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Waves: Trouble with simple Group Velocity derivation

  1. Sep 10, 2012 #1

    K29

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    In my notes on waves (specifically water waves) there is a derivation of Group Velocity.

    They consider two waveforms with the same amplitude, that differ slightly in wavelength and frequency, which are then superimposed to give wave groups.

    [itex]k[/itex]is wavenumber, [itex]\delta k[/itex] is how much the wavenumbers between the two waves differ. Similarly for angular frequency [itex]\omega[/itex]. [itex]a[/itex] is the amplitude of both waves.

    Superimposing both wave equations:
    [itex]\tau(x,t)=a*cos[(k+\delta k)x-(\omega +\delta \omega)t]+ a*cos [kx-\omega t][/itex]

    [itex]=2a*cos[\frac{1}{2}(\delta kx-\delta \omega t)]cos[(k+\frac{1}{2}\delta k)x-(\omega+\frac{1}{2}\delta \omega )t][/itex]


    I am fine with this. This results from a trig identity. So we have a wave group with a varying amplitude given by [itex]2a*cos[\frac{1}{2}(\delta kx-\delta \omega t)][/itex], wavenumber [itex]k+\frac{1}{2}\delta k[/itex], angular frequency[itex]\omega +\frac{1}{2}\delta \omega[/itex]

    The distance between two successive wavegroups is [itex]\Delta x[/itex]

    I am having trouble understanding where the next step comes from:

    (1)
    [itex]\frac{1}{2}\delta k \Delta x = \pi[/itex]

    Thus

    [itex]\Delta x = \frac{2\pi}{\delta k}[/itex]

    They go on with the same thing for [itex]\Delta t[/itex] and then get the Group Velocity.

    Where does (1) come from? I don't see it.

    Any assistance would be appreciated.
     
  2. jcsd
  3. Sep 10, 2012 #2

    rude man

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    1. What is a good way to get rid of t?
    2. How does one classically find minima/maxima of a function?

    Now, there may be a simpler way to do this, but that would be my approach.

    As an aside: do not, unlike my Halliday & Resnick textbook ( 20-7) confuse this superposition process with modulation. No new frequencies are generated with superposition, whereas they most certainly are with modulation.
     
  4. Sep 11, 2012 #3

    K29

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    Thank you. This helped
     
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