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Pierson Moskowitz

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www.codecogs.com/d-ox/engineering/waves/spectra/pierson_moskowitz.php

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Group Description

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 would eventually reach a point of equilibrium with the wind. This is known as a fully developed sea. This spectrum evolved from measurements (and collated data) from Moskowitz (1064) who formed the following graph relating energy distribution to wind:

References:

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

Interface

#include <codecogs/engineering/waves/spectra/pierson_moskowitz.h>

using namespace Engineering::Waves::Spectra;

	double	`PM_Gnnw` (double w, double wp, double alpha=0.0081, double beta=1.25) Defines the Pierson-Moskowitz spectra in the wave-frequency domain
	double	`PM_wp` (double wind) Compute the peak circular frequency for a JONSWAP spectrum based on wind speed and fetch length.
	double	`PM_Gnnk` (double k, double dk, double wp, double depth=0, double alpha=0.0081, double beta=1.25) Defines the Pierson-Moskowitz (PM) spectra in the wave-number domain

Function Documentation

PM Gnnw Calculator

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

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

The underlying equation is:

(1)

$\displaystyle S(\omega) = \frac{\alpha g^2}{(2\pi)^2 \omega^4} exp \left [-\frac{5}{4} \frac{\omega_p}{\omega}^4 \right ]$

where

$\displaystyle \alpha = 0.0081$
$\displaystyle \omega_p = 0.877g/(\pi U_{19.5})$
$\displaystyle U_{19.5}$ is the wind speed at 19.5m above the sea surface.

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

$\graph w=0:1.4, wp=0.5, alpha=0.0081, beta=0.5:1.5:5$

Standards

This function conforms to British Standards (BS 6349-1:2000), 24 July 2003.

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 Wp Calculator

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doublePM_wp( double wind )

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

(2)

$\displaystyle f_p = \frac{0.877\;g}{U_{19.5}}$

where

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

However, $\omega_p = 2\pi f$ , therefore

(3)

$\displaystyle \omega_p = \frac{2 \pi 0.877\;g}{U_{19.5}}$

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

(4)

$\displaystyle U_z = U_w (z/10)^{1/7}$

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