We introduce a new Runge-Kutta discontinuous Galerkin (RKDG) method for problems of wave propagation that achieves full high-order convergence in time and space. For the time integration it uses an mth-order, m-stage, low storage strong stability preserving Runge-Kutta (SSP-RK) scheme which is an extension to a class of non-autonomous linear systems of a recently designed method for autonomous linear systems. This extension allows for a high-order accurate treatment of the inhomogeneous, time-dependent terms that enter the semi-discrete problem on account of the physical boundary conditions. Thus, if polynomials of degree k are used in the space discretization, the (RKDG) method is of overall order m = k + 1, for any k > 0. Numerical results in two space dimensions are presented that confirm the predicted convergence properties.