Solving the Wave equation $u_{tt}=u_{xx}$
If I apply separation of variables, with $u(x,t)=f(x)g(t)$, then we find
$$\frac{f''(x)}{f(x)}=\frac{g''(t)}{g(t)}=k$$
Can $k$ be a complex number here? I read somewhere for the heat equation it is not possible (so it must be real). Is it valid for the wave equation?