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Energy of a struck string

  1. Sep 20, 2015 #1
    1. The problem statement, all variables and given/known data
    An undisturbed string is struck from below by a hammer of width 2a, as shown in the figure.
    upload_2015-9-20_11-28-11.png

    Calculate the total energy of the wave motion and show that it is equal to the energy the string has when it is first struck upload_2015-9-20_11-33-39.png

    3 cases:
    t=0+ (string just got struck)
    t>(a/c) (when the forward and backward travelling waves are resolved)
    0<t<(a/c) (when the waves are not resolved)


    2. Relevant equations
    Initial conditions:
    upload_2015-9-20_11-28-51.png
    D'Alembert's solution:
    upload_2015-9-20_11-34-38.png
    where R is the ramp function, xH(x)
    3. The attempt at a solution
    Here's what I got for the t=0+ case, it isn't the right answer though. The R terms cancel each other out, and the integral is 0.

    upload_2015-9-20_11-48-58.png
    Here's the graph for the t>(a/c) case. I'd like to know what the graph looks like for the case where 0<t<(a/c)
    upload_2015-9-20_11-50-47.png

    Also, here's a hint we got that doesn't really help me but it might help someone else..
    Sketch the waveform as a function of x and label region 1 where 0<x<(a-ct), region 2 where
    (a-ct)<x<(a+ct), and region 3 where x>(a+ct). Note that the energy in region 1 is entirely kinetic (why?), that the wave in region 2 is progressive (why?), and that the energy in region 3 is zero (why?). Add the corresponding energies, then multiply by 2 to account for the region x<0
     
  2. jcsd
  3. Sep 25, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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