Capillary waves are small surface waves or ripples whose dominant restoring force is surface tension.
Suppose a liquid is indefinitely deep and the amplitude of a capillary wave is so small as to play
no role in its behavior. Then its wave speed $v$ can depend only on wavelength $\lambda$, surface tension $\sigma$,
acceleration of gravity $g$, and possibly the mass density $\rho$ of the liquid.
(a) find the two dimensionless products with symbols given in the problem by Buckingham II
theorem. hint:one of the products has $\lambda$ and the other doesn't.
(b) the equation of wave speed can is given below, where $f(\lambda(gp/\sigma)^{1/2})$ is a function of $(gp/\sigma)^{1/2}$.
please derive the equation with the dimensionless products you got from (a).
$v = \sqrt{\frac{\sigma}{\lambda \rho}} \cdot f(\lambda \sqrt{\frac{gp}{\sigma}})$ (1)
hint: $v(\lambda \rho/\sigma)^{1/2}$ and $\lambda(gp/\sigma)^{1/2}$ are both dimensionless.
(c) the dimensionless product $\lambda(gp/\sigma)^{1/2}$ can be used to determine whether it is gravity or surface
tension that dominates. Please calculate the wavelength that separates two regimes for water
at 25 $^\circ$C. Given $g$ 9.8 m/s$^2$, $\rho$ 0.997$\cdot$10$^3$ kg/m$^3$, $\sigma$ 0.072 N/m.
