# Energy and Wave Equation

1. Nov 10, 2014

### joshmccraney

Hi PF!

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!

2. Nov 16, 2014