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Engineering › Fluid Mechanics › Waves › Spectra ›

Pierson Moskowitz

Defines the Pierson Moskowitz spectra in the wave-frequency domain

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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
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Name	HTML Code
PM_Gnnw_U Defines the Pierson-Moskowitz spectra with Wind speed	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=5"></script>
PM_Gnnw_wp Defines the Pierson-Moskowitz spectra with a spectral peak frequency	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=6"></script>
PM_Gnnw Alias to PM_Gnnw_wp(...)	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=7"></script>
PM_Gnnw_Tp Defines the Pierson-Moskowitz spectra with a significant wave height and peak period	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=8"></script>
PM_Gnnw_Tz Defines the Pierson-Moskowitz spectra with a significant wave height and zero crossing period	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=9"></script>
PM_wind_to_wp Peak wave frequency based on wind speed.	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=10"></script>
PM_wind_to_Tp Peak wave period based on wind speed	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=11"></script>
PM_Tp_to_alpha Energy density factor ($\alpha$), based on significant wave height ($H_s$) and peak period ($T_p$)	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=12"></script>
PM_Tz_to_alpha Energy density factor ($\alpha$), based on significant wave height ($H_s$) and zero crossing period ($T_z$)	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=13"></script>
PM_Tz_to_Tp Coverts zero cross wave period $T_z$ to peak wave period $T_p$.	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=14"></script>
PM_Gnnk_wp Defines the Pierson-Moskowitz (PM) spectra in the wave-number domain	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=15"></script>
PM_Gnnk Alias to PM_Gnnk_wp	<script type="text/javascript" src="http://www.codecogs.com/ic_json-466&u=16"></script>

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:

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

(1)

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

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm

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

PM Gnnw U Calculator

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

The original generic PM spectra, defined by wind speed:

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

(2)

Parameters

w	wave-frequency ( $2\pi/s$ )
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 $\alpha=0.0081$
beta	controls the shape factor, $\beta=1.25$

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PM Gnnw Wp

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

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

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

(3)

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

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

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PM Gnnw Tp

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

The PM spectra defined by the significant wave height (

) and the peak wave period (

)):

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

(4)

For sea state with

$\graph w=0:2, Hs=4, Tp=10:6:3$

Parameters

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

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PM Gnnw Tz

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

The PM spectra defined by the significant wave height (

) and the zero crossing period (

)):

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

(5)

For sea state with

$\graph w=0:2, Hs=4, Tz=10:6:3$

Parameters

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

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PM Wind To Wp

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doublePM_wind_to_wp( double wind )

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

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

(6)

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

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

(7)

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

Parameters

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

Returns

peak wave frequency (rad/s)

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PM Wind To Tp

PM Wind To Tp Calculator

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doublePM_wind_to_Tp( double wind )

Converts wind speed to peak wave period:

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

(8)

where $\omega_p$ is defined by (6).

Parameters

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

Returns

peak wave frequency (rad/s)

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PM Tp To Alpha

PM Tp To Alpha Calculator

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doublePM_Tp_to_alpha(	double	`Hs`
	double	`Tp`	)

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

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

(9)

where

is the significant wave heights (m)
is the peak wave period (s)

Parameters

Hs	significant wave height (m), i.e. .
Tp	peak wave period, i.e. .

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PM Tz To Alpha

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doublePM_Tz_to_alpha(	double	`Hs`
	double	`Tz`	)

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

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

(10)

where

is the significant wave heights (m)
is the zero crossing wave period (s)

Parameters

Hs	significant wave height (m), i.e. .
Tz	peak wave period, i.e. .

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PM Tz To Tp

PM Tz To Tp Calculator

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doublePM_Tz_to_Tp( double Tz )

Returns the zero crossing wave period (

) associated with the peak wave period (

). The relationship between these periods comes from:

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

(11)

which reduces to

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

(12)

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

(13)

where

is the peak wave period (s)
is the zero crossing wave period (s)

Parameters

Tz	zero crossing wave period, .

Returns

the peak wave period,

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PM Gnnk Wp

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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

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

(14)

where in deep water

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

(15)

and in shallow water

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

(16)

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

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

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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 ( $\alpha$ ), based on significant wave height () and peak period ()
double	`PM_Tz_to_alpha` (double Hs, double Tz) Energy density factor ( $\alpha$ ), based on significant wave height () and zero crossing period ()
double	`PM_Tz_to_Tp` (double Tz) Coverts zero cross wave period to peak wave period .
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