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Pierson Moskowitz

Defines the Pierson Moskowitz spectra in the wave-frequency domain
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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
  • \alpha is a numerical constant =0.0081
  • \beta is a numerical constant =0.74
  • g is gravity
  • Missing Equation End: $U_{19.4}Missing Equation End: $ is the wind speed at 19.4m above the sea surface.

References

Standards

  • These functions conform to British Standards (BS 6349-1:2000), 24 July 2003.
  • These functions conform to European ISO standards 19901-1:2005

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 (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 \alpha=0.0081
betacontrols the shape factor, \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 (\omega_p): where
  • \alpha = 0.0081
  • \omega_p = 0.877g/(\pi U_{19.5})
  • \beta = 1.25

For a range of typical north sea conditions (where α =0.0081 and \omega_p=2 \pi/12.4=0.5), but with varying peak enhancements the PM spectra has the form
\graph  w=0:1.4, wp=0.5:0.8:4, alpha=0.0081, beta=1.25

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 (H_s) and the peak wave period (T_p)):

For sea state with H_s=4.0m,
\graph  w=0:2, Hs=4, Tp=10:6:3

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 (H_s) and the zero crossing period (T_z)):

For sea state with H_s=4.0m,
\graph  w=0:2, Hs=4, Tz=10:6:3

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
  • U_{19.4} is the wind speed at 19.5m above the sea surface

The relationship between wind speeds at different elevations are given by the expression i.e. U_{19.5}=22.55\;m/s is equivalent to 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 \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 \alpha that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra: where
  • H_s is the significant wave heights (m)
  • T_p is the peak wave period (s)

Parameters

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

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PM Tz To Alpha

 
doublePM_Tz_to_alphadoubleHs
doubleTz )
Returns a factor \alpha that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra: where
  • H_s is the significant wave heights (m)
  • T_z is the zero crossing wave period (s)

Parameters

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

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PM Tz To Tp

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

where
  • T_p is the peak wave period (s)
  • T_z is the zero crossing wave period (s)

Parameters

Tzzero crossing wave period, T_z.

Returns

the peak wave period, T_p
Source Code

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PM Gnnk Wp

PM Gnnk Wp Calculator
 
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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 \omega_p=2 \pi/12.4=0.5), but with varying peak enhancements the PM spectra has the following form in wave-number:
\graph  k=0:0.1, wp=0.5, depth=0, alpha=0.0081, beta=1.25

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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