  This paper is a detailed and self-contained study of the stability properties
of periodic traveling wave solutions of the nonlinear Klein-Gordon equation
$u_{tt}-u_{xx}+V'(u)=0$, where $u$ is a scalar-valued function of $x$ and $t$,
and the potential $V(u)$ is of class $C^2$ and periodic. Stability is
considered both from the point of view of spectral analysis of the linearized
problem (spectral stability analysis) and from the point of view of wave
modulation theory (the strongly nonlinear theory due to Whitham as well as the
weakly nonlinear theory of wave packets). The aim is to develop and present new
spectral stability results for periodic traveling waves, and to make a solid
connection between these results and predictions of the (formal) modulation
theory, which has been developed by others but which we review for
completeness.
