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Fixed String Oscillation

  1. Dec 10, 2012 #1
    This isn't actually a homework or coursework problem, but the style of the question is similar so I'm posting it here. Anyways, here goes. Consider a string of length L clamped at both ends, with one end at x=0 and the other at x=L. The displacement of the oscillating string can be described by the following equation:
    [itex]\frac{\partial^2 \Psi}{\partial t^2}=\frac{\partial^2 \psi}{\partial x^2}[/itex]

    [itex]
    \textrm{Given that at t=0:}\\
    \>\>\>\> \psi(x,0)=\frac{2xh}{L},0\leq x\leq \frac{L}{2}\\
    \>\>\>\> \psi(x,0)=\frac{2xh}{L},(L-x),\frac{L}{2}\leq x\leq L\\
    \textrm{Show:}\\
    \>\>\>\> \psi(x,t)=\sum_{m=1}^\infty\sin\left(\frac{m\pi x}{L}\right)\cdot\cos\omega_mt\cdot\left(\frac{8h}{\pi^2m^2}\right)\cdot\sin\left(\frac{\pi m}{2}\right)
    [/itex]

    So, how do we go about doing that?
     
    Last edited: Dec 10, 2012
  2. jcsd
  3. Dec 10, 2012 #2

    haruspex

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    Do you know the method of separation of variables?
     
  4. Dec 10, 2012 #3
    Yeah, that's just rearranging the equation so that different variables occur on opposite sides of the equation. You can also do this by defining a variable as some expression, substitute, and the separate them.
     
  5. Dec 10, 2012 #4

    haruspex

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    So what does that give you for generic solutions of the PDE?
     
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