  We develop a general theory for linear stability of traveling waves of second
order in time PDE's. More precisely, we introduce an explicitly computable
index $\om^*\in (0, \infty]$ (depending on the self-adjoint part of the
linearized operator) so that the wave is stable if and only if $|c|\geq \om^*$.
The results are applicable both in the periodic case and in the whole line
case. As an application, we consider three classical models - the Boussinesq
equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam
equation. For the Boussinesq model and the KGZ system (and as a direct
application of the main results), we compute explicitly the set of speeds which
give rise to linearly stable traveling waves (and for all powers of $p$ in the
case of Boussinesq). This result is new for the KGZ system, while it
generalizes the results of Alexander-Sachs, which apply to the case $p=2$. For
the beam equation, we provide an explicit formula (depending of the function
$\|\vp_c'\|_{L^2}$), which works for all $p$ and for both the periodic and the
whole line cases. Our results complement (and exactly match, whenever they
exist) the results of a long line of investigation regarding the related notion
of orbital stability of the same waves.

Analysis of PDEs;Spectral Theory