Determining Speed of Water Waves with Dimensional Analysis

[RedX]

From dimensional analysis, the speed of water waves can be determined as follows:

Only the gravitational constant 'g=10 m/s^2' matters. g has units of L/T^2, so the only way to get a velocity is $$\frac{1}{(2\pi)^{\frac{1}{2}}}g^{\frac{1}{2}}*\lambda^{\frac{1}{2}}$$ where $$\lambda$$ is the wavelength.

The problem is that I looked at a small brook which is normally still, but was moving a little from a breeze, and estimated the wavelength (no longer than a foot), and applied the formula. The result is that the speed of the wave comes out a lot faster than what I observed.

So my question is has anyone else observed something similar? Or does this formula only work for big ocean waves?

 

[bm0p700f]

The equation you quoted in an approximation for deep water.

I think it is due to the depth of the water. In shallow water the wave speed is v ~ sqrtgd where d is the depth of water. Depth < lambda/20 in this case.

The formula of all depth is a little more complicated.

See http://hyperphysics.phy-astr.gsu.edu/Hbase/watwav.html#hwav

 

[RedX]

The equation you quoted in an approximation for deep water.

I think it is due to the depth of the water. In shallow water the wave speed is v ~ sqrtgd where d is the depth of water. Depth < lambda/20 in this case.

The formula of all depth is a little more complicated.

See http://hyperphysics.phy-astr.gsu.edu/Hbase/watwav.html#hwav

I guess dimensional analysis can only take you so far.

At least it makes sense that for shallow water, the speed is less. I guess this is due to friction from the floor under the sea. It's interesting that it doesn't depend on the type of floor. For solids, the type of material determines a coefficient of friction. For liquids, the layer touching the ground doesn't move at all (does anyone know why?), and the layers above it have a friction that's the viscosity of the liquid. However, the viscosity doesn't even enter the equation, so maybe it's not because of friction from the floor of the sea!