Energy considerations and the inhomogeneous wave equation

[nickthequick]

Hi,

If I have a forced wave equation

[tex] u_{tt}-c^2u_{xx}= f(x,t) [/tex]

what is my associated energy law?For instance, in the homogeneous case

[tex]\Box u=0 [/tex]

I know that

[tex] E(t)=\frac{1}{2}\int u_t^2 +c^2|u_x|^2 \ dx [/tex]

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

I'm not sure if I should just conclude that

[tex]\frac{d E}{dt}= \int u_t f(x,t) dx [/tex]

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

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

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

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

which leads to a different result than what I quoted above.Any help is appreciated!

Thanks,

Nick

 

[jvicens]

Hello Nick,

The associated energy law for the forced wave equation is given by the Hamiltonian, which is defined as the total energy of the system at any given time t. In this case, the Hamiltonian is given by:

H(t) = \frac{1}{2}\int (u_t^2 + c^2|u_x|^2) + 2uf(x,t) dx

This Hamiltonian represents the total energy of the system, including the energy of the wave and the energy added by the external force f(x,t). The time derivative of the Hamiltonian is equal to the power input from the external force, which is given by:

\frac{dH}{dt} = \int u_t f(x,t) dx

This shows that the energy of the system is changing due to the external force f(x,t). In the homogeneous case, where there is no external force, the Hamiltonian is constant and the energy of the system remains unchanged over time.

I hope this helps clarify the energy law for the forced wave equation.