into Aqwa. User-defined wave spectra are normally employed for input of non-deterministic spectra such as tank spectra, recorded full-scale spectra, or simply where the formulated spectrum is not yet available in Aqwa.

The values of frequencies (in rad/s by default) and the values of the spectral ordinates can also be defined.

Aqwa can import a time history series of wave elevation in order to reproduce model test wave conditions.

In an Aqwa time domain analysis (Aqwa-Drift or Aqwa-Naut analysis), the wave elevation time-history will be reproduced exactly, within the frequency range of the fitted spectrum and subject to the limitations of round-off error. This is achieved by multiplying each of the spectral wave components (as in standard Aqwa) by a different low frequency perturbation (LFP) function:

(2–50)

where is the number of wave components, is the wave amplitude, is the frequency, is the wave number, is the phase angle of wave component , where is the wave direction, and is the low frequency perturbation (LFP) function automatically estimated by Aqwa based on the imported time history record. The starting and finishing wave frequencies are defined by Equation 2–34 and Equation 2–35, and the amplitude of a wave component is defined as where is the fitted JONSWAP wave spectral ordinate and is the frequency band of the j-th wave component.

No spurious low frequency waves are generated by the above method. For any wave component, the minimum frequency present in the wave elevation is , where is the highest frequency present in the LFP function. Note also that there is no frequency overlap for each wave component. Each LFP function can be considered as a frequency spreading function over a limited set of contiguous frequency bands.

The minimum duration of the time history required by the program is 7200s (2 hours). This duration is necessary in order to give sufficient resolution of low frequency resonant responses. You may encounter situations where model tests cannot be performed to cover the 2-hour duration. In such cases, Aqwa will extend the imported wave elevation time history record to 7200 seconds by the following approach.

As shown in Figure 2.2: Extension of Wave Elevation Record, if the original length of time history of wave elevation is assumed to be , it will be first extended to by

(2–51)

where is the number of time steps of the original imported time history records.

If after the extension treatment by Equation 2–51, further extension of wave elevation points will be done by using Equation 2–51 again but replacing the parameters and by and respectively. The described extension procedure will be iterated until the last time step reaches 7200 seconds with wave elevation points in total.

An end-correction treatment will be carried out to enable the Fourier transform to be more accurate. Denoting as the total number of wave elevation points such that

(2–52)

the wave elevation values at the starting and finishing ends of the record are corrected as

(2–53)

where and .

This end-corrected time series of wave elevations can also be employed to generate a user-defined spectrum by using a Fast Fourier Transform. The frequency range is based on a JONSWAP fit of the wave elevation spectral density. This fitted wave spectrum is used in the same way as a normal user-defined spectrum, and is employed by Aqwa in equilibrium prediction, frequency domain dynamic statistic analysis, and time domain analysis of severe waves. As the phases of the spectral wave components are allocated randomly, the imported wave elevation time history cannot be reproduced exactly in these types of Aqwa analyses.

Figure 2.2: Extension of Wave Elevation Record

Aqwa can simulate the effects of cross swell in most types of analyses, with the exception of time history response of severe waves. It can be defined by a JONSWAP, Pierson-Moskowitz, or Gaussian wave spectrum. The default starting and finishing frequencies of the cross swell wave spectrum will be used automatically, which are defined in Equation 2–34 and Equation 2–35 for JONSWAP spectra, Equation 2–38 and Equation 2–39 for Pierson-Moskowitz spectra, and Equation 2–42 and Equation 2–43 for Gaussian spectra.

As the wind over sea surface is turbulent, the induced wave spectrum is often expressed as two-dimensional, of which one dimension corresponds to frequency and the other to wave direction.

For design/analysis purposes representative spectra should be in the form of:

For more information, see [4].

The mean direction is assumed to be constant for all wave frequencies and the spreading function is of the form

(2–55)

Where is the power of the wave spreading function and is selected to ensure that at each frequency, it is satisfied that

(2–56)

Based on this definition, the above equation becomes

(2–57)

which leads to

(2–58)

For the case of ,

(2–59)

When the spreading angle is set to 180 degrees (), Equation 2–58 becomes

(2–60)

Selections of the constant factor are listed in Table 2.2: Weighting Factors with Respect to the Power Number of Spreading Function.

Table 2.2: Weighting Factors with Respect to the Power Number of Spreading Function

n	2	3	4	5	6	7	8	9	10
	2/	3/4	8/(3)	15/16	16/(5)	35/32	128/(35)	315/256	256/(63)

Aqwa employs a Gaussian integral to numerically represent spreading seas. When an -point Gaussian integration scheme is used to represent a spreading sea, the following equations are used:

(2–61)

where is the wave energy spectrum in the m-th sub-direction.

(2–62)

where is the Gaussian integration weighting factor between the standard integration range of [-1, 1] and is the sub-direction corresponding to the m-th standard Gaussian integration point.

For spreading wave functions with a higher power (n>4), the wave energy is mainly distributed over a narrower direction range centered in the mean direction . To increase the numerical accuracy of the spreading wave representation, an effective wave direction range, , is introduced. Outside this range, the spreading wave energy is negligible:

(2–63)

where:

The Gaussian integration is over the effective wave range instead of the wave spreading direction range of . As an example, the effective range is about when n = 250 and .

You may also define spreading seas with user-defined carpet spectra. You can input each of these carpet spectra with each wave spectrum as a series of directions. It is assumed that the directions are in ascending order, as well as the frequencies at which the value of the spectral ordinates are given. Optionally the specification of individual weighting factors for each direction can be defined.

If you do not define weighting factors, the contribution of each directional spectrum is calculated by a simple trapezoidal integral and will add up to unity. The weighting factors are calculated using the following equation:

(2–64)

where m is the direction sequence number in a carpet spectral definition and is the total direction range.

In the case of the first or the last direction, the weighting functions are

(2–65)

It is possible to employ a spectral group to define a specified environmental condition by introducing any combination of the above described wave spectra and/or imported wave elevation time history records.

In Aqwa, each cross swell will be considered as one sub-directional wave spectrum, a spreading sea is represented by sub-directional spectra (by using an -point Gaussian integration scheme). The overall maximum total number of sub-directional spectra in each spectral group is specified by the program. Among each wave spectra group, a limited number of imported wave elevation time history files may be included. These sub-directional spectral numbers may vary with different Aqwa versions. For more information, see the Aqwa User's Manual.

A spectral group may contain only one definition of wind and/or current.