# Water Waves - Group Velocity

1. Feb 22, 2010

### JazzCarrot

1. The problem statement, all variables and given/known data
The motion of short wavelength (about 1 cm or less) ripples on water is controlled by the surface tension S. The phase velocity of such ripples is given by;

$$V_{p}^{2}=2\pi S/\rho \lambda$$

where ρ is the water density, and λ is the wavelength.

(a) Which of the formulae is equal to the group velocity, vg, for a disturbance comprising wavelengths close to a given λ?

25/4(vp)
5/2(vp)
vp
25/4(vp^2)
3/2(vp)

If the group consists of only two wavelengths, λ1 = 0.99 cm and λ2 = 1.05 cm, what is the distance between adjacent crests?

f the group consists of only two wavelengths, λ1 = 0.99 cm and λ2 = 1.05 cm, what is the distance between adjacent beats?

2. Relevant equations

$$V_{g}=\frac{\partial \omega }{\partial x}$$

and

$$V_{g}=V_{p}-\lambda \frac{d V_{p}}{d\lambda }$$

This is the problem really, Im not sure if this is the right way of tackling it?

3. The attempt at a solution

Well, I know the answer is 3/2(vp) (I decided after being stuck that I could try and work backwards from the answer, but still no luck). I have no idea how to get there really, I've tried differentiating the Vp equation, wrt to $$\lambda$$, but it doesn't really help me... but I do get a 3/2 out of it.

I haven’t really attempted the second 2 parts, but looking at them I don't think I understand what to do their either.