 Solution

The characteristic polynomial is $(r+\gamma {)}^2+c^2{k}^2$, with roots $r=-\gamma \pm ick$. For $ck\ne 0$, the transformed general solution isThe repeated-root case $ck=0$ is $e^{-\gamma t}[A_0+B_{0}t]$, rather than two independent copies of the same exponential. An alternative basis regular at $k=0$ is $e^{-\gamma t}[C(k)\cos (ckt)+D(k)\sin (ckt)/(ck)]$, using the limit $t$ for the last factor at zero.

For $c\ne 0$, the Fourier shift theorem turns the two exponential factors into translations. Equivalently, set $v=e^{\gamma t}u$; then $v_{tt}=c^2{v}_{xx}$. The D'Alembert formula gives the general formwhere $F,G$ are arbitrary functions with the required regularity and decay. The arbitrary profiles can accommodate a spatially transformed zero-frequency velocity by their integral components; the isolated repeated root must not be used to discard that degree of freedom. If $c=0$, the equation instead has $u=e^{-\gamma t}[F(x)+tG(x)]$.

This is the exponentially conjugated wave equation. Positive $\gamma$ produces damping, but the algebra also covers negative $\gamma$, which gives exponential amplification. The sign of $c$ is immaterial because the differential equation contains only $c^2$.