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Wave packet spreading

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data

    So I already finished most of this problem, but I'm having trouble figuring out the very last part second part.

    The last part of the problem is:
    "Finally, take one additional term in the Taylor series expression of [tex]\omega(k)[/tex] and show that [tex]\sigma[/tex] is now replaced by a complex quantity. Find the expression of the 1/e width of the packet as a function of time for this case and show that the packet moves with the same group velocity as before but spreads in width as it moves. Illustrate this result with a sketch."

    I found the complex quantity, and it is the second part I'm having issues with.


    2. Relevant equations

    The 1/e width is such that at [tex]k = k_0 \pm \frac{1}{\sqrt{\sigma}}[/tex], the amplitude distribution is 1/e of its maximum value [tex]A(k_0)[/tex].
    The 1/e width is defined as [tex]\frac{2}{\sqrt{\sigma}}[/tex].

    The complex expression for [tex]\sigma[/tex] is [tex]\sigma - \frac{1}{2}i\omega''_0 t[/tex]

    3. The attempt at a solution

    Well, the implication of this is that:
    [tex]\frac{2}{\sqrt{\sigma - \frac{1}{2}i\omega''_0 t}}[/tex]

    Since this is the 1/e width, I had thought that it should be increasing in order to imply spreading, however, when I graph the real component of this equation with respect to time, I always get a decreasing trajectory for t>0. Would this not imply that it is contracting?

    Well, I then went ahead and graphed my wave equation, and I did get some spreading (in that the oscillations remained visible for a larger width, however, the width of each curve was the same, but this is fine due to non-variable frequency).

    How do I get my expression for [tex]\sigma[/tex] to correctly demonstrate the spreading effect?

    thanks,
    Jeans
     
  2. jcsd
  3. Feb 23, 2010 #2

    Gokul43201

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    Please also type out the entire question, so the reader has the correct context.
     
  4. Feb 23, 2010 #3
    Actually, you're timing is impeccable because I just figured it out.

    The 1/e width refers to width in terms of wave number, so if spreading is in terms of the x-coordinate, then it will become larger as opposed to smaller.
     
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