# Capillary waves

## Contents

## Thermal capillary waves

Thermal **capillary waves** are oscillations of an interface
which are thermal in origin. These take place at the molecular level, where only the surface tension
contribution is relevant.

Capillary wave theory (CWT) is a classic account of how thermal
fluctuations distort an interface (Ref. 1). It starts from some **intrinsic surface**
that is distorted. By performing a Fourier analysis treatment, normal modes are easily found.
Each contributes a energy proportional to the square of its amplitude; therefore, according to
classical statistical mechanics, equipartition holds, and the
mean energy of each mode will be . Surprisingly, this result leads to a
**divergent** surface (the width of the interface is bound to diverge with its area) (Ref 2).
This divergence is
nevertheless very mild: even for displacements on the order of meters the deviation of the surface
is comparable to the size of the molecules. Moreover, the introduction of an external field
removes the divergence: the action
of gravity is sufficient to keep the width fluctuation on the order
of one molecular diameter for areas larger than about 1 mm^{2} (Ref. 2).

Recently, a procedure has been proposed to obtain a molecular intrinsic
surface from simulation data (Ref. 3). The density profiles obtained
from this surface are, in general, quite different from the usual
*mean density profiles*.

## Gravity-capillary waves

These are ordinary waves excited in an interface, such as ripples on a water surface. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:

where *ω* is the angular frequency, *g* the acceleration due to gravity, *σ* the surface tension, *ρ* and *ρ‘* the mass density of the two fluids (*ρ > ρ‘*) and *k* the wavenumber.

### Derivation

This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description.

#### Defining the problem

Three contributions to the energy are involved: the surface tension, gravity, and hydrodynamics. The part due to gravity is the simplest: integrating the potential energy density due to gravity, from a reference height to the position of the surface, :

(For simplicity, we are neglecting the density of the fluid above, which is often acceptable.)

An increase in area of the surface causes a proportional increase of energy:

where the fist equality is the area in this (de Monge) representation, and the second applies for small values of the derivatives (surfaces not too rough).

The last contribution involves the kinetic energy of the fluid:

where is the module of the velocity field .

#### Wave solutions

Let us try separation of variables:

where is a two dimensional wave number vector, and the position. In this case,

where a factor of that will appear every integration is dropped for convenience.

To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be potential: , and is a potential (scalar field) which must satisfy Laplace equation . If we try try separation of variables with the potential:

with some function of time , and some function of vertical component (height) Laplace equation then requires on the later This equation can be solved with the proper boundary conditions: first, 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 , with some constant . The less trivial condition is the matching between and : the potential field must correspond to a velocity field that is adjusted to the movement of the surface: . This means that , and that , so that . We may now find , which is . Performing the integration first we are left with

where we have dropped a factor of in the last step.

The problem is thus specified by just a potential energy involving the square of and a kinetic energy involving the square of its time derivative: a regular harmonic oscillator. Its equation of motion will be

whose oscillatory solution is

the same dispersion as above if is neglected.

## References

- 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) - J. S. Rowlinson and B. Widom "Molecular Theory of Capillarity". Dover 2002 (originally: Oxford University Press 1982) ISBN 0486425444
- E. Chacón and P. Tarazona "Intrinsic profiles beyond the capillary wave theory: A Monte Carlo study", Physical Review Letters
**91**166103 (2003) - Samuel Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994
- 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)