What is the solution for a fixed string oscillating at both ends?

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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:
\frac{\partial^2 \Psi}{\partial t^2}=\frac{\partial^2 \psi}{\partial x^2}

<br /> \textrm{Given that at t=0:}\\<br /> \&gt;\&gt;\&gt;\&gt; \psi(x,0)=\frac{2xh}{L},0\leq x\leq \frac{L}{2}\\<br /> \&gt;\&gt;\&gt;\&gt; \psi(x,0)=\frac{2xh}{L},(L-x),\frac{L}{2}\leq x\leq L\\<br /> \textrm{Show:}\\<br /> \&gt;\&gt;\&gt;\&gt; \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)<br />

So, how do we go about doing that?
 
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Do you know the method of separation of variables?
 
haruspex said:
Do you know the method of separation of variables?

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.
 
So what does that give you for generic solutions of the PDE?
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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