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I know this has already been asked (unfortunately without answer)... learning once again for an exam (quantum field theory) I can't figure out a feature of a very central quantity: the total energy of a vibrating string.

Let's start at the string (field) wave equation:

[tex] \frac{1}{c^2} \, \frac{\partial^2 \Phi(x,t)}{\partial t^2} = \frac{\partial^2 \Phi(x,t)}{\partial x^2} [/tex]

With proper boundary conditions you get a set of modes, for example

[tex] \Phi_r(x,t) = A_r(t) \, sin(\frac{r \pi x}{L}) [/tex]

and with that the general motion of the string:

[tex] \Phi(x,t) = \sum\limits_{r=1}^\infty A_r(t) \, sin(\frac{r \pi x}{L}) [/tex]

Now, my book confronts me with the total energy of the vibrating string, declaring it as "analogous to the discrete summation":

[tex]E = \int\limits_0^L \left[ \frac{1}{2} \rho {(\frac{\partial \Phi}{\partial t})}^2 + \frac{1}{2} \rho c^2 {(\frac{\partial \Phi}{\partial x})}^2 \right] \, dx , \,\,where\, the \, "discrete"\, case\, was \,\,\,\,\,\,E = \sum\limits_{r=1}^N \frac{1}{2} m \dot{q}_r ^2 + V(q_1, ..., q_N) [/tex]

I can't work this out. I understand that the first term is the kinetic energy, alright. But the second term I don't get.

The potential Energy / Length on the string is

[tex] E_{pot} / L = - \frac{1}{2 L} D \, \Phi ^2 = - \frac{1}{2} \omega ^2 \, \rho \, \Phi ^2 [/tex]

as far as I can see from classical mechanics ([tex] -D \, \Phi [/tex] is the force pulling the "mass element" backwards), but why the derivative [tex] {(\frac{\partial \Phi}{\partial x})} [/tex] insted of just [tex] \Phi [/tex] ?

Can somebody help me out? I already wasted too much time on this... I hate it when I don't understand stuff from years back that I just should know.

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# Energy of a vibrating string (continuous field)

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