I have this problem from my class of hamiltonian mechanics:
Given the Klein-Gordon equation $u_{tt}=c^2u_{xx}-\omega^2u$ with boundary conditions $u(t,0)=u(t,l)=0$, $u_t(t,0)=u_t(t,l)=0, \forall t\geq0$. Find a solution by separation of variables $u(t,x)=f(t)g(x)$.
Using separation of variables I found $$g_k(x)=b_k \sin(\mu_k x),$$ $$f_k(t)=\alpha_k \cos((\mu_k^2 c^2+\omega^2)t)+\beta_k \sin((\mu_k^2 c^2+\omega^2)t),$$ where $b_k,\alpha_k,\beta_k$ are constants to be found using the conditions and $\mu_k=\frac{2k\pi}{l}$, for $k\geq1$ natural. At this point, ususally, I proceed like this: I define $U_k(t,x)=g_k(x)f_k(x)$ and then I consider the infinite superposition of $U_k$ and use the conditions to find the coefficients.
The problems are:
I don't think it makes much sense to consider a superposition of $U_k$ since there is no initial condition to impose and the conditions on $u_t$ are satisfied by any $U_k$.
The conditions on $u_t$ doesn't really add any useful information to find the coefficients, since $U_k$ satisfies it for every choice of the coefficients.
- Even assuming that $U_k$ is the solution for some $k$, it seems strange that it depends on 3 arbitrary parameters.
My questions are: Am I missing something? Is the problem well posed (i.e. existence and uniqueness of the solution) with this boundary conditions?