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Wave equation - stretched string with mass at each end

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data

    A string is stretched to a tension T, its ends x=0 and x=L are attached to rings with mass M which are able to slide on parallel smooth wires perpendicular to the string. Show:
    1) The transverse displacement must satisfy Mytt = Tyx at x=0, Mytt = -Tyx at x=0
    2) The normal frequencies are
    3) What are the normal frequencies in their limiting cases as M tends to 0 and infinity?


    2. Relevant equations

    y(x,t) = [tex]\sum[/tex] sin(n[tex]\pi[/tex]x/l) (ancos(n[tex]\pi[/tex]ct/l) + bnsin(n[tex]\pi[/tex]ct/l)
    But I think this is only for fixed ends which mine isn't?

    3. The attempt at a solution

    1. I need to use Newtons 2nd Law I think.
    At x=0 Mytt (i+j) = T(i+yxj), so Mytt = Tyx
    At x=L Mytt (i+j) = -T(i+yxj), so Mytt = -Tyx
    This doesn't feel right.

    2. I think I need more conditions to be able to solve this. The ends aren't fixed so I can't say y(0,t)=y(l,t)= 0. But can I say anything about the velocity?
     
  2. jcsd
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