Energy considerations and the inhomogeneous wave equation

1. Dec 5, 2011

nickthequick

Hi,

If I have a forced wave equation

$$u_{tt}-c^2u_{xx}= f(x,t)$$

what is my associated energy law?

For instance, in the homogeneous case

$$\Box u=0$$

I know that

$$E(t)=\frac{1}{2}\int u_t^2 +c^2|u_x|^2 \ dx$$

which implies that $\frac{d E(t)}{d t}$ is equal to zero. (just use integration by parts)

I'm not sure if I should just conclude that

$$\frac{d E}{dt}= \int u_t f(x,t) dx$$

by following the same logic as was used to show that this quantity is zero in the homogeneous case. Or we can more generally see that the forced wave equation has a Lagrangian given by

$$L = \frac{1}{2}\iint -u_t^2 +c^2|u_x|^2 - 2u f(x,t) \ dx \ dt$$

which means that the associated Hamiltonian, or energy, would be

$$E(t) = \frac{1}{2}\int u_t^2 +c^2|u_x|^2 + 2u f(x,t) \ dx$$

which leads to a different result than what I quoted above.

Any help is appreciated!

Thanks,

Nick

Last edited: Dec 5, 2011
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