National Network of Regional Coastal Monitoring Programmes

Author: Thomas Dhoop. Published: 23 August 2022
Please reference as: Channel Coastal Observatory. 2022. Wave Parameter Handbook.
www.coastalmonitoring.org/ccoresources/waveparameterhandbook/


Contents:

Introduction

1. Wave Parameter Summary Table

2. Timestamp and Position Information

3. Zero-Crossing Wave Parameters

4. Wave Parameters Derived from the Wave Spectrum

5. Sea Surface Temperature

6. Battery Status

7. Parameters Calculated by the CCO

8. References

Introduction

This handbook aims to provide both technical and plain-word definitions and descriptions for each wave parameter measured by a Datawell Directional Waverider MkIII buoy and made available by the National Network of Regional Coastal Monitoring Programmes (NNRCMP).

Wave buoys measure waves using an accelerometer. A buoy moves with the waves and in the process measures vertical acceleration. Double integration of the acceleration yields the vertical displacement, in other words the wave height. These displacements, sometimes also called 'heaves', are typically plotted as excursions from the average position (Figure 1). This data is the 'raw' data output from a wave buoy and the closest representation available of individual ocean waves. The displacements from each wave buoy are made available on the coastal monitoring website on the Realtime Data pages of each buoy on the '1Hz Data' tab. From these individual wave heights, wave parameters can be derived through the application of simple statistics. These are commonly referred to as 'statistically-derived' or 'Zero-Crossing' wave parameter and are discussed in Section 3 of this handbook.

Figure 1: Displacement data from the Perranporth wave buoy from 29 February 2008, 08:30 to 09:00 UTC.

However, when looking out to sea, it is easy to see that a lot of information is lost about waves when simply recording the up-and-down-movements of the buoy. The actual sea surface is constantly changing as waves are continually overtaken and crossed by others. To describe the surface more comprehensively, waves can be decomposed into a number of individual wave components, each with their own frequency and direction. This is achieved by applying Fourier analysis to the displacements discussed above, resulting in the distribution of wave energy as a function of wave frequency (where frequency f = 1 / T, where T = wave period). A plot showing the wave energy as a function of frequency or wave period is commonly referred to as a wave spectrum (Figure 2). In short, the wave spectrum shows where along the wave frequencies (or periods) the most wave energy is situated. For example, in the wave spectrum in Figure 2, it is immediately obvious that most energy is situated within the 0.09Hz to 0.10Hz (10s to 11s) and 0.17Hz to 0.20Hz (5s to 6s) ranges. The higher-frequency (shorter-period) range represents the wind-wave component of the spectrum, while the lower-frequency (longer-period) range represents the swell-wave component. The wave spectra from each wave buoy are made available on the coastal monitoring website on the Realtime Data pages of each buoy on the ‘Wave Spectra’ tab. Moreover, several useful wave parameters can be obtained from the wave spectrum. These are discussed in section 4 of this handbook.

Figure 2: Example of a bimodal wave spectrum, measured on 29 February 2008 13:35 UTC by the Perranporth wave buoy. The graph on the left is expressed in frequency (Hz), while the graph on the right is expressed in period (s).

1.      Wave Parameter Summary Table

In the table below, the column numbers correspond to the columns in the 'allwaveparameters' files available from the Advanced Download tab on the Realtime Data page of each wave buoy. Full descriptions of each parameter are provided in sections 2 – 7.

 

Column

Parameter

Definition

Description

Timestamp and position information.

1

-

-

2

-

-

3

-

-

Wave parameters statistically derived from the displacement data (i.e. raw heave data from the buoy).

4

-

Percentage of displacements used in calculating the wave statistics.

100% = full set of displacements with no transmission issues.

5

-

Maximum wave height, i.e. largest zero-upcrossing wave.

6

-

Period of the highest wave, i.e. period of the largest zero-upcrossing wave.

7

-

Mean height of the highest 1/10th of the waves.

8

-

Mean period of the highest 1/10th of the waves.

9

-

Mean height of the highest 1/3rd of the waves.

Significant wave height , statistically-derived.

10

-

Mean period of the highest 1/3rd of the waves.

11

-

Mean height of all waves.

12

-

Mean period of all waves, i.e. the zero-upcross period .

13

 |

Zero-upross bandwidth parameter . 'Eps' is short for 'epsilon'.

14

-

Number of uprcross waves measured over the 30-minute measurement period.

Wave parameters derived from the wave spectrum.

15

Where  is the frequency at which  is maximal

The peak period. The period associated with the most energetic waves in the wave spectrum.  

16

The wave direction associated with the most energetic waves in the wave spectrum.

17

The directional wave spread at the peak frequency.

18

Spectral zero-upcross period

19

Significant wave height, spectrally-derived.

20

Integral period, the average period  of the integral of the wave record.

21

 |

Mean period. The period associated with the mean frequency of the spectrum.

22

Crest period. The average time between successive wave crests.

23

A Datawell-specific period. Estimator for the peak period.

24

A Datawell-specific period. Estimator for the peak period.

25

Calculated peak period. Estimator for the peak period.

26

 |

Spectral width parameter .

: Very narrow bandwidths | : Broad-band wave conditions

'nu' is the Greek lowercase letter , written in Latin/Roman letters.

27

 |

Spectral bandwidth parameter .

: Very narrow bandwidths | : Broad-band wave conditions

'eps' is short for 'epsilon'.

28

 

Goda's spectral peakedness parameter

Large  values: Sharp peak | Small  values: Blunt peak

29

Significant steepness

Large  values: Steep waves | Small  values: Less steep waves

Sea surface temperature measurements.

30

-

Reference temperature for internal checks.

Default = 24.95°C

31

-

Sea surface temperature

Information on the power remaining in the batteries of the buoy.

32

-

Battery status

0 = empty | 7 = full

Parameters calculated by the CCO during the quality-control process.

33

 (s)

The period of an energy equivalent regular wave. In other words, the period corresponding to the weighted average of the wave energy.

34

 

Wave power. The rate of transfer of energy through each metre of wavefront.

Table 1: The wave parameters provided in the 'allwaveparameters' files available from the Advanced Download tab on the real-time data page of each wave buoy, their definition and description.

 


2.      Timestamp and Position Information

The timestamp and position information.

Timestamp

The timestamp has the format: DD-MMM-YYYY hh:mm:ss and is always provided in Greenwich Mean Time (GMT) which is the same as Coordinated Universal Time (UTC). Timestamps represent the start of a 30-minute measurement burst.

Position

The position information of the buoy is provided in WGS 84 coordinates in decimal degrees (DD) format. Positions are provided live using the buoy’s internal GPS.

 

3.      Zero-Crossing Wave Parameters

The wave parameters that were statistically derived via zero-crossing analysis from the wave record (i.e. heave data from the buoy).

The heave measurements are the filtered, double-integrated accelerations measured by Datawell's onboard vertical accelerometer. The sampling rate on a Datawell Directional Waverider MkIII is 3.84Hz. Data are subsequently filtered and down-sampled to 1.28Hz (Datawell 2020: 39). Wave parameters are calculated over a 30 minute sampling period (sampled at 1.28Hz, results in 2304 heave samples).

Percentage of Waves (%)

The percentage of displacement data in the 30-minute wave record without transmission errors.

A percentage of 100% indicates that no errors were encountered.

Maximum wave height  (m)

The maximum wave height in the 30-minute wave record. In other words, the largest zero-upcrossing wave.

Period of the highest wave  (s)

The wave period corresponding to the maximum wave height. In other words, the period of the largest zero-upcrossing wave.

Mean wave height of the highest 10th of waves  (m)

Mean height of the highest 1/10th of the waves in the 30-minute wave record.

Mean period of the highest 10th of waves  (m)

Mean period of the highest 1/10th of the waves in the 30-minute wave record.

Mean wave height of the highest 3rd of waves  (m)

Mean height of the highest 1/3rd of the waves in the 30-minute wave record.

This is the statistically-derived significant wave height It is a commonly used wave height parameter and a close approximation of visually estimated wave height (e.g. Nordenstrøm 1969).

Mean period of the highest 3rd of waves  (m)

Mean period of the highest 1/3rd of the waves in the 30-minute wave record.

This is the statistically-derived significant wave period . In contrast to the reasonable relationship between the statistically-derived significant wave height  () and visually estimated wave height , there is no close agreement between the significant wave period  () and the visually estimated wave period  (e.g. Nordenstrøm 1969).

Mean wave height  (m)

Mean height of all the waves in the 30-minute wave record.

A common alternative notation for mean wave height is .

Mean wave period  (s)

Mean period of all the waves in the 30-minute wave record.

This is the statistically-derived zero-upcross period . Alternative notations are  or .

Zero-upcross bandwidth parameter Eps |

Spectral width parameter evaluated as a function of the number of crests and waves (i.e. zero-crossings) in a 30-minute wave record. It is used to determine the narrowness of a wave spectrum.

A more common notation for Eps is , where the '0' subscript indicates that 'epsilon' was determined using a zero-crossing method.

                                                                                                                3.1

A low number of crests relative to the number of waves indicates a relatively regular heave signal, suggesting a relatively regular sea surface (i.e. long-period swell waves), which is reflected in a narrow wave spectrum. This is indicated by low  values.

A high number of crests relative to the number of waves indicates a relatively irregular heave signal, suggesting a relatively irregular sea surface (i.e. short-period wind waves on top of long-period swell waves), which is reflected in a wide wave spectrum. This is indicated by high  values.

Mathematically, this can be expressed as:

: Very narrow bandwidths

: Broad-band wave conditions

Number of waves

The number of waves (i.e. zero-upcrossings) measured in the 30-minute wave record.

 

4.      Wave Parameters Derived from the Wave Spectrum

The wave parameters derived from the wave spectrum.

To construct the spectrum, Fourier analysis is performed on eight consecutive blocks of 200 seconds worth of wave displacements (sampled at 1.28Hz, resulting in 256 heave samples for each block). The resulting eight spectra are averaged on board the Datawell Directional Waverider MkIII buoy to compute the half-hourly wave spectrum (Datawell 2020: 45).

The frequency range of the MkIII wave spectrum runs from 0.025Hz to 0.1Hz inclusive with a spacing of 0.005Hz and continues from 0.11Hz to 0.58Hz with a frequency spacing of 0.01 Hz.

Definitions of the spectral moments:

 

 

Peak Period Tp (s)

The wave period associated with the most energetic waves in the wave spectrum. 

In more technical terms, the peak period is the inverse of the frequency at which  reaches its maximum value.

                                                                                                                          4.1

where  is the frequency at which  is maximal.

It is worth noting that, because the peak period  is defined as the inverse of the frequency at which  is maximal, the resolution of the parameter is limited by the resolution of the frequencies in the wave spectrum. Moreover, if a wave spectrum is bimodal, it can be the case that the fluctuations between the two energy peaks in the spectrum make the parameter jump between the two peaks, which may be undesirable when analysing the sea state. A number of calculated peak period estimators  and  are included in this handbook which do not exhibit this behaviour.

Direction (degrees)

The wave direction associated with the most energetic waves in the wave spectrum, also referred to as the peak direction.

In other words, peak direction is the wave direction at which  reaches its maximum value.

Wave direction is reported following the nautical convention, meaning that the direction is where the waves come from, measured clockwise from geographic north.

Spread (degrees)

The directional spreading associated with the most energetic waves in the wave spectrum, also referred to as the peak spread.

In other words, peak spread is the spread at which  reaches its maximum value.

Explained in physical terms, waves generated by winds do not necessarily propagate in one direction. Instead wave-generation by winds is a three-dimensional process which means that, although a large part of the wave energy may propagate in a direction very close to the principal wind direction, other wave components will propagate in different directions. The spread indicates the directional spreading of the waves about the principal wind direction.

Spectral zero-upcrossing period  |  (s)

The mean period of all the waves in the 30-minute wave record, estimated from the spectral moments of the wave spectrum.

In other words, the spectrally-derived zero-upcross period .

An alternative notation  is based on the moment equation:

                                                                                                                                               4.2

Although  is a commonly used parameter for various applications, it is worth noting that the parameter is not always the most reliable estimator for the characteristic wave period in a wave record.

One practical reason for this is that the values of higher-order moments are sensitive to noise in the high-frequency range of the spectrum, where noise can be relatively large. This is because, in the definition of the moments of the spectrum , higher-order moments enhance the energy density at higher frequencies more. In utilising the second-order moment , the parameter is dependent on that higher frequency end of the spectrum. The value of , and hence the estimate of , is therefore sensitive to small errors.

Another more abstract reason is that the moments  and  should, strictly speaking, be calculated through integration over a range from . In actual practice, integration takes place from 0.025Hz to 0.58Hz ( the Nyquist frequency) (see also Holthuijsen 2007: 60-2). 

Significant wave height  |  (m)

Significant wave height , derived from the wave spectrum.

The significant wave height is the mean of the highest one-third of wave heights. The parameter can be estimated from the wave spectrum with the zero-order moment.

                                                                                                                                           4.3

Where  represents the total variance of energy in the wave spectrum or, in other words, the variance of the sea surface elevation.

It is worth noting that significant wave height derived from the wave spectrum  is 5%-10% higher than the significant wave height derived from zero-crossing analysis .

Longuet-Higgins (1980), based on data from five hurricanes in the Gulf of Mexico (data from Forristall 1978), suggests that , while Goda (1988), based on linearly simulated wave profiles, arrives at .

One reason for the fact that  waves are slightly smaller than  waves is that, in the theoretical derivation of , waves are assumed to be in deep water with a narrow spectrum where the height of a wave is practically equal to twice the height of the crest. These are a restrictive set of criteria that are not always met in the field. Another reason comes from the fact that the wave spectrum is considered to describe the sea-surface as a stationary, Gaussian process. However, because of non-linear processes such as wave breaking and wave-wave interactions, in the field, these conditions do not always hold (See also Holthuijsen 2007: 68-75).

Integral period  (s)

The  () of the integral of the wave record, defined by Tucker and Pitt (2001: 41) as:

                                                                                                                                                  4.4

Mean period  |  (s)

The period associated with the mean frequency of the spectrum.

                                                                                                                                                  4.5

 is sometimes used as an alternative to the zero-crossing period  because it utilises the first-order moment and is therefore less affected by high-frequency noise in the wave spectrum.

Crest period  |  (s)

The average time between successive wave crests.

                                                                                                                                                    4.6

 is sensitive to noise in the high-frequency range of the spectrum, where noise can be relatively large. This is because, in the definition of the moments of the spectrum , higher-order moments enhance the energy density at higher frequencies more. In utilising the fourth-order moment , the parameter is dependent on that higher frequency end of the spectrum. The values of , and therefore also the estimate of , is sensitive to errors.

Datawell-specific peak period estimator

The peak period, i.e. the wave period associated with the most energetic waves in the wave spectrum, estimated from the moments of the wave spectrum.

                                                                                                                                             4.7

Because the peak period  is defined as the inverse of the frequency at which  is maximal, the resolution of the parameter is limited by the resolution of the frequencies in the wave spectrum. Moreover, if a wave spectrum is bimodal, it can be the case that the fluctuations between the two energy peaks in the spectrum make the parameter jump between the two peaks, which may be undesirable when analysing the sea state.  estimates the peak wave period using the moments of the wave spectrum and therefore avoids such behaviour. It is one of three peak period estimators included in this handbook, the others being  and .

 is a peak period estimator specific to Datawell and does not appear to have found wider acceptance in the field of wave analysis.

 

Datawell-specific peak period estimator

The peak period, i.e. the wave period associated with the most energetic waves in the wave spectrum, estimated from the moments of the wave spectrum.

                                                                                 4.8

Because the peak period  is defined as the inverse of the frequency at which  is maximal, the resolution of the parameter is limited by the resolution of the frequencies in the wave spectrum. Moreover, if a wave spectrum is bimodal, it can be the case that the fluctuations between the two energy peaks in the spectrum make the parameter jump between the two peaks, which may be undesirable when analysing the sea state.  estimates the peak wave period using the moments of the wave spectrum and therefore avoids such behaviour. It is one of three peak period estimators included in this handbook, the others being  and .

 is a peak period estimator specific to Datawell and does not appear to have found wider acceptance in the field of wave analysis.

Calculated peak period  (s)

The peak period, i.e. the wave period associated with the most energetic waves in the wave spectrum, estimated from the moments of the wave spectrum, defined by Tucker and Pitt (2001: 41) as:

                                                                                                                                4.9

Because the peak period  is defined as the inverse of the frequency at which  is maximal, the resolution of the parameter is limited by the resolution of the frequencies in the wave spectrum. Moreover, if a wave spectrum is bimodal, it can be the case that the fluctuations between the two energy peaks in the spectrum make the parameter jump between the two peaks, which may be undesirable when analysing the sea state.  estimates the peak wave period using the moments of the wave spectrum and therefore avoids such behaviour. It is one of three peak period estimators included in this handbook, the others being  and .

Spectral width nu |

Spectral width parameter as a function of the moments of the wave spectrum given by Longuet-Higgins (1975). It is used to determine the narrowness of a wave spectrum. A more common notation for nu is .

                                                                     5.0

 values range from 0 to 1. A low  value indicates a narrow spectrum and a relatively regular sea surface (i.e. long-period swell waves). A high  value indicates a broad spectrum and a relatively irregular sea surface (i.e. short short-period wind waves on top of long-period swell waves).

Mathematically, this can be expressed as: : Very narrow bandwidths; : Broad-band wave conditions.

Because  depends on the second-order moment , it is worth noting that the parameter is relatively sensitive to noise in the high-frequency part of the spectrum, although to a far lesser extent than the spectral bandwidth parameter , which utilises the fourth-order moment  (see below). Nonetheless, because in the definition of the moment of the spectrum , high-order moments are enhanced more, utilising the second-order moment  results in the estimate of  still being sensitive to some small errors due to noise in high-frequency part of the wave spectrum.

For this reason, Goda's spectral peakedness parameter  (see below) is often provided besides the spectral width parameter  (and besides the spectral bandwidth parameter , see below).

Spectral bandwidth eps |  

Spectral width parameter as a function of the moments of the wave spectrum defined by Cartwright and Longuet-Higgins (1956). It is used to determine the narrowness of a wave spectrum. A more common notation for eps is .

                                                                     5.1

 values range from 0 to 1. A low  value indicates a broad spectrum and a relatively irregular sea surface (i.e. long-period swell waves). A high  value indicates a broad spectrum and a relatively irregular sea surface (i.e. short short-period wind waves on top of long-period swell waves).

Mathematically, this can be expressed as: : Very narrow bandwidths; : Broad-band wave conditions.

It is worth noting that  depends not only on the shape of the spectrum, but also, to a significant degree, on the high-frequency cut-off (i.e. Nyquist frequency) and the noise in the high-frequency part of the spectrum.

The value of the fourth-order moment  is often dominated by the upper limit of the integration to determine  (i.e. the Nyquist frequency). This is clearly undesirable. Moreover, values of higher-order moments are sensitive to noise in the high-frequency range of the spectrum, where noise can be relatively large. This is because, in the definition of the moment of the spectrum , higher-order moments enhance the energy density at higher frequencies more. In utilising the fourth-order moment  to estimate , the parameter is dependent on that high frequency end of the spectrum. Therefore, the values of , and hence also the estimate of , are sensitive to errors.

For these reasons, the spectral width parameter  (see above) and Goda's spectral peakedness parameter  (see below) are also commonly provided besides .

Goda's spectral peakedness parameter

Spectral peakedness parameter proposed by Goda (1970) as a function of the moments of the wave spectrum. It is used to determine the sharpness of the peak of a wave spectrum.

                                                                                    5.2

A high  value indicates a sharper peak in the wave spectrum, while a low  value indicates a blunter peak.

The spectral peakedness parameter  is somewhat related to the spectral width parameters  and , but is not a simple function of these. Small values of  and , indicating a narrow spectrum and therefore relatively regular sea surface (e.g. long-period swell), are associated with a large value of . Inversely, high values of  and , indicating a broad-band spectrum and therefore a relatively irregular sea surface (i.e. short short-period wind waves on top of long-period swell waves), are associated with a small value of .

Note that, contrary to  and ,  does not range between 0 and 1. A value of  around 0.7 is associated with a value of  ranging from 0.9 to 3.0 (Goda 1970: 15).

It is also worth noting that Goda (1983) identifies a shortcoming in  in that it is sensitive to the resolution of the spectral analysis.

Significant Steepness Ss |

The steepness of the waves as a measure of how high a wave is relative to its length. Therefore, wave steepness is generally defined as a characteristic wave height divided by a characteristic wave length.

Significant steepness is the wave steepness as a function of the spectrally-derived significant wave height  and the spectrally-derived zero-crossing period .

                                                                                                                                   5.3

A large  value indicates that the wave heights are high relative to the wave lengths and therefore the waves are steep. Inversely, a low  value indicates that the wave heights are low relative to the wave lengths and therefore the waves are less steep.

In practice, due to wave breaking, significant steepness  is limited to  approximately in deep water (e.g. Holthuijsen 2007: 88).

 

5.      Sea Surface Temperature

The sea surface temperature measurements.

Reference temperature Tref (°C)

The reference temperature for internal checks. The default is 24.95 °C.

Sea surface temperature SST (°C)

The sea surface temperature measured by the buoy.

It is worth noting that the temperature sensor sits at the bottom and partly inside the mooring eye at the bottom of a buoy that is 90cm in diameter and sits halfway in the water. This means that the water temperature is measured approximately 45cm below the sea surface. The advantages of this placement are that the sensor is not affected by the warmer topmost water layer, it is unaffected by any warming of the top-half of the buoy and it is to a certain extent protected from marine growth.

 

6.      Battery Status

The power remaining in the batteries of the buoy.

Battery

The status of the power in the batteries on a scale of 1 to 7, where 1 = empty and 7 = full.

 

7.      Parameters calculated by the CCO

The two parameters calculated by the CCO, in addition to the standard output from a Datawell DWR MkIII buoy.

Energy period  (s)

The period of an energy equivalent regular wave. In other words, the period corresponding to the weighted average of the wave energy.

 is derived from the zero and first negative order spectral moments.

                                                                                                        7.1

 is principally provided as an alternative to the peak period  for swell-monitoring purposes. As the ratio of the first negative and zeroth order moments of the wave spectrum, the parameters represents more of the lower frequency energy in the spectrum whilst avoiding marked jumps in the time series, typical of .

At the same time,  is the standard wave period used for wave run-up formulae (EurOtop 2018) and may therefore be of particular use to calibrate and provide a check on overtopping models.

See Dhoop & Thompson 2021 for more detail.

Wave power  (kW/m2)

The rate of transfer of energy through each metre of wavefront.

                                                                                                                        7.2

Where  is the density of seawater, taken as 1025 kg/m3,  is acceleration by gravity at 9.81 m/s2,  is the significant wave height, and  is the wave energy period.

 is provided to account for wave height and wave period, which both have distinct and joint influences on coastal events, in a single parameter. Note that  is proportional to significant wave height squared and the wave energy period, therefore providing significantly more weight to wave heights.  is particularly useful in a coastal monitoring context when swell is combined with a significant amount of wind-generated waves. 

See Dhoop & Thompson 2021 for more detail.


8.      References

Cartwright, D.E. and Longuet-Higgins, M.S. 1956. The statistical distribution of the maxima of a random function. Proceedings of the Royal Society A, 237. 212-232.

Datawell. 2020. Datawell Waverider Reference Manual, DWR-MkIII, DWR-G, WR-SG. Heerhugowaard: Datawell BV.

Dhoop, T. and Thompson, C. 2021. Swell wave progression in the English Channel: implications for coastal monitoring. Anthropocene Coasts, 4.1. 281-305.

EurOtop. 2018. Manual on wave overtopping of sea defences and related structures. An overtopping manual largely based on European research, but for worldwide application. Van der Meer, J.W., Allsop, N.W.H., Bruce, T., De Rouck, J., Kortenhaus, A., Pullen, T., Schüttrumpf, H., Troch, P. and Zanuttigh, B., www.overtopping-manual.com

Forristall, G.Z. 1978. On the statistical distribution of wave heights in a storm. Journal of Geophysical Research, 83.C5, 2353-2358.

Goda, Y. 1970. Numerical Experiments on Wave Statistics with Spectral Simulation. Reports of the Port and Harbour Research Institute, Vol. 9, No. 3. Ministry of Transport.

Goda, Y. 1983. Analysis of Wave Grouping and Spectral of Long-travelled Swell. Reports of the Port and Harbour Research Institute, Vol. 22, No. 1. Ministry of Transport.

Goda, Y. 1988. Statistical variability of sea state parameters as a function of a wave spectrum. Coastal Engineering in Japan, 31.1, 39-52.

Holthuijsen, L.H. 2007. Waves in Oceanic and Coastal Waters. Cambridge: Cambridge University Press.

Longuet-Higgins, M.S. 1975. On the joint distribution of the periods and amplitudes of sea waves. Journal of Geophysical Research, 80.18. 2688-2693.

Longuet-Higgins, M.S. 1980. On the distribution of the heights of sea waves: some effects of nonlinearity and finite band-width. Journal of Geophysical Research, 85.C3, 1519–1523.

Nordenstrøm, N. 1969. Methods for Predicting Long Term Distribution of Wave Loads and Probability of Failure for Ships. Appendix II, Relations between Visually Estimated and Theoretical Wave Heights and Periods. Oslo, Det Norske Veritas, Research Department, Report No. 69-22-S.

Tucker, M.J. and Pitt, E.G. 2001. Waves in Ocean Engineering. Amsterdam: Elsevier.