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Pierson Moskowitz

Defines the Pierson Moskowitz spectra in the wave-frequency domain
Controller:

Dependents

Info

Interface

C++

Overview

The Pierson-Moskowitz (PM) spectra is an empirical relationship that defines the distribution of energy with frequency within the ocean.

Developed in 1964 the PM spectrum is one of the simplest descriptions for the energy distribution. It assumes that if the wind blows steadily for a long time over a large area, then the waves will eventually reach a point of equilibrium with the wind. This is known as a fully developed sea. Pierson and Moskowitz developed their spectrum from measurements in the North Atlantic during 1964, and presented the following relationship between energy distribution and wind:
15422/PMspectrum2.jpg
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where

References

Standards

PM Gnnw U

 
doublePM_Gnnw_Udoublew
doubleU
doublealpha = 0.0081
doublebeta = 1.25 )
The original generic PM spectra, defined by wind speed:

Parameters

wwave-frequency (\inline 2\pi/s)
Uis the wind speed at 19.4m above the sea surface (m/s)
alphacontrols the intensity of the Spectra, the default value is \inline \alpha=0.0081
betacontrols the shape factor, \inline \beta=1.25
Source Code

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PM Gnnw Wp

 
doublePM_Gnnw_wpdoublew
doublewp
doublealpha = 0.0081
doublebeta = 1.25 )
The PM spectra defined by a spectral peak frequency (\inline \omega_p): where

For a range of typical north sea conditions (where α =0.0081 and \inline  \omega_p=2 \pi/12.4=0.5), but with varying peak enhancements the PM spectra has the form
There is an error with your graph parameters for PM_Gnnw_wp with options w=0:1.4 wp=0.5:0.8:4

Error Message:Function PM_Gnnw_wp failed. Ensure that: Invalid C++

Parameters

wwave-frequency (2 π/s)
wpthe peak wave frequency (2 π/s)
alphaThe intensity of the Spectra. Default value = 0.0081
betaA shape factor. Default value = 1.25
Source Code

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PM Gnnw Tp

 
doublePM_Gnnw_Tpdoublew
doubleHs
doubleTp )
The PM spectra defined by the significant wave height (\inline H_s) and the peak wave period (\inline T_p)):

For sea state with \inline H_s=4.0m,
There is an error with your graph parameters for PM_Gnnw_Tp with options w=0:2 Hs=4 Tp=10:6:3

Error Message:Function PM_Gnnw_Tp failed. Ensure that: Invalid C++

Parameters

wwave-frequency (2 π/s)
Hssignificant wave height (m)
Tppeak wave period (s)
Source Code

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PM Gnnw Tz

 
doublePM_Gnnw_Tzdoublew
doubleHs
doubleTz )
The PM spectra defined by the significant wave height (\inline H_s) and the zero crossing period (\inline T_z)):

For sea state with \inline H_s=4.0m,
There is an error with your graph parameters for PM_Gnnw_Tz with options w=0:2 Hs=4 Tz=10:6:3

Error Message:Function PM_Gnnw_Tz failed. Ensure that: Invalid C++

Parameters

wwave-frequency (2 π/s)
Hssignificant wave height (m)
Tzzero crossing period (s)
Source Code

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PM Wind To Wp

 
doublePM_wind_to_wpdoublewind )
The peak frequency of the PM spectrum is based empirically on wind speed,

where

The relationship between wind speeds at different elevations are given by the expression i.e. \inline  U_{19.5}=22.55\;m/s is equivalent to \inline  U_{10}=20.6\;m/s

Parameters

windThe wind speed 19.4m above the sea surface. [m/s]

Returns

peak wave frequency (rad/s)
Source Code

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PM Wind To Tp

 
doublePM_wind_to_Tpdoublewind )
Converts wind speed to peak wave period: where \inline \omega_p is defined by (6).

Parameters

windThe wind speed 19.4m above the sea surface. [m/s]

Returns

peak wave frequency (rad/s)
Source Code

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PM Tp To Alpha

 
doublePM_Tp_to_alphadoubleHs
doubleTp )
Returns a factor \inline \alpha that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra: where

Parameters

Hssignificant wave height (m), i.e. \inline H_s=4m.
Tppeak wave period, i.e. \inline T_p=10s.
Source Code

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PM Tz To Alpha

 
doublePM_Tz_to_alphadoubleHs
doubleTz )
Returns a factor \inline \alpha that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra: where

Parameters

Hssignificant wave height (m), i.e. \inline H_s=4m.
Tzpeak wave period, i.e. \inline T_z=10s.
Source Code

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PM Tz To Tp

 
doublePM_Tz_to_TpdoubleTz )
Returns the zero crossing wave period (\inline T_z) associated with the peak wave period (\inline T_p). The relationship between these periods comes from: which reduces to or

where

Parameters

Tzzero crossing wave period, \inline T_z.

Returns

the peak wave period, \inline T_p
Source Code

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PM Gnnk Wp

 
doublePM_Gnnk_wpdoublek
doublewp
doubledepth = 0
doublealpha = 0.0081
doublebeta = 1.25 )
This function uses the description of the PM spectra described in frequency to obtain an estimate of the distribution in wave-number using the 1st order dispersion relationship give in dispersion.

This conversion is thus

where in deep water and in shallow water

For a range of north sea conditions (where α =0.0081 and \inline  \omega_p=2 \pi/12.4=0.5), but with varying peak enhancements the PM spectra has the following form in wave-number:
There is an error with your graph parameters for PM_Gnnk_wp with options k=0:0.1 dk=0.01 wp=0.5 alpha=0.0081

Error Message:Function PM_Gnnk_wp failed. Ensure that: Invalid C++

Parameters

kWave-number (2 π/m)
wpThe peak wave frequency
depthThe water depth. Default value=0 (infinite depth)
alphaThe intensity of the spectra. Default value = 0.01
betaDefault value = 1.25
Source Code

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Last Modified: 28 Jul 14 @ 21:50     Page Rendered: 2022-03-14 02:32:51