 Solution

For separation of variables, take $u(x,t)=X(x)T(t)$ in the damped wave equation. Dividing by $c^{2}XT$, with $c>0$, gives $X^{\prime \prime }/X=(T^{\prime \prime }+\alpha T^{\prime })/(c^{2}T)=-k^2$. The separated ordinary differential equations are thereforeThe negative separation constant anticipates the positive spatial eigenvalues selected by the fixed endpoints.