Editing Capillary waves
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This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As Richard Feynman put it (Ref. 6): | This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As Richard Feynman put it (Ref. 6): | ||
<blockquote> | <blockquote> | ||
...[water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses... are the worst possible example... they have all the complications that waves can have | |||
</blockquote> | </blockquote> | ||
====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>. | ||
Laplace's equation then requires on the later | Laplace's equation then requires on the later | ||
:<math>f''(z)= q^2 f(z) .</math> | :<math>f''(z)=-q^2 f(z) .</math> | ||
This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in oceanography). Therefore | This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in oceanography). Therefore | ||
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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>. | ||
This means that | |||
This | |||
:<math>\xi(t)=\eta(t)'</math>, and <math>f'(z=h) = 1, </math> | :<math>\xi(t)=\eta(t)'</math>, and <math>f'(z=h) = 1, </math> | ||
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:<math>f(z) = \exp( -|q|(h-z))/|q| </math>. | :<math>f(z) = \exp( -|q|(h-z))/|q| </math>. | ||
[[Image:Wave_v_field.jpg|300px|thumb|right|Velocity field (snapshot)]] | |||
We may now find the velocity field | We may now find the velocity field, 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. For the kinetic energy, we need | ||
<math>v^2=|\nabla\psi|^2</math>, which is: | |||
For the kinetic energy, we need | |||
<math>v^2=|\nabla\ | |||
<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 | ||
[[Category: Classical thermodynamics ]] | [[Category: Classical thermodynamics ]] |