Defines the Pierson Moskowitz spectra in the wave-frequency domain

Contents

Interface Function Documentation Page Comments

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

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

The underlying equation is: where

• 
• 
• is the wind speed at 19.5m above the sea surface.

For a range of typical north sea conditions (where α =0.0081 and =0.5), but with varying peak enhancements the PM spectra has the form

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

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

• is the wind speed at 19.5m above the sea surface

However, , therefore

The relationship between wind speeds at different elevations are given by the expression i.e. is equivalent to

Standards

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

Parameters:

wind	The wind speed 10m above the sea surface. [m/s]

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.

The function uses a 1st order central difference scheme to compute the corresponding value in wave-number, i.e. where

• S() is the original PM spectrum defined in the frequency domain.
• T(k) is the PM spectrum defined in the wave-number domain.

For a range of north sea conditions (where α =0.0081 and =0.5), but with varying peak enhancements the PM spectra has the following form in wave-number:

Parameters:

k	Wave-number (2 π/m)
dk	The discretization in wave-number (representing accuracy of conversion)
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