Capillary waves - Revision history

Carl McBride: Reverted edits by 70.135.118.126 (talk) to last revision by Carl McBride

2012-01-04T14:31:57Z

Reverted edits by 70.135.118.126 (talk) to last revision by Carl McBride

← Older revision		Revision as of 16:31, 4 January 2012
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	*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]		*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
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70.135.118.126 at 05:57, 4 January 2012

2012-01-04T05:57:58Z

← Older revision		Revision as of 07:57, 4 January 2012
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	*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]		*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
			[http://www.prlog.org/11289974-phone-number-lookup-verizon-phone-number-reverse-lookup-to-get-information-you-need-quickly.html phone address lookup]

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Carl McBride: Undo revision 11384 by RamiroRoberts (talk)

2011-05-12T10:01:24Z

Undo revision 11384 by RamiroRoberts (talk)

← Older revision		Revision as of 12:01, 12 May 2011
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	*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]		*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
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RamiroRoberts at 09:51, 12 May 2011

2011-05-12T09:51:30Z

← Older revision		Revision as of 11:51, 12 May 2011
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	*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]		*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
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Dduque: Link to (new) hydrodynamics

2010-05-13T09:40:59Z

Link to (new) hydrodynamics

← Older revision		Revision as of 11:40, 13 May 2010
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	</blockquote>		</blockquote>
	====Defining the problem====		====Defining the problem====
	Three contributions to the energy are involved: the [[surface tension]], gravity, and hydrodynamics. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above.		Three contributions to the energy are involved: the [[surface tension]], gravity, and [[hydrodynamics]]. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above.
	The new contribution involves the [[kinetic energy]] of the fluid:		The new contribution involves the [[kinetic energy]] of the fluid:

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	where <math>v</math> is the module of the velocity field <math>\vec{v}</math>.		where <math>v</math> is the module of the velocity field <math>\vec{v}</math>.
	(Again, we are neglecting the flow of the gas above for simplicity.)		(Again, we are neglecting the flow of the gas above for simplicity.)

	====Wave solutions====		====Wave solutions====
	Let us suppose the surface of the liquid is described by a traveling plane wave:		Let us suppose the surface of the liquid is described by a traveling plane wave:

Nice and Tidy: Placed external links at foot of page.

2008-12-11T18:30:50Z

Placed external links at foot of page.

← Older revision		Revision as of 20:30, 11 December 2008
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	This derivation makes clear the assumptions introduced. In particular, the linearization will only hold for smooth waves, the ones for which the wave amplitude, <math>a</math> is smaller than the wavelength. Mathematically, the limit is <math>q a \ll 1</math>. For ocean waves, this happens when waves approach the shore and the amplitude grows (in this limit, a bottom boundary condition <math>v_z (z=-H) =0</math> must be employed, and waves are not dispersive, see Ref 7.)		This derivation makes clear the assumptions introduced. In particular, the linearization will only hold for smooth waves, the ones for which the wave amplitude, <math>a</math> is smaller than the wavelength. Mathematically, the limit is <math>q a \ll 1</math>. For ocean waves, this happens when waves approach the shore and the amplitude grows (in this limit, a bottom boundary condition <math>v_z (z=-H) =0</math> must be employed, and waves are not dispersive, see Ref 7.)

	~~==External links==~~
	*[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in wikipedia]
	*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in wikipedia]
	==References==		==References==
	#[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)]		#[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)]
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	#R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153		#R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153
	#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5		#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5
			#[http://dx.doi.org/10.1088/0143-0807/25/1/014 F Behroozi "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy", European Journal of Physics '''25''' 115 (2004)]
			==External links==
			*[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in Wikipedia]
			*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
	#[http://dx.doi.org/10.1088/0143-0807/25/1/014 F Behroozi "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy", European Journal of Physics '''25''' 115 (2004)]

Dduque: added Ref

2008-11-04T10:21:55Z

added Ref

← Older revision		Revision as of 12:21, 4 November 2008
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	#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5		#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5
	[[Category: Classical thermodynamics ]]		[[Category: Classical thermodynamics ]]
			#[http://dx.doi.org/10.1088/0143-0807/25/1/014 F Behroozi "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy", European Journal of Physics '''25''' 115 (2004)]

Dduque: Completed Alternative derivation

2008-11-04T10:18:50Z

Completed Alternative derivation

← Older revision		Revision as of 12:18, 4 November 2008
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	\frac{\partial h}{\partial t}+v_x \frac{\partial h}{\partial x}+v_y \frac{\partial h}{\partial y}= v_z .		\frac{\partial h}{\partial t}+v_x \frac{\partial h}{\partial x}+v_y \frac{\partial h}{\partial y}= v_z .
	</math>		</math>
	A simpler condition follows from linearization: <math> \partial h /\partial t =v_z </math>, as in the previous derivation.		A simpler condition follows from linearization: <math> \partial h /\partial t =v_z </math>, as in the previous derivation. There is an additional boundary condition at the bottom of the fluid, which we take here as <math>v_z=0</math> when <math>z\rightarrow -\infty</math>.

	The other boundary condition is a [[Bernoulli equation]], stating that the pressure just below the surface, <math>p_-</math>, must equal the [[saturation pressure]] of coexistence, minus a contribution due to the surface:		In the same fashion as above, we seek surface wave solutions, of the form <math>h(x,y,t)=a e^{i (qx-\omega t)}</math>. We may guess a solution of the form
			:<math>\phi=-i\omega h(x,y;t) f(z).</math>

			This first condition implies <math>f'(z=h)=1</math>. Together with Laplace's equation, this leads to a function
			:<math>f=(1/q) \exp(q(z-h)). </math>
			(see Ref 8 for a discussion on when Laplace's equation admits wave solutions.)


			The other surface boundary condition is a [[Bernoulli equation]], stating that the pressure just below the surface, <math>p_-</math>, must equal the [[saturation pressure]] of coexistence, minus a contribution due to the surface:
	:<math>		:<math>
	p_-=p -		p_-=p -
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	</math>		</math>

	In the same ~~fashion~~ as above, ~~we seek surface~~ wave ~~solutions~~, of the ~~form~~ <math>h(~~x,y~~,~~t)=~~a ~~e^{i~~ (qx-~~\omega t~~)}</math>.		This second condition, when applied to the surface wave above, establishes that <math>f(z=h)=(g+\sigma/\rho q^2)/\omega^2</math>.

			For the two conditions to hold, <math>1/q</math> must equal <math>(g+\sigma/\rho q^2)/\omega^2</math>, which is precisely the same dispersion relation as the one above.

			This derivation makes clear the assumptions introduced. In particular, the linearization will only hold for smooth waves, the ones for which the wave amplitude, <math>a</math> is smaller than the wavelength. Mathematically, the limit is <math>q a \ll 1</math>. For ocean waves, this happens when waves approach the shore and the amplitude grows (in this limit, a bottom boundary condition <math>v_z (z=-H) =0</math> must be employed, and waves are not dispersive, see Ref 7.)

	==External links==		==External links==

Dduque: New derivation, work in progress...

2008-11-04T09:06:08Z

New derivation, work in progress...

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 ====Alternative derivation====
-In Reference 6 the dispersion relation is derived in a somewhat different manner. The same assumptions are made regarding the fluid (it is inviscid, irrotational, and incompressible), so Laplace's Equation is to be satisfied: <math>\nabla^2\phi=0</math>. The boundary conditions, on the other hand, are sufficient to solve the problem.
+In Reference 7 the dispersion relation is derived in a somewhat different manner. The same assumptions are made regarding the fluid (it is inviscid, irrotational, and incompressible), so Laplace's Equation is to be satisfied: <math>\nabla^2\phi=0</math>, with <math>\vec{v}=\nabla\phi</math>. The boundary conditions, on the other hand, are sufficient to solve the problem.
 ==External links==

Dduque: psi -> phi, new section (to be expanded in a minute)

2008-11-04T08:50:25Z

psi -> phi, new section (to be expanded in a minute)

← Older revision		Revision as of 10:50, 4 November 2008
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	where <math>A=L_x\times L_y</math> is the area of the system.		where <math>A=L_x\times L_y</math> is the area of the system.

	To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow\|potential]]: <math>\vec{v}=\nabla\~~psi~~</math>, and <math>\~~psi~~</math> is a potential (scalar field) which must satisfy [[Laplace's equation]] <math>\nabla^2\~~psi~~=0</math>.		To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow\|potential]]: <math>\vec{v}=\nabla\phi</math>, and <math>\phi</math> is a potential (scalar field) which must satisfy [[Laplace's equation]] <math>\nabla^2\phi=0</math>.
	If we try try separation of variables with the potential:		If we try try separation of variables with the potential:

	:<math>\~~psi~~(x,y,z,t)=\xi(t) f(z) e^{i q x},</math>		:<math>\phi(x,y,z,t)=\xi(t) f(z) e^{i q x},</math>

	with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math>.		with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math>.
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	:<math>f(z) = a \, \exp(\|q\| z) </math>,		:<math>f(z) = a \, \exp(\|q\| z) </math>,

	with some constant <math>a</math>. The less trivial condition is the proper matching between <math>\~~psi~~</math> and <math>h</math>: the potential field must correspond to a velocity field that is adjusted to the movement of the surface:		with some constant <math>a</math>. The less trivial condition is the proper matching between <math>\phi</math> and <math>h</math>: the potential field must correspond to a velocity field that is adjusted to the movement of the surface:

	:<math>v_z (z=h) =\partial h/\partial t</math>.		:<math>v_z (z=h) =\partial h/\partial t</math>.

	(Actually, this is the linearized version of a more general expression, see ~~Ref [6]~~).		(Actually, this is the linearized version of a more general expression, see below.)

	This implies that:		This implies that:
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	We may now find the velocity field, <math>\vec{v}=\nabla\~~psi~~</math>, which shows the well-known circles: the elements of fluid undergo circular motion in the <math>x,z</math> plane, with the circles getting smaller at deeper levels. The displacement of a fluid element is given be <math>\partial\vec{\~~phi~~}/\partial t= \vec{v}</math>, and is plotted in Figure 1.		We may now find the velocity field, <math>\vec{v}=\nabla\phi</math>, which shows the well-known circles: the elements of fluid undergo circular motion in the <math>x,z</math> plane, with the circles getting smaller at deeper levels. The displacement of a fluid element is given by <math>\partial\vec{\psi}/\partial t= \vec{v}</math>, and is plotted in Figure 1.

	[[Image:Wave_v_field.jpg\|300px\|thumb\|right\|Figure 1. Displacement of particle (snapshot)]]		[[Image:Wave_v_field.jpg\|300px\|thumb\|right\|Figure 1. Displacement of particle (snapshot)]]

	For the kinetic energy, we need		For the kinetic energy, we need
	<math>v^2=\|\nabla\~~psi~~\|^2</math>, which is:		<math>v^2=\|\nabla\phi\|^2</math>, which is:

	:<math>v^2= (\eta')^2 e^{ -2 \|q\|(h-z)}, </math>		:<math>v^2= (\eta')^2 e^{ -2 \|q\|(h-z)}, </math>
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	the same dispersion as above if <math>\rho'</math> is neglected.		the same dispersion as above if <math>\rho'</math> is neglected.

			====Alternative derivation====

			In Reference 6 the dispersion relation is derived in a somewhat different manner. The same assumptions are made regarding the fluid (it is inviscid, irrotational, and incompressible), so Laplace's Equation is to be satisfied: <math>\nabla^2\phi=0</math>. The boundary conditions, on the other hand, are sufficient to solve the problem.

	==External links==		==External links==