The JONSWAP spectra in the wave-frequency domain

Contents

Interface Function Documentation Page Comments

Group Description

The JONSWAP (Joint North Sea Wave Project) spectra is an empirical relationship that defines the distribution of energy with frequency within the ocean.

The JONSWAP spectrum is effectively a fetch-limited version of the Pierson-Moskowitz spectrum, except that the wave spectrum is never fully developed and may continue to develop due to non-linear wave-wave interactions for a very long time. Therefore in the JONSWAP spectrum, waves continues to grow with distance (or time) as specified by the α (alpha) term, and the peak in the spectrum is more pronounced, as specified by the γ (gamma) term. Hasselmann (1966) found the latter to be particularly important as it lead to enhanced non-linear interactions.

References:

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

Interface

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

using namespace Engineering::Waves::Spectra;

	double	JONSWAP_Gnnw (double w, double wp, double alpha=0.0081, double gamma=3.3, double beta=1.25) Defines the JONSWAP spectra in the wave-frequency domain
	double	JONSWAP_wp (double wind, double length) Compute the peak frequency for a JONSWAP spectrum based on wind speed and fetch length.
	double	JONSWAP_alpha (double wind, double length) Compute the constant α that defines the intensity of the JONSWAP spectra from wind speed and fetch length.
	double	JONSWAP_Gnnk (double k, double dk, double wp, double depth=0, double alpha=0.0081,double gamma=3.3, double beta=1.25) Defines the Jonswap spectra in the wave-number domain

Function Documentation

The JONSWAP (Joint North Sea Wave Project) spectra is an empirical relationship that defines the distribution of energy with frequency within the ocean.

The underlying equation is: where

• 
• 
• 
• α is a constant that relates to the wind speed and fetch length, see below. Typical values in the northern north sea are in the range of 0.0081 to 0.01
• ω is the wave frequency
• is the peak wave-frequency

Most problems is the literature are expressed in the above form. However if a particular wind speed and fetch length are known, then α and can be estimated using the subsequent two functions.

For a range of typical north sea conditions (where α =0.0081 and =0.5), but with varying peak enhancements the JONSWAP 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
gamma	Peak enhancement factor. Default value = 3.3
beta	A shape factor (Rarely changed). Default value = 1.25

The peak of the JONSWAP spectrum is empirically define by where

• is the wind speed at 10m above the sea surface
• is the fetch length

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]
length	The fetch length. [m]

The overall energy within the JONSWAP spectrum is controlled by the α constant and is related to wind speed and the peak frequency by: where

• is the wind speed at 10m above the sea surface
• is the peak frequency calculated using equation (2)

This function uses JONSWAP_wp (above) to obtain w_p for a given fetch length and wind speed.

Parameters:

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

This function uses the description of the JONSWAP 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 JONSWAP spectrum defined in the frequency domain.
• T(k) is the JONSWAP 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 JONSWAP 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
gamma	The peak enhancement factor. Default value = 3.3
beta	A shape factor (Rarely changed). Default value = 1.25