Derivation of the ordinary time derivative of the energy associated with a wave

1. Oct 21, 2008

ben_trovato

Okay, so I'm at a loss for words to describe my irritation and curiousity on how this is solved.
Given the one-dimensional wave equation
(i.e. u_tt=c^2*u_xx 0<x<L, t>0) with no source and constant velocity, we define the energy associated with the wave to be
E=integral from 0 to L of (1/2)*(u_t)^2 with respect to x plus the integral from 0 to L of (c^2/2)*(u_x)^2 with respect to x. I get that E is the sum of the kinetic and potential energies. I am, however, having a hard time grasping how the ordinary time derivative of the energy function was derived, where dE/dt=c^2*u_t(L,t)*u_x(L,t)-c^2*u_t(0,t)*u_x(0,t).
The problem I am having is that I don't know what an ordinary derivative of a function of two variables is. What I'm trying to say is that I don't know what d/dt [(u_x)^2] or d/dt[(u_t)^2] are.

If you could throw me a bone here if I'm going in the correct direction in the derivation or tell me before I hit the wall of wrongness, that would be golden!

2. Oct 22, 2008

HallsofIvy

Where did you see the phrase "ordinary time derivative"?

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook