  We investigate the linear telegraph equation $$ u_{tt}-u_{xx}+2\mu u_t=f(x,t)
$$ with periodic Neumann boundary conditions. We prove that the operator of the
problem is modeled as a Fredholm operator of index zero in the scale of
Sobolev-type spaces of periodic functions. This result extends to small
perturbations of the equation where $\mu$ becomes variable and discontinuous or
an additional zero-order term appears. We also show that the solutions to the
problem are smoothing.
