Nonlinear Waves - Revision history

Meylan: /* Introduction */

2006-10-30T10:45:31Z

Introduction

← Older revision		Revision as of 10:45, 30 October 2006
Line 6:		Line 6:
	</center>		</center>

	The phases of the components for the Fourier series expansion of <math>\zeta</math> ~~in (16.14)~~ are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{\mathrm{max}} = 0.07L </math> <math> (ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also [[Lamb 1945]], <math>\S \;</math>250).		The phases of the components for the Fourier series expansion of <math>\zeta</math> are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{\mathrm{max}} = 0.07L </math> <math> (ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also [[Lamb 1945]], <math>\S \;</math>250).

	Knowledge of non-linear waves came slowly until Hasselmann ([[Hasselmann 1961]], [[Hasselmann 1963a]], [[Hasselmann 1963b]], [[Hasselmann 1966]]), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, [[Phillips 1960]], and [[Longuet-Higgins and Phillips 1962]] showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:		Knowledge of non-linear waves came slowly until Hasselmann ([[Hasselmann 1961]], [[Hasselmann 1963a]], [[Hasselmann 1963b]], [[Hasselmann 1966]]), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, [[Phillips 1960]], and [[Longuet-Higgins and Phillips 1962]] showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:
Line 22:		Line 22:
	</center>		</center>

	where we allow waves to travel in any direction, and <math>k_i</math> is the vector wave number giving wave-length and direction. ~~(16.15a, b)~~ are general requirements for any interacting waves. The fewest number of waves that meet the conditions ~~of (16.15)~~ are three waves which interact to produce a fourth. The interaction is weak; waves must interact for hundreds of wave-lengths and periods to produce a fourth wave with amplitude comparable to the interacting waves. The Stokes wave does not meet ~~the~~ criteria ~~of (16.15)~~ and the wave components are not free waves; the higher harmonics are bound to the primary wave.		where we allow waves to travel in any direction, and <math>k_i</math> is the vector wave number giving wave-length and direction. These equations are general requirements for any interacting waves. The fewest number of waves that meet the conditions are three waves which interact to produce a fourth. The interaction is weak; waves must interact for hundreds of wave-lengths and periods to produce a fourth wave with amplitude comparable to the interacting waves. The Stokes wave does not meet these criteria and the wave components are not free waves; the higher harmonics are bound to the primary wave.

	= Wave Momentum =		= Wave Momentum =

Meylan: /* Introduction */

2006-10-30T10:43:56Z

Introduction

← Older revision		Revision as of 10:43, 30 October 2006
Line 6:		Line 6:
	</center>		</center>

	The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{max} = 0.07L ( ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also [[Lamb 1945]], <math>\S \;</math>250).		The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{\mathrm{max}} = 0.07L </math> <math> (ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also [[Lamb 1945]], <math>\S \;</math>250).

	Knowledge of non-linear waves came slowly until Hasselmann ([[Hasselmann 1961]], [[Hasselmann 1963a]], [[Hasselmann 1963b]], [[Hasselmann 1966]]), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, [[Phillips 1960]], and [[Longuet-Higgins and Phillips 1962]] showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:		Knowledge of non-linear waves came slowly until Hasselmann ([[Hasselmann 1961]], [[Hasselmann 1963a]], [[Hasselmann 1963b]], [[Hasselmann 1966]]), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, [[Phillips 1960]], and [[Longuet-Higgins and Phillips 1962]] showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:

Meylan: /* Solitary Waves */

2006-10-28T23:35:08Z

Solitary Waves

← Older revision		Revision as of 23:35, 28 October 2006
Line 45:		Line 45:

	You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China ([[Pugh 1987]]: 249).		You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China ([[Pugh 1987]]: 249).

			=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]].

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

Syan077 at 09:13, 26 October 2006

2006-10-26T09:13:54Z

← Older revision		Revision as of 09:13, 26 October 2006
Line 28:		Line 28:

	= Solitary Waves =		= Solitary Waves =
	Solitary waves are another class of non-linear waves, and they have very interesting properties. They propagate without change of shape, and two solitons can cross without interaction. The first soliton was discovered by John Scott Russell (1808-1882), who followed a solitary wave generated by a boat in Edinburgh's Union Canal in 1834.		Solitary waves are another class of non-linear waves, and they have very interesting properties. They propagate without change of shape, and two solitons can cross without interaction. The first soliton was discovered by [http://en.wikipedia.org/wiki/John_Scott_Russell John Scott Russell] (1808-1882), who followed a solitary wave generated by a boat in Edinburgh's Union Canal in 1834.

	Scott witnessed such a wave while watching a boat being drawn along the Union Canal by a pair of horses. When the boat stopped, he noticed that water around the vessel surged ahead in the form of a single wave, whose height and speed remained virtually unchanged. Russell pursued the wave on horseback for more than a mile before returning home to reconstruct the event in an experimental tank in his garden. - Nature 376, 3 August 1995: 373.		Scott witnessed such a wave while watching a boat being drawn along the Union Canal by a pair of horses. When the boat stopped, he noticed that water around the vessel surged ahead in the form of a single wave, whose height and speed remained virtually unchanged. Russell pursued the wave on horseback for more than a mile before returning home to reconstruct the event in an experimental tank in his garden. - Nature 376, 3 August 1995: 373.

Syan077 at 09:06, 26 October 2006

2006-10-26T09:06:59Z

← Older revision		Revision as of 09:06, 26 October 2006
Line 44:		Line 44:
	</center>		</center>

	You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China ([[Pugh, 1987]]: 249).		You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China ([[Pugh 1987]]: 249).

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

Syan077 at 09:05, 26 October 2006

2006-10-26T09:05:30Z

← Older revision		Revision as of 09:05, 26 October 2006
Line 1:		Line 1:
	= Introduction =		= Introduction =
	We derived the properties of an ocean surface wave assuming waves were infinitely small <math>ka = O</math> (0). If the waves are small <math>ka << 1</math> but not infinitely small, the wave properties can be expanded in a power series of <math>ka</math> (Stokes, 1847). He calculated the properties of a wave of finite amplitude and found:		We derived the properties of an ocean surface wave assuming waves were infinitely small <math>ka = O</math> (0). If the waves are small <math>ka << 1</math> but not infinitely small, the wave properties can be expanded in a power series of <math>ka</math> ([[Stokes 1847]]). He calculated the properties of a wave of finite amplitude and found:

	<center>		<center>
Line 6:		Line 6:
	</center>		</center>

	The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{max} = 0.07L ( ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also Lamb, 1945, <math>\S \;</math>250).		The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{max} = 0.07L ( ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also [[Lamb 1945]], <math>\S \;</math>250).

	Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b, 1966), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, Phillips (1960), and Longuet-Higgins and Phillips (1962) showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:		Knowledge of non-linear waves came slowly until Hasselmann ([[Hasselmann 1961]], [[Hasselmann 1963a]], [[Hasselmann 1963b]], [[Hasselmann 1966]]), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, [[Phillips 1960]], and [[Longuet-Higgins and Phillips 1962]] showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:

	<center>		<center>
Line 25:		Line 25:

	= Wave Momentum =		= Wave Momentum =
	The concept of wave momentum has caused considerable confusion (McIntyre, 1981). In general, waves do not have momentum, a mass flux, but they do have a momentum flux. This is true for waves on the sea surface. Ursell (1950) showed that ocean swell on a rotating Earth has no mass transport. His proof seems to contradict the usual textbook discussions of steep, non-linear waves such as Stokes waves. Water particles in a Stokes wave move along paths that are nearly circular, but the paths fail to close, and the particles move slowly in the direction of wave propagation. This is a mass transport, and the phenomena is called Stokes drift. But the forward transport near the surface is balanced by an equal transport in the opposite direction at depth, and there is no net mass flux.		The concept of wave momentum has caused considerable confusion [[McIntyre 1981]]. In general, waves do not have momentum, a mass flux, but they do have a momentum flux. This is true for waves on the sea surface. [[Ursell 1950]] showed that ocean swell on a rotating Earth has no mass transport. His proof seems to contradict the usual textbook discussions of steep, non-linear waves such as Stokes waves. Water particles in a Stokes wave move along paths that are nearly circular, but the paths fail to close, and the particles move slowly in the direction of wave propagation. This is a mass transport, and the phenomena is called Stokes drift. But the forward transport near the surface is balanced by an equal transport in the opposite direction at depth, and there is no net mass flux.

	= Solitary Waves =		= Solitary Waves =

Meylan: /* Solitary Waves */

2006-10-05T23:56:52Z

Solitary Waves

← Older revision		Revision as of 23:56, 5 October 2006
Line 35:		Line 35:

	<center>		<center>
	<math>\zeta = a\;sech^2\left[\left(\frac{3a}{4d^3}\right)^{1/2}(x-ct)\right]</math>		<math>\zeta = a\;\mathrm{sech}^2\left[\left(\frac{3a}{4d^3}\right)^{1/2}(x-ct)\right]</math>
	</center>		</center>

Syan077 at 10:41, 5 October 2006

2006-10-05T10:41:25Z

← Older revision		Revision as of 10:41, 5 October 2006
Line 44:		Line 44:
	</center>		</center>

	You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China (Pugh, 1987: 249).		You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China ([[Pugh, 1987]]: 249).

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

Syan077 at 04:14, 30 September 2006

2006-09-30T04:14:45Z

← Older revision		Revision as of 04:14, 30 September 2006
Line 3:		Line 3:

	<center>		<center>
	<math> \zeta = a\cos(kx-\omega t) + \frac{1}{2}k a^2\cos2(kx - \omega t)+\frac{3}{8}k^2 a^3\cos3(kx - \omega t)+...</math>		<math> \zeta = a\cos(kx-\omega t) + \frac{1}{2}k a^2\cos2(kx - \omega t)+\frac{3}{8}k^2 a^3\cos3(kx - \omega t)+... \,\!</math>
	</center>		</center>

Line 11:		Line 11:

	<center>		<center>
	<math>\omega_1 \pm \omega_2 \pm \omega_3 \pm ... \omega_n = 0</math>		<math>\omega_1 \pm \omega_2 \pm \omega_3 \pm ... \omega_n = 0 \,\!</math>
	</center>		</center>

	<center>		<center>
	<math>k_1 \pm k_2 \pm k_3 \pm ... k_n = 0</math>		<math>k_1 \pm k_2 \pm k_3 \pm ... k_n = 0 \,\!</math>
	</center>		</center>

	<center>		<center>
	<math>\omega_i^2 = g k_i</math>		<math>\omega_i^2 = g k_i \,\!</math>
	</center>		</center>

Syan077 at 02:53, 30 September 2006

2006-09-30T02:53:15Z

← Older revision		Revision as of 02:53, 30 September 2006
Line 6:		Line 6:
	</center>		</center>

	The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>~~\mathrm{a}_~~{max} = 0.~~07\;L\;~~(~~k\mathrm{a}~~ = 0.44)</math>. Such steep waves in deep water are called Stokes waves (See also Lamb, 1945, ~~?50~~).		The phases of the components for the Fourier series expansion of <math>\zeta</math> in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is <math>a_{max} = 0.07L ( ka = 0.44 ) </math>. Such steep waves in deep water are called Stokes waves (See also Lamb, 1945, <math>\S \;</math>250).

	Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b, 1966), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, Phillips (1960), and Longuet-Higgins and Phillips (1962) showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:		Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b, 1966), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, Phillips (1960), and Longuet-Higgins and Phillips (1962) showed that <math>n</math> free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:
Line 13:		Line 13:
	<math>\omega_1 \pm \omega_2 \pm \omega_3 \pm ... \omega_n = 0</math>		<math>\omega_1 \pm \omega_2 \pm \omega_3 \pm ... \omega_n = 0</math>
	</center>		</center>

	<center>		<center>
	<math>k_1 \pm k_2 \pm k_3 \pm ... k_n = 0</math>		<math>k_1 \pm k_2 \pm k_3 \pm ... k_n = 0</math>
	</center>		</center>

	<center>		<center>
	<math>\omega_i^2 = g k_i</math>		<math>\omega_i^2 = g k_i</math>