Klein-Gordon Equation with boundary conditions

[dsaun777]

I am trying to find solutions for the Klien-Gordon equations in 1-d particle in a box. The difference here is the box itself oscillating and has boundary conditions that are time dependent, something like this L(t)=L0+ΔLsin(ωt). My initial approach is to use a homogeneous solution and use Eigenfunction expansion to get a solution. But I can't make progress. Are there other methods that are easier? maybe numerical methods...

 

[pasmith]

Do you mean that you have homogenous BCs at $x = 0$ and $x = L(t)$?

Set $\epsilon = \Delta L/L_0$ so that the boundary condition is applied at $x = L_0(1 + \epsilon \sin \omega t)$. Then for small $\epsilon$, you can seek an asymptotic expansion $\psi = \psi_0 + \epsilon \psi_1 + \dots$ where the boundary conditions on the $\psi_n$ are shifted to $x = L_0$ by expanding $\psi(L_0(1 + \epsilon \sin \omega t),t)$ in Taylor series in $x$ about $x = L_0$ and comparing coefficients of powers of $\epsilon$. This gives you a series of problems which can be solved by eigenfunction expansion.

EDIT: It might be better to solve for each $\psi_n$ using a Laplace transform in time. The change of variable $\tilde x = x/L_0$ may help to simplify the algebra. Also note that if $\omega$ is a natural frequency of any of these problems then the resulting resonance will cause the asymptotic assumption to break down, since eventually the amplitude will exceed $\epsilon^{-1}$.