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SO we have defined energy per unit mass as $$E(t) = \int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx$$. We are given a vibrating string that exhibits ##u_x(0,t) = 0## and ##u(L,t)=0##. I am trying to figure out what is happening with total energy, ##E(t)##. My work is $$\int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx = \int_0^L \frac{1}{2} u_t^2 dx + \int_0^L \frac{c^2}{2} u_x^2 dx=\int_0^L \frac{1}{2} u_t^2 dx + \frac{c^2}{2} u_x u \bigg|_0^L - \int_0^L u u_{xx} dx\\

=\int_0^L \frac{1}{2} u_t^2 dx - \frac{c^2}{2}\int_0^L u u_{xx} dx = \int_0^L \frac{1}{2} u_t^2 - \frac{1}{2} u u_{tt} dx$$ where the last quantity appears from ##u## satisfying the wave eq, ##u_{tt} = c^2 u_{xx}##. Is there anything more I can/should do?

Thanks!

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# Energy and Wave Equation

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