Editing Capillary waves
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====Defining the problem==== | ====Defining the problem==== | ||
Three contributions to the energy are involved: the [[surface tension]], gravity, and | 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: | ||
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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\ | 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>. | ||
If we try try separation of variables with the potential: | If we try try separation of variables with the potential: | ||
:<math>\ | :<math>\psi(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>\ | 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: | ||
:<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 | (Actually, this is the linearized version of a more general expression, see Ref [6]). | ||
This implies that: | This implies that: | ||
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We may now find the velocity field, <math>\vec{v}=\nabla\ | 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. | ||
[[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\ | <math>v^2=|\nabla\psi|^2</math>, which is: | ||
:<math>v^2= | :<math>v^2= 2 (h')^2 e^{ -2 |q|(h-z)}, </math> | ||
with no dependence on <math>x</math> or <math>y</math>; the other integration provides: | with no dependence on <math>x</math> or <math>y</math>; the other integration provides: | ||
:<math>T= \frac{\rho A }{2|q|} ( | :<math>T= \frac{\rho A }{2|q|} ( h' )^2 \int_{-\infty}^h e^{ -2 |q|(h-z)} = \frac{A}{2} \frac{\rho }{2|q|} ( h' )^2 . | ||
</math> | </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. | ||
== | ==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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#[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)] | #[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)] | ||
#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 | ||
[[Category: Classical thermodynamics ]] | [[Category: Classical thermodynamics ]] |