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Wave equation - stretched string with mass at each end watch

    • Thread Starter

    This is the question:
    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?

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

    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?

    But I think this is only for fixed ends which mine isn't?
    You're right, the ends are free to move... In this case the boundary condition you want is the free end boundary condition, which is that the gradient of the displacement is 0 at x = 0 and x= L.

    See here for more details - this site pretty much has the exact problem you're dealing with, except it considers massless rings.

    • Thread Starter

    Thanks, think I've got it now. Not too sure about the last bit but at least the main part of the question is done.

    Sorry to bring up an old thread but I'm having trouble getting my head around this problem, even after reading the link.

    I have started my separation of variables with y=F(x)G(t)

    With some persuading I am willing to believe that y_x(0,t)=y_x(L,t)=0 as the ends are free to move.

    Now when I try separation of variables I end up with

    y(x,t) = \cos(\frac{n \pi x}{L}) \left( a \cos(\frac{n \pi c t}{L}) + (b \sin(\frac{n \pi c t}{L}) \right) and I'm not sure how to get a w/2Pi which are the normal frequencies into this equation.

    Also, I am wondering whether I should have multiple constants a and b or just the two there i.e. should it be a big sum with lots of different constants for different values of n?
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Updated: March 20, 2010

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