Ocean-Wave Spectra - Revision history

Kri: /* JONSWAP Spectrum */ Why substitute x for F?? F is a variable just like x.

2012-12-09T20:48:28Z

JONSWAP Spectrum: Why substitute x for F?? F is a variable just like x.

← Older revision		Revision as of 20:48, 9 December 2012
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	The energy of the waves increases with fetch:		The energy of the waves increases with fetch:

	<center><math>\left \langle \zeta^2\right \rangle =1.67 \times 10^{-7} \frac{(U_{10})^2}{g}x</math></center>		<center><math>\left \langle \zeta^2\right \rangle =1.67 \times 10^{-7} \frac{(U_{10})^2}{g}F.</math></center>

	~~where x is fetch.~~

	The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].		The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].

Kri: /* JONSWAP Spectrum */ This is incorrect (alpha is defined differently for the two different spectra)

2012-12-09T20:37:16Z

JONSWAP Spectrum: This is incorrect (alpha is defined differently for the two different spectra)

← Older revision		Revision as of 20:37, 9 December 2012
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	<center><math>r=\exp\left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2}\right]</math></center>		<center><math>r=\exp\left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2}\right]</math></center>

	or

	~~<center><math> \,\!S_j(f) = S_p(f) \gamma^r </math></center>~~

	~~<center><math> r = \exp\left[-\frac{(f-f_m)^2}{2\sigma^2f_m^2}\right] </math></center>~~

	<center>		<center>

Adi Kurniawan: /* JONSWAP Spectrum */

2008-12-25T07:07:35Z

JONSWAP Spectrum

← Older revision		Revision as of 07:07, 25 December 2008
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	or		or

	<center><math> S_j(f) = S_p(f) \gamma^r </math></center>		<center><math> \,\!S_j(f) = S_p(f) \gamma^r </math></center>

	<center><math> r = \exp\left[-\frac{(f-f_m)^2}{2\sigma^2f_m^2}\right] </math></center>		<center><math> r = \exp\left[-\frac{(f-f_m)^2}{2\sigma^2f_m^2}\right] </math></center>

Adi Kurniawan: /* Pierson-Moskowitz Spectrum */

2008-12-25T07:06:52Z

Pierson-Moskowitz Spectrum

← Older revision		Revision as of 07:06, 25 December 2008
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	<center>		<center>
	<math>\left \langle \zeta^2\right \rangle=\int_{0}^{\infty}S(\omega)\mbox{ }d\omega=2.74 \times 10^{-3}\frac{(U_{10.5})^4}{g^2}</math>		<math>\left \langle \zeta^2\right \rangle=\int_{0}^{\infty}S(\omega)\mbox{ }\mathrm{d}\omega=2.74 \times 10^{-3}\frac{(U_{10.5})^4}{g^2}</math>
	</center>		</center>

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	Practical wave analysis of uses the frequency, <math> f </math>, instead of the angular frequency <math> \omega </math>. If we have a frequency spectrum, <math> S(\omega) </math>, then the corresponding frequency spectrum, <math> S'(f) </math>, will be		Practical wave analysis of uses the frequency, <math> f </math>, instead of the angular frequency <math> \omega </math>. If we have a frequency spectrum, <math> S(\omega) </math>, then the corresponding frequency spectrum, <math> S'(f) </math>, will be

	<center><math> S'(f) = S(2\pi f) \frac{d\omega}{df} = 2\pi S(2\pi f) </math></center>		<center><math> S'(f) = S(2\pi f) \frac{\mathrm{d}\omega}{\mathrm{d}f} = 2\pi S(2\pi f) </math></center>
	(the factor of <math> 2\pi </math> insures that the area under the curve remains the same).		(the factor of <math> 2\pi </math> insures that the area under the curve remains the same).

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	</math>		</math>
	</center>		</center>
	where <math> \alpha </math> is Phillips constant and <math> f_m </math> is the peak frequency. ~~Rememeber~~ that		where <math> \alpha </math> is Phillips constant and <math> f_m </math> is the peak frequency. Remember that
	these expressions are an approximation to the real sea spectrum.		these expressions are an approximation to the real sea spectrum.

Adi Kurniawan at 04:14, 23 September 2008

2008-09-23T04:14:47Z

← Older revision		Revision as of 04:14, 23 September 2008
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	= Introduction =		== Introduction ==

	Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?		Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?
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	The concept of a wave spectrum can be quite abstract and is described in [[Waves and the Concept of a Wave Spectrum]]		The concept of a wave spectrum can be quite abstract and is described in [[Waves and the Concept of a Wave Spectrum]]

	= Pierson-Moskowitz Spectrum =		== Pierson-Moskowitz Spectrum ==
	Various idealized spectra are used to answer the question in oceanography and ocean engineering. Perhaps the simplest is that proposed by [[Pierson and Moskowitz 1964]]. They assumed that if the wind blew steadily for a long time over a large area, the waves would come into equilibrium with the wind. This is the concept of a '''fully developed sea''' (a sea produced by winds blowing steadily over hundreds of miles for several days).Here, a long time is roughly ten-thousand wave periods, and a "large area" is roughly five-thousand wave-lengths on a side.		Various idealized spectra are used to answer the question in oceanography and ocean engineering. Perhaps the simplest is that proposed by [[Pierson and Moskowitz 1964]]. They assumed that if the wind blew steadily for a long time over a large area, the waves would come into equilibrium with the wind. This is the concept of a '''fully developed sea''' (a sea produced by winds blowing steadily over hundreds of miles for several days).Here, a long time is roughly ten-thousand wave periods, and a "large area" is roughly five-thousand wave-lengths on a side.
	<center>		<center>
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	</center>		</center>

	= JONSWAP Spectrum =		== JONSWAP Spectrum ==
	[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor		[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor
	<math> \gamma^r </math>		<math> \gamma^r </math>
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	The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].		The JONSWAP spectrum is similar to the Pierson-Moskowitz spectrum except that waves continues to grow with distance (or time) as specified by the a term, and the peak in the spectrum is more pronounced, as specified by the g term. The latter turns out to be particularly important because it leads to enhanced non-linear interactions and a spectrum that changes in time according to the theory of [[Hasselmann 1966]].

	= Generation of Waves by Wind =		== Generation of Waves by Wind ==
	We have seen in the last few paragraphs that waves are related to the wind. We have, however, put off until now just how they are generated by the wind. Suppose we begin with a mirror-smooth sea (Beaufort Number 0). What happens if the wind suddenly begins to blow steadily at say 8m/s? Three different physical processes begin:		We have seen in the last few paragraphs that waves are related to the wind. We have, however, put off until now just how they are generated by the wind. Suppose we begin with a mirror-smooth sea (Beaufort Number 0). What happens if the wind suddenly begins to blow steadily at say 8m/s? Three different physical processes begin:

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	# Finally, the waves begin to interact among themselves to produce longer waves ([[Hasselmann et al. 1973]]). The interaction transfers wave energy from short waves generated by Miles mechanism to waves with frequencies slightly lower than the frequency of waves at the peak of the spectrum. Eventually, this leads to waves going faster than the wind, as noted by Pierson and Moskowitz.		# Finally, the waves begin to interact among themselves to produce longer waves ([[Hasselmann et al. 1973]]). The interaction transfers wave energy from short waves generated by Miles mechanism to waves with frequencies slightly lower than the frequency of waves at the peak of the spectrum. Eventually, this leads to waves going faster than the wind, as noted by Pierson and Moskowitz.

	=Acknowledgement=		==Acknowledgement==

	The material in this page has been derived from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].		The material in this page has been derived from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

Meylan: /* Acknowledgement */

2006-11-09T09:02:12Z

Acknowledgement

← Older revision		Revision as of 09:02, 9 November 2006
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	=Acknowledgement=		=Acknowledgement=

	The material in this page has ~~come~~ from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].		The material in this page has been derived from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].


	[[Category:Geophysics]]		[[Category:Geophysics]]

Meylan: /* Introduction */

2006-11-09T09:01:33Z

Introduction

← Older revision		Revision as of 09:01, 9 November 2006
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	time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two		time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two
	peaks, one from distance swell and the other generated by the local wind.		peaks, one from distance swell and the other generated by the local wind.

			The concept of a wave spectrum can be quite abstract and is described in [[Waves and the Concept of a Wave Spectrum]]

	= Pierson-Moskowitz Spectrum =		= Pierson-Moskowitz Spectrum =

Meylan: /* Introduction */

2006-11-09T08:58:15Z

Introduction

← Older revision		Revision as of 08:58, 9 November 2006
Line 3:		Line 3:
	Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?		Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?

	It is ~~importand~~ to realise that the spectra presented in the section are attempts to describe the ocean wave spectra		It is important to realise that the spectra presented in the section are attempts to describe the ocean wave spectra
	in very special conditions, namely the conditions after a wind with constant velocity has been blowing for a long		in very special conditions, namely the conditions after a wind with constant velocity has been blowing for a long
	time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two		time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two

Meylan: /* Introduction */

2006-11-09T08:57:26Z

Introduction

← Older revision		Revision as of 08:57, 9 November 2006
Line 2:		Line 2:

	Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?		Ocean waves are produced by the wind. The faster the wind, the longer the wind blows, and the bigger the area over which the wind blows, the bigger the waves. In designing ships or offshore structures we wish to know the biggest waves produced by a given wind speed. Suppose the wind blows at 20m/s for many days over a large area of the North Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?

			It is importand to realise that the spectra presented in the section are attempts to describe the ocean wave spectra
			in very special conditions, namely the conditions after a wind with constant velocity has been blowing for a long
			time. A typical ocean wave spectrum wil be much more complicated and variable. For example it may have two
			peaks, one from distance swell and the other generated by the local wind.

	= Pierson-Moskowitz Spectrum =		= Pierson-Moskowitz Spectrum =

Meylan: /* JONSWAP Spectrum */

2006-11-09T08:53:27Z

JONSWAP Spectrum

← Older revision		Revision as of 08:53, 9 November 2006
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	= JONSWAP Spectrum =		= JONSWAP Spectrum =
	[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor ~~(Figure 3):~~		[[Hasselmann et al. 1973]], after analyzing data collected during the Joint North Sea Wave Observation Project JONSWAP, found that the wave spectrum is never fully developed. It continues to develop through non-linear, wave-wave interactions even for very long times and distances. Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements. The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor
			<math> \gamma^r </math>

	<center><math>S_j(\omega)=\frac{\alpha g^2}{\omega^5}\exp\left[-\frac{5}{4}\left(\frac{\omega_p}{\omega}\right)^4\right]\gamma^r</math></center>		<center><math>S_j(\omega)=\frac{\alpha g^2}{\omega^5}\exp\left[-\frac{5}{4}\left(\frac{\omega_p}{\omega}\right)^4\right]\gamma^r</math></center>
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	</center>		</center>

			Figure 3 show wave spectra as mearured from {[Hasselmann et al. 1973]] and Figure 4 shows a comparison
			of the JONSWAP and Pierson-Moskowitz spectra.
	<center>		<center>
	[[Image:SpSj.jpg]]		[[Image:SpSj.jpg]]
	</center>		</center>
	<center>		<center>
	Figure 4 The JONSWAP and Pierson-Moskowitz ~~spectrums~~		Figure 4 The JONSWAP and Pierson-Moskowitz spectra.
	</center>		</center>