How Does Surface Tension Affect the Group Velocity of Water Ripples?

[JazzCarrot]

Homework Statement

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?

Homework 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, I am not sure if this is the right way of tackling it?

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 V_p 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.

 

[anathema]

Thank you for your post. I am a scientist and I would like to help you with your questions.

Firstly, to find the group velocity, we can use the formula Vg = dω/dx. In this case, we can substitute the given formula for Vp into this equation and differentiate it with respect to λ. This will give us Vg in terms of λ. Then, we can substitute the given wavelengths (λ1=0.99 cm and λ2=1.05 cm) into the equation to find the group velocity for this specific case.

For the second part, we can use the formula Vg = Vp - λ(dVp/dλ). Again, we can substitute the given formula for Vp and differentiate it with respect to λ to get Vg in terms of λ. Then, we can substitute the given wavelengths into the equation to find the group velocity for this specific case.

As for the distance between adjacent crests and beats, we can use the formula d = λ1 - λ2. In this case, the distance between adjacent crests will be the same as the distance between adjacent beats, since there are only two wavelengths present in the group.

I hope this helps you with your problem. Let me know if you have any further questions.