= Solution
The characteristic polynomial is $(r+\gamma)^2+c^2k^2$, with roots $r=-\gamma\pm ick$. For $ck\ne0$, the transformed general solution is
$$
\boxed{\widetilde u(k,t)=e^{-\gamma t}\big[A(k)e^{-ickt}+B(k)e^{ickt}\big].}
$$
The repeated-root case $ck=0$ is $e^{-\gamma t}[A_0+B_0t]$, 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 $c\ne0$, the <Fourier shift theorem> turns the two exponential factors into translations. Equivalently, set $v=e^{\gamma t}u$; then $v_{tt}=c^2v_{xx}$. The <D'Alembert formula> gives the general form
$$
\boxed{u(x,t)=e^{-\gamma t}[F(x-|c|t)+G(x+|c|t)],\qquad c\ne0,}
$$
where $F,G$ 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 $c^2$.
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