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Pierson Moskowitz

Defines the Pierson Moskowitz spectra in the wave-frequency domain

Controller:

Contents

Interface
PM Gnnw U
PM Gnnw Wp
PM Gnnw Tp
PM Gnnw Tz
PM Wind To Wp
PM Wind To Tp
PM Tp To Alpha
PM Tz To Alpha
PM Tz To Tp
PM Gnnk Wp
Page Comments

Dependents

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:

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$S_{PM}(\omega) = \frac{\alpha g^2}{\omega^5} exp \left [-\beta \left (\frac{g}{\omega U_{19.4}} \right )^4 \right ]$

(2)

where

$\inline \alpha$ is a numerical constant =0.0081
$\inline \beta$ 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

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm
Massel, S.R. 1996. Ocean Surface Waves: Their Physics and Predictions. Advanced Series on Coastal Engineering. 11. London

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_U(	double	`w`
	double	`U`
	double	`alpha` `= 0.0081`
	double	`beta` `= 1.25`	)

The original generic PM spectra, defined by wind speed:

$S_{PM}(\omega) = \frac{\alpha g^2}{\omega^5} exp \left [-\beta \left (\frac{g}{\omega U} \right )^4 \right ]$

(3)

Parameters

w	wave-frequency ( $\inline 2\pi/s$ )
U	is the wind speed at 19.4m above the sea surface (m/s)
alpha	controls the intensity of the Spectra, the default value is $\inline \alpha=0.0081$
beta	controls the shape factor, $\inline \beta=1.25$

Source Code

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PM Gnnw Wp

doublePM_Gnnw_wp(	double	`w`
	double	`wp`
	double	`alpha` `= 0.0081`
	double	`beta` `= 1.25`	)

The PM spectra defined by a spectral peak frequency ( $\inline \omega_p$ ):

$S_{PM}(\omega) = \frac{\alpha g^2}{\omega^5} exp \left [-\beta} \left (\frac{\omega_p}{\omega} \right )^4 \right ]$

(4)

where

$\inline \alpha = 0.0081$
$\inline \omega_p = 0.877g/(\pi U_{19.5})$
$\inline \beta = 1.25$

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

w	wave-frequency (2 π/s)
wp	the peak wave frequency (2 π/s)
alpha	The intensity of the Spectra. Default value = 0.0081
beta	A shape factor. Default value = 1.25

Source Code

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PM Gnnw Tp

doublePM_Gnnw_Tp(	double	`w`
	double	`Hs`
	double	`Tp`	)

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

$S_{PM}(\omega) = 5 \pi^4 \frac{H_s^2}{T_p^4} \cdot \frac{1}{\omega^5} exp \left [-\frac{20 \pi^4}{T_p^4}\cdot \frac{1}{\omega^4} \right ]$

(5)

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

w	wave-frequency (2 π/s)
Hs	significant wave height (m)
Tp	peak wave period (s)

Source Code

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PM Gnnw Tz

doublePM_Gnnw_Tz(	double	`w`
	double	`Hs`
	double	`Tz`	)

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

$S_{PM}(\omega) = 4 \pi^3 \frac{H_s^2}{T_z^4} \cdot \frac{1}{\omega^5} exp \left [-\frac{16 \pi^3}{T_z^4}\cdot \frac{1}{\omega^4} \right ]$

(6)

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

w	wave-frequency (2 π/s)
Hs	significant wave height (m)
Tz	zero crossing period (s)

Source Code

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PM Wind To Wp

doublePM_wind_to_wp( double wind )

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

$\omega_p = \frac{4 \beta}{5}^{\tfrac{1}{4}} \frac{g}{U_{19.4}}$

(7)

where

$\inline U_{19.4}$ is the wind speed at 19.5m above the sea surface

The relationship between wind speeds at different elevations are given by the expression

$U_z = U_w \cdot (z/w)^{1/7}$

(8)

i.e. $\inline U_{19.5}=22.55\;m/s$ is equivalent to $\inline U_{10}=20.6\;m/s$

Parameters

wind	The 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_Tp( double wind )

Converts wind speed to peak wave period:

$T_p = \frac{2\pi}{\omega_p}$

(9)

where $\inline \omega_p$ is defined by (7).

Parameters

wind	The 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_alpha(	double	`Hs`
	double	`Tp`	)

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

$\alpha = \frac{5\pi^4}{g^2} \frac{H_s^2}{T_p^4}$

(10)

where

$\inline H_s$ is the significant wave heights (m)
$\inline T_p$ is the peak wave period (s)

Parameters

Hs	significant wave height (m), i.e. $\inline H_s=4m$ .
Tp	peak wave period, i.e. $\inline T_p=10s$ .

Source Code

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PM Tz To Alpha

doublePM_Tz_to_alpha(	double	`Hs`
	double	`Tz`	)

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

$\alpha = \frac{4\pi^3}{g^2} \frac{H_s^2}{T_z^4}$

(11)

where

$\inline H_s$ is the significant wave heights (m)
$\inline T_z$ is the zero crossing wave period (s)

Parameters

Hs	significant wave height (m), i.e. $\inline H_s=4m$ .
Tz	peak wave period, i.e. $\inline T_z=10s$ .

Source Code

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PM Tz To Tp

doublePM_Tz_to_Tp( double Tz )

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:

$5\pi^4\frac{{H_s}^2}{{T_p}^4} = 4\pi^3\frac{{H_s}^2}{{T_z}^4}$

(12)

which reduces to

${T_p}^4=\frac{5\pi}{4}{T_z}^4$

(13)

${T_p}=\left(\frac{5\pi}{4}{T_z}^4 \right )^\tfrac{1}{4}$

(14)

where

$\inline T_p$ is the peak wave period (s)
$\inline T_z$ is the zero crossing wave period (s)

Parameters

Tz	zero 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_wp(	double	`k`
	double	`wp`
	double	`depth` `= 0`
	double	`alpha` `= 0.0081`
	double	`beta` `= 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

$G_{\eta\eta}(k) = G_{\eta\eta}(\omega) \frac{\partial \omega}{\partial k}$

(15)

where in deep water

$\frac{\partial \omega}{\partial k} = \frac{g}{2\omega}$

(16)

and in shallow water

$\frac{\partial \omega}{\partial k} = \frac{g}{2\omega} \left [ k sech^2(k d) + tanh(k d) \right ]$

(17)

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

k	Wave-number (2 π/m)
wp	The peak wave frequency
depth	The water depth. Default value=0 (infinite depth)
alpha	The intensity of the spectra. Default value = 0.01
beta	Default value = 1.25

Source Code

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double	`PM_Gnnw_U` (double w, double U, double alpha=0.0081, double beta=1.25) Defines the Pierson-Moskowitz spectra with Wind speed
double	`PM_Gnnw_wp` (double w, double wp, double alpha=0.0081, double beta=1.25) Defines the Pierson-Moskowitz spectra with a spectral peak frequency
double	`PM_Gnnw` (double w, double wp, double alpha=0.0081, double beta=1.25) Alias to PM_Gnnw_wp(...)
double	`PM_Gnnw_Tp` (double w, double Hs, double Tp) Defines the Pierson-Moskowitz spectra with a significant wave height and peak period
double	`PM_Gnnw_Tz` (double w, double Hs, double Tz) Defines the Pierson-Moskowitz spectra with a significant wave height and zero crossing period
double	`PM_wind_to_wp` (double wind) Peak wave frequency based on wind speed.
double	`PM_wind_to_Tp` (double wind) Peak wave period based on wind speed
double	`PM_Tp_to_alpha` (double Hs, double Tp) Energy density factor ( $\inline \alpha$ ), based on significant wave height ( $\inline H_s$ ) and peak period ( $\inline T_p$ )
double	`PM_Tz_to_alpha` (double Hs, double Tz) Energy density factor ( $\inline \alpha$ ), based on significant wave height ( $\inline H_s$ ) and zero crossing period ( $\inline T_z$ )
double	`PM_Tz_to_Tp` (double Tz) Coverts zero cross wave period $\inline T_z$ to peak wave period $\inline T_p$ .
double	`PM_Gnnk_wp` (double k, double wp, double depth=0, double alpha=0.0081, double beta=1.25) Defines the Pierson-Moskowitz (PM) spectra in the wave-number domain
double	`PM_Gnnk` (double k, double wp, double depth=0, double alpha=0.0081, double beta=1.25) Alias to PM_Gnnk_wp