Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / Section A / A6

The dimensions are $[\rho ]=M{L}^{-3}$, $[\sigma ]=M{T}^{-2}$ and $[\lambda ]=L$. Equating exponents in $L{T}^{-1}=M^{\alpha +\beta }{L}^{-3\alpha +\gamma }{T}^{-2\beta }$ yieldsThus the model has $v_{\mathrm{ph}}\propto \sqrt{\sigma /(\rho \lambda )}$. Its common dimensionless factor cancels in the ratio:They are nearly equal within this capillary-wave model. The assumed power law omits gravitational restoring forces and finite-depth effects; dimensional analysis cannot establish those omissions for an arbitrary real surface wave.