Linear Theory of Ocean Surface Waves - Revision history

Kri: /* Group Velocity */ Framed image

2010-12-11T16:15:03Z

Group Velocity: Framed image

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	Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).		Each wave crest moves at twice the speed of the group. Do real ocean waves move in groups governed by the dispersion relation? Yes. [[Munk et al. 1963]] in a remarkable series of experiments in the 1960s showed that ocean waves propagating over great distances are dispersive, and that the dispersion could be used to track storms. They recorded waves for many days using an array of three pressure gauges just offshore of San Clemente Island, 60 miles due west of San Diego, California. Wave spectra were calculated for each day's data. (The concept of a spectra is discussed below.) From the spectra, the amplitudes and frequencies of the low-frequency waves and the propagation direction of the waves were calculated. Finally, they plotted contours of wave energy on a frequency-time diagram (Figure 1).
	~~<center>~~
	~~[[image:Fig16-1s.jpg]]~~
	~~</center>~~
	Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].

			[[image:Fig16-1s.jpg\|frame\|center\|Figure 1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high wave energy show the arrival of dispersed wave trains from distant storms. The slope of the ridge is inversely proportional to distance to the storm. D is distance in degrees, <math>\theta</math> is direction of arrival of waves at California. From [[Munk et al. 1963]].]]

	To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.		To understand the figure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest w) travel the fastest and they arrive before other, higher-frequency waves. The further away the storm, the longer the delay between arrivals of waves of different frequencies. The ridges of high wave energy seen in the Figure are produced by individual storm <math> \theta </math>. The slope of the ridge gives the distance to the storm in degrees <math> \Delta </math> along a great circle; and the phase information from the array gives the angle to the storm. The two angles give the storm's location relative to San Clemente. Thus waves arriving from 15 to 18 September produce a ridge indicating the storm was 115° away at an angle of 205° which is south of new Zealand near Antarctica.

Kri: /* Wave Energy */ clarified

2010-12-11T16:13:43Z

Wave Energy: clarified

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	== Wave Energy ==		== Wave Energy ==

	~~Wave~~ energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:		The wave energy density <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

	<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>		<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

	where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a ~~space and~~ time average.		where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time average. Note that this formula requires that there is quasi steady state so that the average kinetic and potential energies are equal and is only valid for linear waves. Although the formula in theory will give different energy densities for different locations (e.g. for a standing wave there will be locations where the displacement, hence also the energy density, will always be zero), it will in practice give a good result which doesn't vary much from location to location.
	~~Not~~ that this formula requires that there is quasi steady state so that the average kinetic and potential energies
	are equal and is only valid for linear waves.		[[image:Fig16-2s.jpg\|frame\|center\|Figure 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.]]
	~~<center>~~
	[[image:Fig16-2s.jpg]]
	</center>
	~~FIgure~~ 2. A short record of wave amplitude measured by a wave buoy in the North Atlantic.

	= Significant Wave-Height =		= Significant Wave-Height =

Kri: /* Wave Energy */ + space average

2010-12-11T15:51:58Z

Wave Energy: + space average

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	<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>		<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

	where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time average.		where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a space and time average.
	Not that this formula requires that there is quasi steady state so that the average kinetic and potential energies		Not that this formula requires that there is quasi steady state so that the average kinetic and potential energies
	are equal and is only valid for linear waves.		are equal and is only valid for linear waves.

Meylan: /* Wave Energy */

2010-10-17T22:03:46Z

Wave Energy

← Older revision		Revision as of 22:03, 17 October 2010
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	Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:		Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

	<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> ~~(this formula in incorrect, see [[Talk:Linear Theory of Ocean Surface Waves#Equation for water energy incorrect\|talk page]])~~</center>		<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>

	where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time ~~or space~~ average.		where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time average.
			Not that this formula requires that there is quasi steady state so that the average kinetic and potential energies
			are equal and is only valid for linear waves.
	<center>		<center>
	[[image:Fig16-2s.jpg]]		[[image:Fig16-2s.jpg]]

Kri: /* Wave Energy */ I will let this comment be there for so long, untill anyone clears out the error or explains on the talk page why the equation is not wrong.

2010-10-17T00:08:41Z

Wave Energy: I will let this comment be there for so long, untill anyone clears out the error or explains on the talk page why the equation is not wrong.

← Older revision		Revision as of 00:08, 17 October 2010
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	Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:		Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

	<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> </center>		<center> <math>E = p_w \ g < \zeta^2 > \,\! </math> (this formula in incorrect, see [[Talk:Linear Theory of Ocean Surface Waves#Equation for water energy incorrect\|talk page]])</center>

	where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time or space average.		where <math>p_w</math> is water density, <math>g</math> is gravity, and the brackets denote a time or space average.

Meylan: /* Acknowledgement */

2010-02-08T19:20:18Z

Acknowledgement

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	where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements		where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

	== ~~Acknowledgement~~ ==		== Acknowledgment ==

	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 come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

	[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]		[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Meylan: /* Acknowledgement */

2010-02-08T19:20:03Z

Acknowledgement

← Older revision		Revision as of 19:20, 8 February 2010
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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 come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by Robert Stewart.

	[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]		[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Meylan at 19:19, 8 February 2010

2010-02-08T19:19:47Z

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	= Introduction =		{{complete pages}}

			== Introduction ==
	Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.		Looking out to sea from the shore, we can see waves on the sea surface. Looking carefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of a trough and the top of a nearby crest. The wavelength, which we might take to be the distance between prominent crests, is around 50m - 100m. Watching the waves for a few minutes, we notice that wave-height and wave-length are not constant. The heights vary randomly in time and space, and the statistical properties of the waves, such as the mean height averaged for a few hundred waves, change from day to day. These prominent offshore waves are generated by wind. Sometimes the local wind generates the waves, other times distant storms generate waves which ultimately reach the coast. For example, waves breaking on the Southern California coast on a summer day may come from vast storms offshore of Antarctica 10,000km away.

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	The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.		The stated limits for <math>h / \lambda</math> give a dispersion relation accurate within 10%. Because many wave properties can be measured with accuracies of 5-10%, the approximations are useful for calculating wave properties. Later we will learn to calculate wave properties as the waves propagate from deep to shallow water.

	= Phase Velocity =		== Phase Velocity ==
	The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:		The phase velocity c is the speed at which a particular phase of the wave propagates, for example, the speed of propagation of the wave crest. In one wave period <math>T</math> the crest advances one wave-length \lambda and the phase speed is <math>c = \lambda / T = \omega / k</math>. Thus, the definition of phase speed is:

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	In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.		In deep water, the phase speed depends on wave-length or wave frequency. Longer waves travel faster. Thus, deep-water waves are said to be dispersive. In shallow water, the phase speed is independent of the wave; it depends only on the depth of the water. Shallow-water waves are non-dispersive.

	= Group Velocity =		== Group Velocity ==
	The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.		The concept of group velocity <math>c_g</math> is fundamental for understanding the propagation of linear and nonlinear waves. First, it is the velocity at which a group of waves travels across the ocean. More importantly, it is also the propagation velocity of wave energy. [[Whitham 1974]] ( §1.3 and §11.6) gives a clear derivation of the concept and the fundamental equation.

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	The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.		The locations of the storms producing the waves recorded from June through October 1959 were compared with the location of storms plotted on weather maps and in most cases the two agreed well.

	= Wave Energy =		== Wave Energy ==

	Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:		Wave energy <math>E</math> in Joules per square meter is related to the variance of sea-surface displacement <math> \zeta </math> by:

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	where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements		where <math>< \zeta^2 >^{1/2}</math> is the standard deviation of surface displacement. This relationship is much more useful, and it is now the accepted way to calculate wave-height from wave measurements

	=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 come from [http://oceanworld.tamu.edu/resources/ocng_textbook/contents.html Introduction to Physical Oceanography] by [[Robert Stewart]].

	[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]		[[Category:Geophysics]][[Category:Linear Water-Wave Theory]]

Meylan: /* Dispersion Relation */

2009-09-11T10:15:52Z

Dispersion Relation

← Older revision		Revision as of 10:15, 11 September 2009
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	# [[Infinite Depth\|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.		# [[Infinite Depth\|Deep-water]] approximation is valid if the water depth <math>h</math> is much greater than the wave-length <math>L</math>. In this case, <math>h</math> >> <math>\lambda</math>, <math>kh</math> >> 1, and <math>\tanh (kh) = 1</math>.
	# [[Shallow Depth\|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.		# [[:Category:Shallow Depth\|Shallow-water]] approximation is valid if the water depth is much less than a wavelength. In this case, <math>h << \lambda</math>, <math>kh</math> << 1, and <math>\tanh (kh) = kh</math>.

	For these two limits of water depth compared with wavelength the dispersion relation reduces to:		For these two limits of water depth compared with wavelength the dispersion relation reduces to:

Adi Kurniawan: /* Group Velocity */

2009-03-17T05:50:53Z

Group Velocity

← Older revision		Revision as of 05:50, 17 March 2009
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	<center> <math> c_g = \sqrt {gd} = c </math> Shallow-water group velocity </center>		<center> <math> c_g = \sqrt {gh} = c </math> Shallow-water group velocity </center>

	For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.		For ocean-surface waves, the direction of propagation is perpendicular to the wave crests in the positive x direction. In the more general case of other types of waves, such as [http://en.wikipedia.org/wiki/Kelvin_Wave Kelvin] and [http://en.wikipedia.org/wiki/Rossby_wave Rossby] waves, the group velocity is not necessarily in the direction perpendicular to wave crests.