Pierson_Moskowitz — Spectra

Defines the Pierson Moskowitz spectra in the wave-frequency domain

Interface

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

using namespace Engineering::Fluid_Mechanics::Waves::Spectra;

double PM_Gnnw_U(double w, double U, double alpha=0.0081, double beta=0.74)
double PM_Gnnw_wp(double w, double wp, double alpha=0.0081, double beta=1.25)
double PM_Gnnw(double w, double wp, double alpha=0.0081, double beta=1.25)
double PM_Gnnw_Tp(double w, double Hs, double Tp)
double PM_Gnnw_Tz(double w, double Hs, double Tz)
double PM_wind_to_wp(double wind)
double PM_wind_to_Tp(double wind)
double PM_Tp_to_alpha(double Hs, double Tp)
double PM_Tz_to_alpha(double Hs, double Tz)
double PM_Tz_to_Tp(double Tz)
double PM_Gnnk_wp(double k, double wp, double depth=0, double alpha=0.0081, double beta=1.25)
double PM_Gnnk(double k, double wp, double depth=0, double alpha=0.0081, double beta=1.25)

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:

This can be generalised by:

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

(1)

where

$\alpha$ is a numerical constant =0.0081
$\beta$ is a numerical constant =0.74
g is gravity
$U_{19.4}$ is the wind speed at 19.4m above the sea surface.

References

http://oceanworld.tamu.edu/resources/ocng_textbook/chapter16/chapter16_04.htm
Massel, S.R. 1996. Ocean Surface Waves: Their Physics and Predictions. Advanced Series on Coastal Engineering. 11. London

Standards

These functions conform to British Standards (BS 6349-1:2000), 24 July 2003.
These functions conform to European ISO standards 19901-1:2005

FUNCTION

PM_Gnnw_U

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

wave-frequency ($2\pi/s$)

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, the default value is $\beta=0.74$

Interactive Calculator

alpha

beta

Result –

FUNCTION

PM_Gnnw_wp

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

$\alpha = 0.0081$
$\omega_p = 0.877g/U_{19.5}$
$\beta = 1.25$

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

Parameters

wave-frequency (2 π/s)

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

Returns

The PM spectral density at frequency w

Interactive Calculator

alpha

beta

Result –

FUNCTION

PM_Gnnw

Parameters

wave-frequency (2 π/s)

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

Returns

The PM spectral density at frequency w

Interactive Calculator

alpha

beta

Result –

FUNCTION

PM_Gnnw_Tp

The PM spectra defined by the significant wave height ($H_s$) and the peak wave period ($T_p$)):

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

(4)

For sea state with $H_s=4.0m$, \graph w=0:2 Hs=4 Tp=10:6:3

Parameters

wave-frequency (2 π/s)

significant wave height (m)

peak wave period (s)

Returns

The PM spectral density at frequency w

Interactive Calculator

Result –

FUNCTION

PM_Gnnw_Tz

The PM spectra defined by the significant wave height ($H_s$) and the zero crossing period ($T_z$)):

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

(5)

For sea state with $H_s=4.0m$, \graph w=0:2 Hs=4 Tz=10:6:3

Parameters

wave-frequency (2 π/s)

significant wave height (m)

zero crossing period (s)

Returns

The PM spectral density at frequency w

Interactive Calculator

Result –

FUNCTION

PM_wind_to_wp

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)

Interactive Calculator

wind

Result –

FUNCTION

PM_wind_to_Tp

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 period (s)

Interactive Calculator

wind

Result –

FUNCTION

PM_Tp_to_alpha

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

$H_s$ is the significant wave heights (m)
$T_p$ is the peak wave period (s)

Parameters

significant wave height (m), i.e. $H_s=4m$.

peak wave period, i.e. $T_p=10s$.

Returns

The energy density factor $\alpha$

Interactive Calculator

Result –

FUNCTION

PM_Tz_to_alpha

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

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

Parameters

significant wave height (m), i.e. $H_s=4m$.

zero crossing wave period, i.e. $T_z=10s$.

Returns

The energy density factor $\alpha$

Interactive Calculator

Result –

FUNCTION

PM_Tz_to_Tp

Returns the zero crossing wave period ($T_z$) associated with the peak wave period ($T_p$). 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

$T_p$ is the peak wave period (s)
$T_z$ is the zero crossing wave period (s)

Parameters

zero crossing wave period, $T_z$.

Returns

the peak wave period, $T_p$

Interactive Calculator

Result –

FUNCTION

PM_Gnnk_wp

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

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

(15)

and in shallow water

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

(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 dk=0.01 wp=0.5 alpha=0.0081

Parameters

Wave-number (2 π/m)

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

Returns

The PM spectral density in the wave-number domain

Interactive Calculator

depth

alpha

beta

Result –

FUNCTION

PM_Gnnk

Parameters

Wave-number (2 π/m)

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

Returns

The PM spectral density in the wave-number domain

Interactive Calculator

depth

alpha

beta

Result –