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Water Waves - Group Velocity

  1. Feb 22, 2010 #1
    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;

    [tex] V_{p}^{2}=2\pi S/\rho \lambda [/tex]

    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


    [tex]V_{g}=\frac{\partial \omega }{\partial x} [/tex]

    and

    [tex]V_{g}=V_{p}-\lambda \frac{d V_{p}}{d\lambda }[/tex]

    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 [tex]\lambda[/tex], 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.
     
  2. jcsd
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