- #1
ben_trovato
- 2
- 0
Okay, so I'm at a loss for words to describe my irritation and curiosity 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!
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!