An $H^1$-Galerkin mixed finite element method is discussed for a class of second order hyperbolic problems. It is proved that the Galerkin approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition and quasi-uniformity requirement on the finite element mesh. Compared to the results proved for one space variable, the $L^∞(L^2)$-estimate of the stress is not optimal with respect to the approximation property for the problems in two and three space dimensions. It is further noted that if the Raviart- Thomas spaces are used for approximating the stress, then optimal estimate in $L^∞(L^2)$-norm is achieved using the new formulation. Finally, without restricting the approximating spaces for the stress, a modification of the method is proposed and analyzed. This confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed procedure under an extra regularity assumption on the exact solution.