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Homework Help: Phase vs. Group Velocity.

  1. Oct 10, 2004 #1
    If a question says: The phase velocity of ripples on the liquid surface is (2πS / λp)^(1/2), where S is the surface tension and p is the density of the liquid. Find the group velocity of the ripples. I know that the phase velocity = omega/k, and group velocity = delta omega/delta k. Do I just take the deriviative of the phase velocity with respect to S?
     
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  3. Oct 10, 2004 #2

    arildno

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    1. Is S k?????????????
    2.[tex]\frac{d}{dk}(\frac{\omega}{k})=\frac{d\omega}{dk}[/tex]??????????????
    is that what you're saying?
    Then think again.
     
  4. Oct 10, 2004 #3
    S is the surface tension of the liquid. Is there another way to find the group velocity?
     
  5. Oct 10, 2004 #4

    arildno

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    You have the definition; wherever have you gotten the idea that the surface tension S is the wavenumber "k"??????
     
  6. Oct 10, 2004 #5
    I realize that S is not k. Yet how do I do the problem?
     
  7. Oct 10, 2004 #6

    arildno

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    Since you have the phase velocity, you may find the frequency [tex]\omega[/tex]
    The group velocity is then, by your definition, the derivative of [tex]\omega[/tex] with respect to "k".
     
  8. Oct 10, 2004 #7
    How can we find omega if we don't know what k is?
     
  9. Oct 10, 2004 #8

    arildno

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    Multiply your phase velocity with k.
     
  10. Oct 10, 2004 #9
    then isn't (2πS / λp)^(1/2) a constant?
     
  11. Oct 10, 2004 #10

    arildno

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    No, because your wavelength satisfies identically the relation:
    [tex]\lambda{k}=2\pi[/tex]
    since your expression for ph.vel. is proportional to the square root of the wavelength, your frequency will be proportional to the square root of the wavenumber
     
  12. Oct 10, 2004 #11
    this probably sounds really idiotic on my part. I just need to take (dw/dk) of k(2πS / λp)^(1/2), right?
     
  13. Oct 10, 2004 #12
    And that will give me the group velocity?
     
  14. Oct 10, 2004 #13

    arildno

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    Yes, it will
    Differentiate, if you dare..:wink:
     
  15. Oct 10, 2004 #14
    product rule?
     
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