I posed the question to find by dimensional analysis the dispersion
relation w(k) for surface waves on water. We considered deep water,
so the wavelength is short compared to the depth. In that case the
depth can't affect the waves. If the wavelength is long enough however,
then surface tension can be neglected, and the wave is called a
"gravity wave", since the restoring force is gravity. For short
enough wavelengths, the surface tension becomes dominant. Those waves
are called capillary waves. In the two limits, the class found the
results:

w = Sqrt[gk]             gravity
w = Sqrt[sigma k^3/rho]  capillary

where g is the acceleration of gravity at the earth's surface,
sigma is the surface tension, with dimensions energy/area,
or force/length, and rho is the mass density.

The dispersion relation that combines them correctly is

w^2 = gk + (sigma/rho)k^3.

I suggested you could guess this since Newton's law m x_tt = F_1 + F_2
is second order in time derivatives, and additive in forces, suggesting
the above form.

The question was now to use this to understand the phenomenon
of standing ripples upstream from a rock in a river. If the flow
is fast enough, such a pattern forms, and has a wavelength that
is shorter the higher the speed. You can think of these waves
as travelling upstream, but getting nowhere since the water speed
is equal and opposite to the phase velocity. So, use the above dispersion
relation to find the critical water speed, and the corresponding
wavelength of the ripples. Find a formula, and find the numerical
values in cm/s and cm. You may need to make an approximation or
use a computer or root finding calculator (although I think the
numbers are such that one can get a very pretty very accurate analytic
approximation). Also find how the ripple wavelength depends on the
speed of the water.