I'm interested in solving the very basic wave equation with Neumann boundary conditions, except I have defined on some $ [-L, L]$ domain instead of the more common $[0,L]$ domain.
$$ u_{tt} - c^2 u_{xx} = 0, \ \ \ \ \ u_x(\pm L) = 0 $$
if I simply take the usual solution kind for the spatial dependence (I have a simple cosinusoidal temporal dependence which can be easily treated, so I won't refer to it anymore): $$ u = A \cos(kx) + B\sin(kx) $$ and I inject it in the boundary conditions, I find that $$ A \sin(kL) = B \cos(kL) $$ $$-A \sin(kL) = B \cos(kL) $$ which gives me that $ B \cos(kL) = 0 $ but doesn't allow me to find the usual fact that $B = 0$ and simply have a condition for cos functions to exist. I am unsure how to get out of this. Is this normal and supposed to happen for the symmetric domain that I work with? Or is there something obvious that I'm not seeing?
Thanks a lot for your help