Showing that the energy of a wave doesn't change with time
This is the question:
For the wave equation problem Utt = c^2 Laplace(U) in a regular bounded region with boundary conditions being 0 and initial conditions U(x, 0) = f(x), Ut(x, 0) = g(x), a combination of the total displacement and gradient gives a useful measure of the energy of the wave. So a solution U, let:
E(U, t) = Integral over the boundary(Ut^2 + c^2*div(U)^2))
Show that the derivate of E with respect to t is 0.
I have no idea how to start, any hints would be great. Sorry about it being hard to read.