Difference between revisions of "Statistical description of wave parameters"
Dronkers J (talk  contribs) 
Dronkers J (talk  contribs) 

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[[image:timeseries.jpg400pxthumbleftFig. 1. Timeseries of surface elevations by individual waves for a certain sea state.]]  [[image:timeseries.jpg400pxthumbleftFig. 1. Timeseries of surface elevations by individual waves for a certain sea state.]]  
−  An example of a wave record representative for a certain sea state is shown in Fig. 1. The significant wave height, <math>H_s</math>, is the mean of the highest third of the waves; instead of <math>H_s</math> the notation <math>H_{1/3}</math> is also often used. <math>H_s</math>  +  An example of a wave record representative for a certain sea state is shown in Fig. 1. The ''significant wave height'', <math>H_s</math>, is the mean of the highest third of the waves; instead of <math>H_s</math> the notation <math>H_{1/3}</math> is also often used. <math>H_s</math> represents well the average height of the highest waves in a wave group. The significant wave height can also be computed from the wave energy. For nonbreaking waves it appears that <math>H_s \approx H_{m0} = 4 [{\overline E} / (g \rho)]^{1/2}, </math> where <math>H_{m0}</math> is the ''spectral significant wave height''. An explanation, definitions and formulas are given in appendix A. 
===Mean wave period===  ===Mean wave period===  
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−  The peak wave period, <math>T_p</math>, is the wave period with the highest energy. The analysis of the distribution of the wave energy as a function of wave frequency <math>f=1/T</math> for a timeseries of individual waves is referred to as a spectral analysis. Wind wave periods (frequencies) often follow the socalled JONSWAP or PiersonMoskowitz spectra (see appendix B). The peak wave period is extracted from the spectra. As a rule of thumb the following relation can be used:  +  The peak wave period, <math>T_p</math>, is the wave period with the highest energy. The analysis of the distribution of the wave energy as a function of wave frequency <math>f=1/T</math> for a timeseries of individual waves is referred to as a spectral analysis. Wind wave periods (frequencies) often follow the socalled JONSWAP or PiersonMoskowitz spectra (see appendix B). The peak wave period is extracted from the spectra. As a rule of thumb the following relation can be used, see Fig. 5<ref> Mangor, K., Drønen, N.K., Kaergaard, K.H. and Kristensen, S.E. 2017. Shoreline Management Guidelines. DHI Water and Environment, 451pp.</ref>: 
<math>T_p \approx 5 \sqrt{H_{m0}}. \qquad (1) </math>  <math>T_p \approx 5 \sqrt{H_{m0}}. \qquad (1) </math>  
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==Description of wave conditions==  ==Description of wave conditions==  
−  These parameters are often calculated from continuous or periodic timeseries of the surface elevations; typically the parameters are calculated once every one or three hours, whereby a new discrete timeseries of the statistical wave parameters is constructed. This timeseries is thereafter  +  These various wave parameters are often calculated from continuous or periodic timeseries of the surface elevations; typically the parameters are calculated once every one or three hours, whereby a new discrete timeseries of the statistical wave parameters is constructed. This timeseries is thereafter analyzed statistically to arrive at a condensed description of the wave conditions as follows: 
*Wave height distribution represented by <math>H_s</math> vs. percentage of exceedance. This often follows a Weibulldistribution (see appendix A and the example in Fig. 3);  *Wave height distribution represented by <math>H_s</math> vs. percentage of exceedance. This often follows a Weibulldistribution (see appendix A and the example in Fig. 3);  
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−  Analyses of extreme wave conditions are performed on the basis of max. wave heights in single storm events or on the basis of annual max. wave heights. These analyses are often presented as  +  Analyses of extreme wave conditions are performed on the basis of max. wave heights in single storm events or on the basis of annual max. wave heights. These analyses are often presented as exceedance probability vs. wave heights, see Fig. 6 for an example. 
[[image:extreme value.jpgthumbcenter500pxFig. 6. Extreme value analysis of wave height by Weibull distribution. Threshold = 2 m.]]  [[image:extreme value.jpgthumbcenter500pxFig. 6. Extreme value analysis of wave height by Weibull distribution. Threshold = 2 m.]]  
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<math>p_R(H) = \Large\frac{2H}{H_{rms}^2}\normalsize \exp\Large (–(\frac{H}{H_{rms}})^2)\normalsize . \qquad (A3)</math>  <math>p_R(H) = \Large\frac{2H}{H_{rms}^2}\normalsize \exp\Large (–(\frac{H}{H_{rms}})^2)\normalsize . \qquad (A3)</math>  
−  The mean square  +  The root mean square wave height (also called mean energy wave height) <math>H_{rms}</math> is related to the average wave energy <math>\overline E</math>: 
<math> H_{rms}^2 = \int_0^{\infty} p_R(H) H^2 dH =\Large\frac{8}{g \rho}\normalsize \overline E. \qquad (A4)</math>  <math> H_{rms}^2 = \int_0^{\infty} p_R(H) H^2 dH =\Large\frac{8}{g \rho}\normalsize \overline E. \qquad (A4)</math>  
−  The average wave height <math>\overline H</math> is related to the mean square  +  The average wave height <math>\overline H</math> is related to the root mean square wave height <math>H_{rms}</math> by 
<math>\overline H= \int_0^{\infty} p_R(H) H dH = \Large\frac{\sqrt{\pi}}{2}\normalsize H_{rms}  <math>\overline H= \int_0^{\infty} p_R(H) H dH = \Large\frac{\sqrt{\pi}}{2}\normalsize H_{rms}  
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<math>P_R(H)=\int_0^H p_R(H')dH' = 1\exp\Large ((\frac{H}{H_{rms}})^2)\normalsize .\qquad (A6) </math>  <math>P_R(H)=\int_0^H p_R(H')dH' = 1\exp\Large ((\frac{H}{H_{rms}})^2)\normalsize .\qquad (A6) </math>  
−  For  +  Assuming that wave heights are Reynolds distributed, relations can be derived between different wave parameters that are often used in practice. For the derivation of the significant wave height <math>H_s \equiv H_{1/3}</math> (the mean of the highest third of the waves) we first determine the lowest of the largest third waves, <math>H_3</math>, from the condition <math>P_R(H_3)=2/3</math>, yielding <math>H_3=H_{rms} \ln(3)</math>. The significant wave height is then related to the root mean square wave height by 
−  <math>H_s=\Large\frac{\int_{  +  <math>H_s=\Large\frac{\int_{H_3}^{\infty} p_R(H)HdH}{\int_{H_3}^{\infty} p_R(H)dH }\normalsize \approx 1.6 \overline H = 1.42 H_{rms}. \qquad (A7)</math> 
−  From (A4  +  From Eqs. (A4) and (A7), it follows that the significant wave height <math>H_s</math> is related to the average wave energy <math>\overline E</math> by 
<math>H_s \approx H_{m0} \equiv 4 \Large \sqrt{ \frac{\overline E}{g \rho}}\normalsize . \qquad(A8)</math>  <math>H_s \approx H_{m0} \equiv 4 \Large \sqrt{ \frac{\overline E}{g \rho}}\normalsize . \qquad(A8)</math>  
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Extreme wave heights can be derived from the Rayleigh distribution in a similar way. For example, the mean of the 1% highest waves is given by  Extreme wave heights can be derived from the Rayleigh distribution in a similar way. For example, the mean of the 1% highest waves is given by  
−  <math>H_{1/100} \approx  +  <math>H_{1/100} \approx 1.52 H_s . \qquad (A9)</math> 
In spite of the restrictive conditions for which the Rayleigh distribution has been derived, it appears that in many cases it corresponds reasonably well with wave height statistics obtained from field observations, even if the conditions (a) and (b) are not well satisfied.  In spite of the restrictive conditions for which the Rayleigh distribution has been derived, it appears that in many cases it corresponds reasonably well with wave height statistics obtained from field observations, even if the conditions (a) and (b) are not well satisfied.  
−  However, the Rayleigh distribution does not put a limit on the wave height, which is physically unrealistic and leads to overestimation of the highest waves. Therefore often the Weibull distribution is used instead of the Rayleigh distribution. The Weibull distribution <ref>https://en.wikipedia.org/wiki/Weibull_distribution</ref>  +  However, the Rayleigh distribution does not put a limit on the wave height, which is physically unrealistic and leads to overestimation of the highest waves. Therefore often the Weibull distribution is used instead of the Rayleigh distribution. The Weibull distribution reads <ref>https://en.wikipedia.org/wiki/Weibull_distribution</ref> : 
<math>p_W(H)=\Large\frac{m}{\lambda}(\frac{H}{\lambda})^{(m1)}\normalsize \exp\Large (–(\frac{H}{\lambda})^{m}) \normalsize . \qquad (A10)</math>  <math>p_W(H)=\Large\frac{m}{\lambda}(\frac{H}{\lambda})^{(m1)}\normalsize \exp\Large (–(\frac{H}{\lambda})^{m}) \normalsize . \qquad (A10)</math>  
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The Rayleigh distribution corresponds to the Weibull distribution for <math>m=2, \; \lambda=H_{rms}</math>. The Weibull distribution has an additional parameter (<math>m</math>) that allows suppression of the highest waves for <math>m>2</math> and an optimum adjustment to the observed wave data.  The Rayleigh distribution corresponds to the Weibull distribution for <math>m=2, \; \lambda=H_{rms}</math>. The Weibull distribution has an additional parameter (<math>m</math>) that allows suppression of the highest waves for <math>m>2</math> and an optimum adjustment to the observed wave data.  
−  This is especially relevant for shallowwater waves, which are truncated due to depthinduced wave breaking. For this situation, alternative distributions have been proposed, for example by Battjes and Groenendijk (2000)<ref> Battjes, J.A. and Groenendijk, H.W. 2000. Wave height distributions on shallow foreshores. Coastal Engineering 40: 161182</ref>. According to this study, a Weibull distribution with <math>m=3.6</math> should be used above a certain threshold, <math>H_{tr}</math> (threshold for depthinduced wave breaking).  +  This is especially relevant for shallowwater waves, which are truncated due to depthinduced wave breaking (see [[Shallowwater wave theory]]). For this situation, alternative distributions have been proposed, for example by Battjes and Groenendijk (2000)<ref name=BG> Battjes, J.A. and Groenendijk, H.W. 2000. Wave height distributions on shallow foreshores. Coastal Engineering 40: 161182</ref>. According to this study, a Weibull distribution with <math>m=3.6</math> should be used above a certain threshold, <math>H_{tr}</math> (threshold for depthinduced wave breaking). This implies that the relationships (A7A9) are not valid in shallow water. For example, if <math>H_{tr}<H_3</math> (very shallow water), Eq. (A9) should be replaced by <ref name=BG></ref> 
+  
+  <math>H_{1/100} \approx 1.28 H_s . \qquad (A11)</math>  
+  
==Appendix B: Frequency spectrum==  ==Appendix B: Frequency spectrum== 
Revision as of 17:05, 18 March 2020
Because of the random nature of natural waves, a statistical description of the waves is normally always used. Observed wave heights often follow the Rayleigh distribution. Statistical wave parameters are calculated based on this distribution. The most commonly used variables in coastal engineering are described below.
Contents
Most commonly used variables in coastal engineering
Significant wave height
An example of a wave record representative for a certain sea state is shown in Fig. 1. The significant wave height, [math]H_s[/math], is the mean of the highest third of the waves; instead of [math]H_s[/math] the notation [math]H_{1/3}[/math] is also often used. [math]H_s[/math] represents well the average height of the highest waves in a wave group. The significant wave height can also be computed from the wave energy. For nonbreaking waves it appears that [math]H_s \approx H_{m0} = 4 [{\overline E} / (g \rho)]^{1/2}, [/math] where [math]H_{m0}[/math] is the spectral significant wave height. An explanation, definitions and formulas are given in appendix A.
Mean wave period
The mean wave period, [math]T_m[/math], is the mean of all wave periods in a timeseries representing a certain sea state.
Peak wave period
The peak wave period, [math]T_p[/math], is the wave period with the highest energy. The analysis of the distribution of the wave energy as a function of wave frequency [math]f=1/T[/math] for a timeseries of individual waves is referred to as a spectral analysis. Wind wave periods (frequencies) often follow the socalled JONSWAP or PiersonMoskowitz spectra (see appendix B). The peak wave period is extracted from the spectra. As a rule of thumb the following relation can be used, see Fig. 5^{[1]}:
[math]T_p \approx 5 \sqrt{H_{m0}}. \qquad (1) [/math]
Mean wave direction
The mean wave direction, [math]\theta_m[/math], is defined as the mean of all the individual wave directions in a timeseries representing a certain sea state.
Description of wave conditions
These various wave parameters are often calculated from continuous or periodic timeseries of the surface elevations; typically the parameters are calculated once every one or three hours, whereby a new discrete timeseries of the statistical wave parameters is constructed. This timeseries is thereafter analyzed statistically to arrive at a condensed description of the wave conditions as follows:
 Wave height distribution represented by [math]H_s[/math] vs. percentage of exceedance. This often follows a Weibulldistribution (see appendix A and the example in Fig. 3);
 Directional distribution of the wave heights, which is often presented in the form of a wave rose (see appendix B and the example in Fig. 4);
 Scatter diagram of [math]T_p[/math] vs. [math]H_s[/math] (example in Fig. 5).
Analyses of extreme wave conditions are performed on the basis of max. wave heights in single storm events or on the basis of annual max. wave heights. These analyses are often presented as exceedance probability vs. wave heights, see Fig. 6 for an example.
Further reading
 Mangor, K., Drønen, N.K., Kaergaard, K.H. and Kristensen, S.E. 2017. Shoreline Management Guidelines. DHI Water and Environment, 451pp. ^{[2]}
 Coastal Engineering Manual, part II, chapter 1. US Army Corps of Engineers (USACE), 2008
Appendix A: Rayleigh distribution
The wave contribution to the ocean sea level [math]\eta(x,y,t)[/math] at a certain location [math]x,y[/math] is generally a superposition of a large number [math]n[/math] of random waves with amplitudes [math]a_j[/math], radial frequencies [math]\omega_j[/math] and random phases [math]\phi_j[/math], originating from different nearby and remote regions. This superposition can be represented by
[math]\eta=Re[\sum_{j=1}^{n} a_j \exp(i\omega_j t + i\phi_j)]. \qquad (A1)[/math]
The statistical distribution of wave heights was derived by LonguetHiggins (1952)^{[3]} under a few specific conditions: (a) the random numbers [math]a=\sum_{j=1}^{n} a_j \cos\phi_j, \, b=\sum_{j=1}^{n} a_j \sin\phi_j[/math] are statistically independent and normally (Gaussian) distributed; (b) the radial frequencies [math]\omega_j[/math] of the random waves are grouped in a single narrow band around a central frequency [math]\omega[/math] such that [math]\omega_j \omega_j'/ \omega \lt \lt 1[/math] for each [math]j, j'[/math]. Under this last condition the expression (A1) may be approximated for the time interval [[math]\pi / \omega \lt t\lt \pi / \omega[/math]] by
[math]\eta \approx Re[ \exp(i \omega t) \sum_{j=1}^{n} a_j \exp(i\phi_j)] \equiv \frac{1}{2} H \, Re[\exp(i \omega t+i \phi)] , \quad H = 2 \sqrt{a^2 + b^2} . \qquad (A2)[/math]
A wellknown mathematical theorem^{[4]} states that the length of a vector with Gaussian distributed components follows the Rayleigh distribution. In this case the vector length is the wave height [math]H[/math] and the components are the random numbers [math]2a, 2b[/math]. The Rayleigh probability density function [math]p_R(H)[/math] for the wave height [math]H[/math] reads:
[math]p_R(H) = \Large\frac{2H}{H_{rms}^2}\normalsize \exp\Large (–(\frac{H}{H_{rms}})^2)\normalsize . \qquad (A3)[/math]
The root mean square wave height (also called mean energy wave height) [math]H_{rms}[/math] is related to the average wave energy [math]\overline E[/math]:
[math] H_{rms}^2 = \int_0^{\infty} p_R(H) H^2 dH =\Large\frac{8}{g \rho}\normalsize \overline E. \qquad (A4)[/math]
The average wave height [math]\overline H[/math] is related to the root mean square wave height [math]H_{rms}[/math] by
[math]\overline H= \int_0^{\infty} p_R(H) H dH = \Large\frac{\sqrt{\pi}}{2}\normalsize H_{rms} \approx 0.89 H_{rms} .\qquad (A5)[/math]
The cumulative Rayleigh distribution (probability of wave height [math]\lt H[/math]) is given by
[math]P_R(H)=\int_0^H p_R(H')dH' = 1\exp\Large ((\frac{H}{H_{rms}})^2)\normalsize .\qquad (A6) [/math]
Assuming that wave heights are Reynolds distributed, relations can be derived between different wave parameters that are often used in practice. For the derivation of the significant wave height [math]H_s \equiv H_{1/3}[/math] (the mean of the highest third of the waves) we first determine the lowest of the largest third waves, [math]H_3[/math], from the condition [math]P_R(H_3)=2/3[/math], yielding [math]H_3=H_{rms} \ln(3)[/math]. The significant wave height is then related to the root mean square wave height by
[math]H_s=\Large\frac{\int_{H_3}^{\infty} p_R(H)HdH}{\int_{H_3}^{\infty} p_R(H)dH }\normalsize \approx 1.6 \overline H = 1.42 H_{rms}. \qquad (A7)[/math]
From Eqs. (A4) and (A7), it follows that the significant wave height [math]H_s[/math] is related to the average wave energy [math]\overline E[/math] by
[math]H_s \approx H_{m0} \equiv 4 \Large \sqrt{ \frac{\overline E}{g \rho}}\normalsize . \qquad(A8)[/math]
Extreme wave heights can be derived from the Rayleigh distribution in a similar way. For example, the mean of the 1% highest waves is given by
[math]H_{1/100} \approx 1.52 H_s . \qquad (A9)[/math]
In spite of the restrictive conditions for which the Rayleigh distribution has been derived, it appears that in many cases it corresponds reasonably well with wave height statistics obtained from field observations, even if the conditions (a) and (b) are not well satisfied.
However, the Rayleigh distribution does not put a limit on the wave height, which is physically unrealistic and leads to overestimation of the highest waves. Therefore often the Weibull distribution is used instead of the Rayleigh distribution. The Weibull distribution reads ^{[5]} :
[math]p_W(H)=\Large\frac{m}{\lambda}(\frac{H}{\lambda})^{(m1)}\normalsize \exp\Large (–(\frac{H}{\lambda})^{m}) \normalsize . \qquad (A10)[/math]
The Rayleigh distribution corresponds to the Weibull distribution for [math]m=2, \; \lambda=H_{rms}[/math]. The Weibull distribution has an additional parameter ([math]m[/math]) that allows suppression of the highest waves for [math]m\gt 2[/math] and an optimum adjustment to the observed wave data.
This is especially relevant for shallowwater waves, which are truncated due to depthinduced wave breaking (see Shallowwater wave theory). For this situation, alternative distributions have been proposed, for example by Battjes and Groenendijk (2000)^{[6]}. According to this study, a Weibull distribution with [math]m=3.6[/math] should be used above a certain threshold, [math]H_{tr}[/math] (threshold for depthinduced wave breaking). This implies that the relationships (A7A9) are not valid in shallow water. For example, if [math]H_{tr}\lt H_3[/math] (very shallow water), Eq. (A9) should be replaced by ^{[6]}
[math]H_{1/100} \approx 1.28 H_s . \qquad (A11)[/math]
Appendix B: Frequency spectrum
A wave record can further be characterized by its frequency spectrum. The energy density spectrum of a sea state is generally designated by [math]E(f)[/math]. The total energy is given by
[math]\overline E=\int_0^{\infty} E(f)df . \qquad (B1)[/math]
The wave frequency spectrum can be determined from a wave record by using a Fourier transform. It can also be determined by modelling the windinduced wave field in a large source area. Empirical formulas have been established for fully developed wave fields under constant wind stress. For deep water without fetch restriction, it is recommended to use the adapted PiersonMoskowitz frequency distribution [math]E_{PM}[/math] ^{[7]}:
[math]E_{PM}(f)=3.26 \Large\frac{\overline E}{f_p}(\frac{f_p}{f})^4 e^{(\frac{f_p}{f})^4}\normalsize . \qquad (B2)[/math]
For the average wave energy [math]\overline E[/math] and the peak frequency [math]f_p[/math] the following empirical expressions are found:
[math]\overline E \approx 0.005 \rho g^{1} U_{10}^4\; , \; f_p \approx 0.123 g U_{10}^{1} \;,[/math]
where [math]g[/math] is the gravitational acceleration and [math]U_{10}[/math] the wind velocity at 10 m above the sea surface.
For fetchlimited seas, the spectrum is more strongly peaked around the peak frequency. For this situation, the empirical JONSWAP spectrum can be used. It has the form
[math]E_{WONSJAP}=\Large\frac{\alpha \rho g^3}{f_p (2 \pi f_p)^4} (\frac{f_p}{f})^4 e^{(\frac{f_p}{f})^4}\normalsize \gamma^\delta \;, \quad \delta =e^{\Large\frac{1}{2}\Large(\frac{(f/f_p)1}{\sigma})^2}\normalsize , \qquad (B3)[/math],
where the parameters [math]\alpha, \gamma, \sigma, f_p[/math] depend on the fetch length and should be fitted to the wave data. For [math]\gamma=1[/math] the PiersonMoskowitz and JONSWAP spectra are the same.
Different characteristic wave periods can be derived from the wave spectrum: the significant wave period [math]T_{01}[/math], the mean wave period [math]T_{02}[/math] and the mean energy period [math]T_E[/math]. They are given by the expressions
[math]T_{01}=\Large\frac{\int_0^{\infty} E(f)df}{\int_0^{\infty} E(f)fdf }\normalsize, \quad T_{02}=\Large \sqrt{\frac{\int_0^{\infty} E(f)df}{\int_0^{\infty} E(f) f^2 df }}\normalsize, \quad T_E=\Large\frac{\int_0^{\infty} E(f) f^{1} df}{\int_0^{\infty} E(f)df }\normalsize \; .\qquad (B4) [/math]
For the PiersonMoskowitz distribution (B2) we have
[math]T_{01}=0.69 \, T_p, \; T_{02}=0.58 \, T_p, \; T_E=0.82 \, T_p [/math].
A closer estimate of the peak period [math]T_p=1/f_p[/math] can be obtained by using a higher power of the energy density spectrum, for example
[math]\Large\frac{\int_0^{\infty} E^5(f)df}{\int_0^{\infty} E^5(f)fdf }\normalsize \approx 0.95 T_p[/math].
In the fetchlimited case with a JONSWAPtype spectrum the value of [math]T_E[/math] is generally found close to the peak period [math]T_p[/math].
In an irregular wave field, waves may come from different directions. The wave incidence direction is an important parameter for sediment transport in the coastal zone. Waves originating from different areas may have different spectra. The directional spread of incoming waves for a particular wave frequency can be represented by a distribution function [math]D(f,\theta)[/math], where [math]\theta[/math] is the wave incidence angle. We then have
[math]\overline E \equiv \int_0^{\infty} \int_0^{2 \pi} S(f, \theta) df d \theta \equiv \int_0^{\infty} \int_0^{2 \pi} E(f) \, D(f, \theta) df d \theta \; ,\quad \int_0^{2 \pi} D(f, \theta) d \theta =1 \; .\qquad (B5) [/math]
The directional wave spectrum [math]S(f, \theta)[/math] can be derived from directional wave buoys. In practice, it is often obtained by numerical modelling of the wave field in the major source area.
References
 ↑ Mangor, K., Drønen, N.K., Kaergaard, K.H. and Kristensen, S.E. 2017. Shoreline Management Guidelines. DHI Water and Environment, 451pp.
 ↑ https://www.dhigroup.com/upload/campaigns/ShorelineManagementGuidelines_Feb2017.pdf
 ↑ LonguetHiggins, M.S. 1952. On the statistical distribution of sea waves. Journal of Marine Research 11: 245266
 ↑ https://en.wikipedia.org/wiki/Rayleigh_distribution
 ↑ https://en.wikipedia.org/wiki/Weibull_distribution
 ↑ ^{6.0} ^{6.1} Battjes, J.A. and Groenendijk, H.W. 2000. Wave height distributions on shallow foreshores. Coastal Engineering 40: 161182
 ↑ The Rock Manual. The use of rock in hydraulic engineering (2nd edition).CIRIA. London, 2007
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