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13/09/2020
1
Waves
3. Statistics and Irregular Waves
Statistics and Irregular Waves
Statistics and Irregular Waves
โ Real wave fields are not regular
โ Combination of many periods, heights, directions
โ Design and simulation require realistic wave statistics:
โ probability distribution of heights
โ energy spectrum of frequencies (and directions)
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3. STATISTICS AND IRREGULAR WAVES
3.1 Measures of height and period
3.2 Probability distribution of wave heights
3.3 Wave spectra
3.4 Reconstructing a wave field
3.5 Prediction of wave climate
Statistics and Irregular Waves
Measures of Wave Height
๐ป๐๐๐ฅ Largest wave height in the sample ๐ป๐๐๐ฅ = max๐(๐ป๐)
๐ป๐๐ฃ๐ Mean wave height ๐ป๐๐ฃ๐ =1
๐๐ป๐
๐ป๐๐๐ Root-mean-square wave height ๐ป๐๐๐ =1
๐๐ป๐
2
๐ป ฮค1 3 Average of the highest ๐/3 waves ๐ป ฮค1 3 =1
๐/3
1
๐/3
๐ป๐
๐ป๐0 Estimate based on the rms surface elevation ๐ป๐0 = 4 ฮท2ฮค1 2
๐ป๐ Significant wave height Either ๐ป ฮค1 3 or ๐ป๐0
Measures of Wave Period
๐๐ Significant wave period (average of highest ๐/3waves)
๐๐ Peak period (from peak frequency of energy spectrum)
๐๐Energy period (period of a regular wave with same significant wave height andpower density; used in wave-energy prediction; derived from energy spectrum)
๐๐ง Mean zero up-crossing period
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3. STATISTICS AND IRREGULAR WAVES
3.1 Measures of height and period
3.2 Probability distribution of wave heights
3.3 Wave spectra
3.4 Reconstructing a wave field
3.5 Prediction of wave climate
Statistics and Irregular Waves
Probability Distribution of Wave Heights
For a narrow-banded frequency spectrum the Rayleigh probability distribution is appropriate:
๐ height > ๐ป = exp โ ฮค(๐ป )๐ป๐๐๐ 2
Cumulative distribution function:
๐น ๐ป = ๐ height < ๐ป = 1 โ eโ ฮค๐ป ๐ป๐๐๐ 2
Probability density function:
๐ ๐ป =d๐น
d๐ป= 2
๐ป
๐ป๐๐๐ 2 eโ ฮค๐ป ๐ป๐๐๐
2
Rayleigh Distribution
๐ป๐๐ฃ๐ โก ๐ธ ๐ป = เถฑ0
โ
๐ป ๐ ๐ป d๐ป
๐ป๐๐๐ 2 โก ๐ธ ๐ป2 = เถฑ
0
โ
๐ป2 ๐ ๐ป d๐ป
๐ป๐๐ฃ๐ =ฯ
2๐ป๐๐๐ = 0.886 ๐ป๐๐๐
๐ป ฮค1 3 = 1.416 ๐ป๐๐๐
๐ป ฮค1 10 = 1.800 ๐ป๐๐๐
๐ป ฮค1 100 = 2.359 ๐ป๐๐๐
๐ height > ๐ป = exp โ ฮค(๐ป )๐ป๐๐๐ 2
Single parameter: ๐ป๐๐๐
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Example
Near a pier, 400 consecutive wave heights are measured.Assume that the sea state is narrow-banded.
(a) How many waves are expected to exceed 2๐ป๐๐๐ ?
(b) If the significant wave height is 2.5 m, what is๐ป๐๐๐ ?
(c) Estimate the wave height exceeded by 80 waves.
(d) Estimate the number of waves with a height between1.0 m and 3.0 m.
Near a pier, 400 consecutive wave heights are measured. Assume that the sea state isnarrow-banded.(a) How many waves are expected to exceed 2๐ป๐๐๐ ?(b) If the significant wave height is 2.5 m, what is ๐ป๐๐๐ ?(c) Estimate the wave height exceeded by 80 waves.(d) Estimate the number of waves with a height between 1.0 m and 3.0 m.
๐ height > ๐ป = eโ ฮค๐ป ๐ป๐๐๐ 2
๐ height > 2๐ป๐๐๐ = eโ4(a)
= 0.01832
In 400 waves, ๐ = 400 ร 0.01832
๐ป ฮค1 3 = 1.416 ๐ป๐๐๐ (b)
2.5 = 1.416 ๐ป๐๐๐
๐ฏ๐๐๐ = ๐. ๐๐๐ ๐ฆ
(c) ๐ height > ๐ป = eโ ฮค๐ป ๐ป๐๐๐ 2=
80
400= 0.2
๐ป/๐ป๐๐๐ 2 = โ ln 0.2
๐ป = ๐ป๐๐๐ ร โ ln0.2 = ๐. ๐๐๐ ๐ฆ
๐ 1.0 < height < 3.0 = ๐ height > 1.0 โ ๐ height > 3.0
= eโ 1.0/๐ป๐๐๐ 2โ eโ 3.0/๐ป๐๐๐
2
= 0.6699In 400 waves, ๐ = 400 ร 0.6699
(d)
= ๐. ๐๐๐
= ๐๐๐. ๐
3. STATISTICS AND IRREGULAR WAVES
3.1 Measures of height and period
3.2 Probability distribution of wave heights
3.3 Wave spectra
3.4 Reconstructing a wave field
3.5 Prediction of wave climate
Statistics and Irregular Waves
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Regular vs Irregular Waves
Regular wave:- single frequency
Irregular wave:- many frequencies
Wave energy depends on ฮท2
Energy Spectrum
โ โSpectralโ means โby frequencyโ
โ A spectrum is usually determined by a Fourier transform
โ This splits a signal up into its component frequencies
โ Energy in a wave is proportional to ฮท2, where ฮท is surface displacement
โ The energy spectrum, or power spectrum, is the Fourier transform of ฮท2
Energy Spectrum
For regular waves (single frequency):
๐ธ =1
2ฯ๐๐ด2 = ฯ๐ )ฮท2(๐ก =
1
8ฯ๐๐ป2
For irregular waves (many frequencies):
๐ ๐ d๐
เถฑ๐1
๐2
๐ ๐ d๐
is the โenergyโ in a small interval d๐ near frequency ๐
is the โenergyโ between frequencies ๐1 and ๐2
(strictly: energy/ฯ๐)
The energy spectrum ๐(๐) is determined by Fourier transforming ฮท2
ฮท = ๐ด cos ๐๐ฅ โ ฯ๐ก
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Model SpectraOpen ocean: Bretschneider spectrum
Fetch-limited seas: JONSWAP spectrum
Key parameters: peak frequency ๐๐ ( =1/(peak period, ๐๐) )
significant wave height ๐ป๐0
Model Spectra
Bretschneider spectrum:
JONSWAP spectrum:
๐ ๐ =5
16๐ป๐02
๐๐4
๐5exp โ
5
4
๐๐4
๐4
๐ ๐ = ๐ถ ๐ป๐02
๐๐4
๐5exp โ
5
4
๐๐4
๐4ฮณ๐
๐ = เต0.07 ๐ < ๐๐0.09 ๐ > ๐๐
๐ = exp โ1
2
ฮค๐ ๐๐ โ 1
ฯ
2
ฮณ = 3.3
Significant Wave Height, ๐ฏ๐๐
Bretschneider spectrum: ๐ ๐ =5
16๐ป๐02
๐๐4
๐5exp โ
5
4
๐๐4
๐4
Total energy:๐ธ
ฯ๐โก เถฑ
0
โ
๐ ๐ d๐ =1
16๐ป๐02
๐ป๐0 = 4 ฮค๐ธ ฯ ๐ = 4 )ฮท2(๐ก
Total energy for a regular wave:๐ธ
ฯ๐=1
8๐ป๐๐๐ 2
Same energy if ๐ป๐0 = 2๐ป๐๐๐ = 1.414๐ป๐๐๐
Rayleigh distribution: ๐ป๐ = ๐ป ฮค1 3 = 1.416๐ป๐๐๐
๐ฏ๐๐ and ๐ฏ ฮค๐ ๐ can be used synonymously for ๐ฏ๐ ...
โฆ but ๐ป๐0 is easier to measure!
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Finding Wave Parameters From a Spectrum
โ Surface displacement ฮท(๐ก) is measured (e.g. wave buoy)
โ ฮท2(๐ก) is Fourier-transformed to get energy spectrum ๐(๐)
โ Height and period parameters are deduced from the peak and the momentsof the spectrum:
๐๐ = เถฑ0
โ
๐๐๐ ๐ d๐
Peak period: ๐๐ =1
๐๐
Significant wave height: ๐ป๐ = ๐ป๐0 = 4 ๐0
Energy period: ๐๐ =๐โ1
๐0
Multi-Modal Spectra
frequency f (Hz)
spect
ral d
ensi
ty S
(m
^2 s
)
swell
wind
3. STATISTICS AND IRREGULAR WAVES
3.1 Measures of height and period
3.2 Probability distribution of wave heights
3.3 Wave spectra
3.4 Reconstructing a wave field
3.5 Prediction of wave climate
Statistics and Irregular Waves
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Using a Spectrum To Generate a Wave Field
ฮท ๐ก = )๐๐ cos(๐๐๐ฅ โ ฯ๐๐ก โ ๐๐
amplitude
๐ ๐๐ d๐ = ๐ธ๐ =1
2๐๐2 ๐๐ = 2๐ ๐๐ d๐
(angular) frequency
ฯ๐ = 2ฯ๐๐
wavenumber
๐๐2 = ๐๐๐ tanh๐๐โ
random phase
Simulated Wave Field๐๐ = 8 s
๐ป๐ = 1.0 m
โ = 30 mRegular waves:
Irregular waves:
Focused waves:
Example
An irregular wavefield at a deep-water location is characterisedby peak period of 8.7 s and significant wave height of 1.5 m.
(a) Provide a sketch of a Bretschneider spectrum, labelling bothaxes with variables and units and indicating the frequenciescorresponding to both the peak period and the energy period.
Note: Calculations are not needed for this part.
(b) Determine the power density (in kW mโ1) of a regular wavecomponent with frequency 0.125 Hz that represents thefrequency range 0.12 to 0.13 Hz of the irregular wave field.
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An irregular wavefield at a deep-water location is characterised by peak period of 8.7 s andsignificant wave height of 1.5 m.
(a) Provide a sketch of a Bretschneider spectrum, labelling both axes with variables and unitsand indicating the frequencies corresponding to both the peak period and the energyperiod.
frequency f (Hz)
spect
ral d
ensi
ty S
(m
^2 s
)
fp fe
An irregular wavefield at a deep-water location is characterised by peak period of 8.7 s andsignificant wave height of 1.5 m.
(b) Determine the power density (in kW mโ1) of a regular wave component with frequency0.125 Hz that represents the frequency range 0.12 to 0.13 Hz of the irregular wave field.
๐ = 0.125 Hz
ฮ๐ = 0.01 Hz
๐๐ = 8.7 s ๐ป๐ = 1.5 m
๐๐ =1
๐๐
๐ ๐ = 1.645 m2 s
๐ธ = ฯ๐ ร ๐ ๐ ฮ๐
๐ = ๐ธ๐๐ = ๐ธ(๐๐) Deep water: ๐ =1
2
๐ =๐๐
2ฯ
๐ =1
๐= 8 s
๐ท = ๐. ๐๐๐ ๐ค๐๐ฆโ๐
= 0.1149 Hz
= 165.4 J mโ2
= 12.49 m sโ1
๐ ๐ =5
16๐ป๐ 2๐๐4
๐5ex p( โ
5
4
๐๐4
๐4)
3. STATISTICS AND IRREGULAR WAVES
3.1 Measures of height and period
3.2 Probability distribution of wave heights
3.3 Wave spectra
3.4 Reconstructing a wave field
3.5 Prediction of wave climate
Statistics and Irregular Waves
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Terminology
A wave climate or sea state is a model wave spectrum, usually defined by a representative height and period, for use in:
โ forecasting (from a weather forecast in advance)
โ nowcasting (from ongoing weather)
โ hindcasting (reconstruction from previous event)
Prediction of Sea State
Require: significant wave height, ๐ป๐ significant wave period, ๐๐ (or peak period, ๐๐)
Depend on: wind speed, ๐fetch, ๐นduration, ๐กgravity, ๐
Dimensional analysis:
Empirical functions: JONSWAP or SMB curves
เตฐ๐๐ป๐ ๐2 = ๐๐ข๐๐๐ก๐๐๐(
๐๐น
๐2 ,๐๐ก
๐
แ๐๐๐๐
= ๐๐ข๐๐๐ก๐๐๐(๐๐น
๐2 ,๐๐ก
๐
Fetch-Limited vs Duration-Limited
distance travelled by wave energy greater than fetch F
F
distance travelled by wave energy less than fetch F
F
Feff
FETCH-LIMITED
DURATION-LIMITED
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JONSWAP CurvesFor fetch-limited waves:
๐๐ป๐ ๐2
= 0.0016๐๐น
๐2
ฮค1 2
(up to maximum 0.2433)
๐๐๐๐
= 0.286๐๐น
๐2
ฮค1 3
(up to maximum 8.134)
Waves are fetch-limited provided the storm has blown for a minimum time๐ก๐๐๐ given by
๐๐ก
๐ ๐๐๐= 68.8
๐๐น
๐2
ฮค2 3
up to maximum 7.15 ร 104
Otherwise the waves are duration-limited, and the fetch used to determineheight and period is an effective fetch determined by inversion using the actualstorm duration ๐ก:
๐๐น
๐2๐๐๐
=1
68.8
๐๐ก
๐
ฮค3 2
JONSWAP Curves (reprise)
๐ป๐ = 0.0016 ๐น ฮค1 2
๐๐ = 0.286 ๐น ฮค1 3
ฦธ๐ก๐๐๐ = 68.8 ๐น ฮค2 3
๐น โก๐๐น
๐2 , ฦธ๐ก โก๐๐ก
๐, ๐ป๐ โก
๐๐ป๐ ๐2 , ๐๐ โก
๐๐๐๐
JONSWAP and SMB Curves
Solid lines: JONSWAP
Dashed lines: SMB
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Example
(a) Wind has blown at a consistent ๐ = 20m sโ1
over a fetch ๐น = 100 km for ๐ก = 6 hrs .Determine๐ป๐ and ๐๐ using the JONSWAP curves.
(b) If the wind blows steadily for another 4 hourswhat are ๐ป๐ and ๐๐?
(a) Wind has blown at a consistent ๐ = 20 m sโ1 over a fetch ๐น = 100 km for ๐ก = 6 hrs.Determine ๐ป๐ and ๐๐ using the JONSWAP curves.
๐ = 20 m sโ1
๐น = 105 m
๐ก = 6 ร 3600 = 21600 s
๐น โก๐๐น
๐2
ฦธ๐ก =๐๐ก
๐
ฦธ๐ก๐๐๐ = 68.8 ๐น ฮค2 3
ฦธ๐ก๐๐๐ = 68.8 ๐น ฮค2 3 = 12510
๐ป๐ = 0.0016 ๐น1/2
๐๐ = 0.286 ๐น1/3
๐น โก๐๐น
๐2ฦธ๐ก โก
๐๐ก
๐๐ป๐ โก
๐๐ป๐ ๐2
Duration-limited
๐น๐๐๐ =ฦธ๐ก
68.8
ฮค3 2
๐ป๐ = 0.0016 ๐น๐๐๐ฮค1 2
๐๐ = 0.286 ๐น๐๐๐ฮค1 3
๐ป๐ = ๐ป๐ ร๐2
๐
๐๐ = ๐๐ ร๐
๐
= 1910
= 0.06993
= 3.548
= ๐. ๐๐๐ ๐ฆ
= ๐. ๐๐๐ ๐ฌ
= 2453
= 10590
(b) If the wind blows steadily for another 4 hours what are ๐ป๐ and ๐๐?
๐ = 20 m sโ1
๐น = 105 m
๐ก = 10 ร 3600 = 36000 s
๐น = 2453
ฦธ๐ก =๐๐ก
๐
ฦธ๐ก๐๐๐ = 68.8 ๐น ฮค2 3
ฦธ๐ก๐๐๐ = 12510
๐ป๐ = 0.0016 ๐น1/2
๐๐ = 0.286 ๐น1/3
๐น โก๐๐น
๐2ฦธ๐ก โก
๐๐ก
๐๐ป๐ โก
๐๐ป๐ ๐2
Fetch-limited
๐ป๐ = 0.0016 ๐น1/2
๐๐ = 0.286 ๐น1/3
๐ป๐ = ๐ป๐ ร๐2
๐
๐๐ = ๐๐ ร๐
๐
= 0.07924
= 3.857
= ๐. ๐๐๐ ๐ฆ
= ๐. ๐๐๐ ๐ฌ
= 17660
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Examples on Blackboard
Try examples 10-13
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