Dimensional Analysis - waves and surface tension question

[kapitan90]

Homework Statement

In some cases waves on deep water are driven by surface tension effects and not by gravity. The surface tension is the energy per unit area of the water surface.
The area of the water surface increases by a factor proportional to $$(Ak)^2$$ in the presence of a wave amplitude A and wavenumber k. Say whether you would expect surface tension effects to dominate over gravitational effects when the wavenumber is small or large. Give your reasons.


2. Homework Equations and the attempted solution
In the previous part of the question I found the dimension of the surface tension to be $$M*T^{-2}$$. I also found the phase velocity to be equal to $$v=c*\sqrt{\frac{y*k}{d}}$$ where y - surface tension k - wavenumber, d - density, c - some constant.

 

[Nate810]

For small wavenumbers, the phase velocity is large which means that the surface tension effects would dominate over gravitational effects. This is because for large phase velocities, the energy associated with the wave is mainly due to surface tension and not gravity. On the other hand, for large wavenumbers, the phase velocity is small. This means that the energy associated with the wave is mainly due to gravity and not surface tension. Therefore, in this case, gravitational effects would dominate over the surface tension effects.