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Harmonics and Telegrapher's Equation

  1. Dec 12, 2007 #1
    The problem statement, all variables and given/known data

    Consider a small (sub-threshold) signal on a nerve fiber or a telegraph cable. The signal can be written as a sum of "harmonics"

    V(x,t) = [tex]\sum(V_{n}cos(k_{n}x)f_{n}(t)[/tex]

    where n is some index identifying the terms, the V[tex]_{n}[/tex] are constants, the k[tex]_{n}[/tex] are wave numbers for the harmonics, and the f[tex]_{n}[/tex] are time-dependence functions to be determined [with f[tex]_{n}[/tex] = 1 for all n]. We have seen that a uniform signal (k[tex]_{n}[/tex] = 0) decays exponentially in time. For the above more general signal, if the harmonics all decay at the same rate, the signal preserves its shape as time passes. If harmonics with larger k[tex]_{n}[/tex] values decay faster, the signal smears out with time; if those with smaller k[tex]_{n}[/tex] values decay faster, the shape of the signal sharpens with time. Which happens?

    Sorry, I accidentally put all the subscripts as superscripts in the paragraph above, I'm not sure how to change it so I'm sorry. So please assume all the superscripts in the above paragraph only are subscripts. Thanks.


    Relevent equations

    I think the telegrapher's equation is relevent, but I'm not sure how to manipulate it.


    Attempt at a solution

    I have tried solving this problem but the only way I have been able to so far is to read it over and over and try and understand everything that is asked. I'm sorry I haven't come up with a definite track to try and solve the problem because I honestly haven't been able to do it.

    I have also posted this in the advanced physics section because to me it seems like it's advanced physics, but maybe to others it may be introductory physics, so i posted it here as well.

    Thanks!
     
    Last edited: Dec 12, 2007
  2. jcsd
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