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# Pierson Moskowitz

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

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:
$S_{PM}(\omega)&space;=&space;\frac{\alpha&space;g^2}{\omega^5}&space;exp&space;\left&space;[-\beta&space;\left&space;(\frac{g}{\omega&space;U_{19.4}}&space;\right&space;)^4&space;&space;\right&space;]$
where
• $\inline&space;\alpha$ is a numerical constant =0.0081
• $\inline&space;\beta$ 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.

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

### Parameters

 w wave-frequency ($\inline&space;2\pi/s$) 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 $\inline&space;\alpha=0.0081$ beta controls the shape factor, $\inline&space;\beta=1.25$
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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 ($\inline&space;\omega_p$):
$S_{PM}(\omega)&space;=&space;\frac{\alpha&space;g^2}{\omega^5}&space;exp&space;\left&space;[-\beta}&space;\left&space;(\frac{\omega_p}{\omega}&space;\right&space;)^4&space;&space;\right&space;]$
where
• $\inline&space;\alpha&space;=&space;0.0081$
• $\inline&space;\omega_p&space;=&space;0.877g/(\pi&space;U_{19.5})$
• $\inline&space;\beta&space;=&space;1.25$

For a range of typical north sea conditions (where α =0.0081 and $\inline&space;&space;\omega_p=2&space;\pi/12.4$=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
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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 ($\inline&space;H_s$) and the peak wave period ($\inline&space;T_p$)):
$S_{PM}(\omega)&space;=&space;5&space;\pi^4&space;\frac{H_s^2}{T_p^4}&space;\cdot&space;\frac{1}{\omega^5}&space;exp&space;\left&space;[-\frac{20&space;\pi^4}{T_p^4}\cdot&space;\frac{1}{\omega^4}&space;\right&space;]$

For sea state with $\inline&space;H_s=4.0m$,
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 ($\inline&space;H_s$) and the zero crossing period ($\inline&space;T_z$)):
$S_{PM}(\omega)&space;=&space;4&space;\pi^3&space;\frac{H_s^2}{T_z^4}&space;\cdot&space;\frac{1}{\omega^5}&space;exp&space;\left&space;[-\frac{16&space;\pi^3}{T_z^4}\cdot&space;\frac{1}{\omega^4}&space;\right&space;]$

For sea state with $\inline&space;H_s=4.0m$,
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,
$\omega_p&space;=&space;\frac{4&space;\beta}{5}^{\tfrac{1}{4}}&space;\frac{g}{U_{19.4}}$

where
• $\inline&space;U_{19.4}$ is the wind speed at 19.5m above the sea surface

The relationship between wind speeds at different elevations are given by the expression
$U_z&space;=&space;U_w&space;\cdot&space;(z/w)^{1/7}$
i.e. $\inline&space;&space;U_{19.5}=22.55\;m/s$ is equivalent to $\inline&space;&space;U_{10}=20.6\;m/s$

### Parameters

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

### Returns

##### Source Code

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## PM Wind To Tp

 doublePM_wind_to_Tp( double wind )
Converts wind speed to peak wave period:
$T_p&space;=&space;\frac{2\pi}{\omega_p}$
where $\inline&space;\omega_p$ is defined by (6).

### Parameters

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

### Returns

##### Source Code

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## PM Tp To Alpha

 doublePM_Tp_to_alpha( double Hs double Tp )
Returns a factor $\inline&space;\alpha$ that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra:
$\alpha&space;=&space;\frac{5\pi^4}{g^2}&space;\frac{H_s^2}{T_p^4}$
where
• $\inline&space;H_s$ is the significant wave heights (m)
• $\inline&space;T_p$ is the peak wave period (s)

### Parameters

 Hs significant wave height (m), i.e. $\inline&space;H_s=4m$. Tp peak wave period, i.e. $\inline&space;T_p=10s$.
##### Source Code

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## PM Tz To Alpha

 doublePM_Tz_to_alpha( double Hs double Tz )
Returns a factor $\inline&space;\alpha$ that provides a linear scaling of the wave energy within both the PM and JONSWAP spectra:
$\alpha&space;=&space;\frac{4\pi^3}{g^2}&space;\frac{H_s^2}{T_z^4}$
where
• $\inline&space;H_s$ is the significant wave heights (m)
• $\inline&space;T_z$ is the zero crossing wave period (s)

### Parameters

 Hs significant wave height (m), i.e. $\inline&space;H_s=4m$. Tz peak wave period, i.e. $\inline&space;T_z=10s$.
##### Source Code

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## PM Tz To Tp

 doublePM_Tz_to_Tp( double Tz )
Returns the zero crossing wave period ($\inline&space;T_z$) associated with the peak wave period ($\inline&space;T_p$). The relationship between these periods comes from:
$5\pi^4\frac{{H_s}^2}{{T_p}^4}&space;=&space;4\pi^3\frac{{H_s}^2}{{T_z}^4}$
which reduces to
${T_p}^4=\frac{5\pi}{4}{T_z}^4$
or
${T_p}=\left(\frac{5\pi}{4}{T_z}^4&space;\right&space;)^\tfrac{1}{4}$

where
• $\inline&space;T_p$ is the peak wave period (s)
• $\inline&space;T_z$ is the zero crossing wave period (s)

### Parameters

 Tz zero crossing wave period, $\inline&space;T_z$.

### Returns

the peak wave period, $\inline&space;T_p$
##### 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
$G_{\eta\eta}(k)&space;=&space;G_{\eta\eta}(\omega)&space;\frac{\partial&space;\omega}{\partial&space;k}$

where in deep water
$\frac{\partial&space;\omega}{\partial&space;k}&space;=&space;\frac{g}{2\omega}$
and in shallow water
$\frac{\partial&space;\omega}{\partial&space;k}&space;=&space;\frac{g}{2\omega}&space;\left&space;[&space;k&space;sech^2(k&space;d)&space;+&space;tanh(k&space;d)&space;\right&space;]$

For a range of north sea conditions (where α =0.0081 and $\inline&space;&space;\omega_p=2&space;\pi/12.4$=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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