Capillary wave

 ( Fluid Dynamics ) 

Three contributions to the energy are involved: the potential energy $V\_\{g\}$ due to standard gravity, the potential energy $V\_\{st\}$ due to the surface tension and the kinetic energy $T$ of the flow. The part $V\_\{g\}$ due to gravity is the simplest: integrating the potential energy density due to gravity, $\backslash rho\; g\; z$ (or $\backslash rho\text{'}\; g\; z$) from a reference height to the position of the surface, $z\; =\; \backslash eta(x,y,t)$:Lamb (1994), §174 and §230.

 V_\mathrm{g}
   = \iint dx\, dy\; \int_0^\eta dz\; (\rho - \rho') g z
   = \frac{1}{2} (\rho-\rho') g \iint dx\, dy\; \eta^2,
     

assuming the mean interface position is at $z=0$.

An increase in area of the surface causes a proportional increase of energy due to surface tension:Lamb (1994), §266.

 V_\mathrm{st}
 = \sigma \iint dx\, dy\;
   \left[
     \sqrt{ 1 + \left( \frac{\partial \eta}{\partial x} \right)^2
              + \left( \frac{\partial \eta}{\partial y} \right)^2}
     - 1
   \right]
 \approx \frac{1}{2} \sigma \iint dx\, dy\;
   \left[
     \left( \frac{\partial \eta}{\partial x} \right)^2
     +
     \left( \frac{\partial \eta}{\partial y} \right)^2
   \right],
     

where the first equality is the area in this (Gaspard Monge's) 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:Lamb (1994), §61.

 T=
 \frac{1}{2} \iint dx\, dy\;
 \left[
   \int_{-\infty}^\eta dz\; \rho\,  \left

Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both $\backslash phi(x,y,z,t)$ and $\backslash phi\text{'}(x,y,z,t)$ must satisfy the Laplace equation:Lamb (1994), §20

$\backslash nabla^2\; \backslash Phi\; =\; 0$   and   $\backslash nabla^2\; \backslash Phi\text{'}\; =\; 0.$

These equations can be solved with the proper boundary conditions: $\backslash phi$ and $\backslash phi\text{'}$ must vanish well away from the surface (in the "deep water" case, which is the one we consider).

Using Green's identity, and assuming the deviations of the surface elevation to be small (so the z–integrations may be approximated by integrating up to $z=0$ instead of $z\; =\; \backslash eta$), the kinetic energy can be written as:

 T \approx
 \frac{1}{2} \iint dx\, dy\;
 \left[
   \rho\,  \Phi\,  \frac{\partial \Phi }{\partial z}\;
   -\;
   \rho'\, \Phi'\, \frac{\partial \Phi'}{\partial z}
 \right]_{\text{at } z=0}.
     

To find the dispersion relation, it is sufficient to consider a sinusoidal wave on the interface, propagating in the x–direction:

$\backslash eta\; =\; a\backslash ,\; \backslash cos\backslash ,\; (\; kx\; -\; \backslash omega\; t)\; =\; a\backslash ,\; \backslash cos\backslash ,\; \backslash theta\; ,$

with amplitude $a$ and wave phase $\backslash theta\; =\; kx\; -\; \backslash omega\; t$. The kinematic boundary condition at the interface, relating the potentials to the interface motion, is that the vertical velocity components must match the motion of the surface:

$\backslash frac\{\backslash partial\backslash Phi\}\{\backslash partial\; z\}\; =\; \backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; t\}$   and   $\backslash frac\{\backslash partial\backslash Phi\text{'}\}\{\backslash partial\; z\}\; =\; \backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; t\}$   at $z\; =\; 0$.

To tackle the problem of finding the potentials, one may try separation of variables, when both fields can be expressed as:

 \Phi(x,y,z,t) & = + \frac{1}{