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    A string is stretched to a tension T and its ends x = 0 and x = L are
    attached to rings of mass M which are free to slide on parallel smooth
    wires which are perpendicular to the string. Show that the transverse
    displacement must satisfy the conditions:

    My_{tt} = Ty_x at x=0 and  My_{tt}=-Ty_x at x=L

    and that the normal frequencies are the numbers \frac{\omega}{2\pi}

    where

    \cot (\dfrac{\omega l}{c}) = \dfrac{T}{2M\omega c} \left( ( \dfrac{M \omega c}{T})^2-1 \right)


    Now I think I can show the first bits but it is getting the normal frequencys bit which I am strugglind with. The example in the notes which is almost similar is that of a weighted string and in this they split it the problem into two halves - the left and right of the mass. I don't know how to split this one up though.

    Thanks in advance for any help.
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    Wnat is c here? Does your string have mass?
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    (Original post by DFranklin)
    Wnat is c here? Does your string have mass?
    The string doesn't have a mass and I think c is from the equation

    c^2y_{xx}=y_{tt}
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    (Original post by The Muon)
    The string doesn't have a mass and I think c is from the equation

    c^2y_{xx}=y_{tt}
    But the derivation of that equation usually involves the mass of the string - specifically, c=\sqrt{T/\rho}, where T is the tension and \rho the mass per unit length.
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    (Original post by DFranklin)
    But the derivation of that equation usually involves the mass of the string - specifically, c=\sqrt{T/\rho}, where T is the tension and \rho the mass per unit length.
    Just gone through my notes and looking at the earlier stuff I have

    Flexible string
    Stretched to tension T
    mass density \rho
    small vibrations
    Gravity and air resistance ignored.
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    Not sure, but I would guess a solution of form y = sin(Ax+B) cos Ct and solve for A, B and C.

    You can relate A and C immediately by the standard wave equation (which applies when you're not at the end points). And then the end point conditions give you conditions on A and B, from which you should be able to get the equation they give. I hope.

    (Of course, the guess about the forum of solution might be wrong as well).
 
 
 
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