  An $hp$-discontinuous Galerkin (DG) method is applied to a class of second
order linear hyperbolic integro-differential equations. Based on the analysis
of an expanded mixed type Ritz-Volterra projection, {\it a priori} $hp$-error
estimates in $L^{\infty}(L^2)$-norm of the velocity as well as of the
displacement, which are optimal in the discretizing parameter $h$ and
suboptimal in the degree of polynomial $p$ are derived. For optimal estimates
of the displacement in $L^{\infty}(L^2)$-norm with reduced regularity on the
exact solution, a variant of Baker's nonstandard energy formulation is
developed and analyzed. Results on order of convergence which are similar in
spirit to linear elliptic and parabolic problems are established for the
semidiscrete case after suitably modifying the numerical fluxes. For the
completely discrete scheme, an implicit-in-time procedure is formulated,
stability results are derived and {\it a priori} error estimates are discussed.
Finally, numerical experiments on two dimensional domains are conducted which
confirm the theoretical results.
