Consider waves in resistant medium that satisfy the problem
$U_{tt}=c^2U_{xx}-rU_t$ for $0<x<l$ .
$u=0$ at both ends.
$u(x,0)=\phi(x)$
$u_t(x,0)=\theta(x)$, where r is a constant.
Using separation of variables method
$X''(x)=\lambda X(x)$
$T''(t)+rT'(t)=\lambda c^2T(t)$
I solved the problem for $0<r<{2\pi c\over l}$.
There $\omega={n\pi\over l}$ for the $\lambda<0$ case setting $\lambda=-\omega^2$.
I wanted the condition $0<r<{2\pi c\over l}$ to determine the roots in order to solve $T''(t)+rT'(t)=\lambda c^2T(t)$.Since the discriminant was <0 in this case there was complex roots.
But now I have to do this same problem for ${2\pi c\over l}<r<{4\pi c\over l}$. I think here also I have to use the condition to determine the type of root I would get.But with this given condition I am unable to come up with any condition to the roots.
Can someone please help me to do this