We consider the inverse coefficient problem of identifying the principal coefficient in the damped wave equation , subject to the boundary conditions , , from the Neumann boundary output f (t): = r(0)u(x)(0,t), . We propose detailed microlocal analysis of the regularity of the solution of the wave equation in each subdomain defined by the characteristics of the wave equation. This analysis allows us to derive sufficient conditions for the regularity of the weak solution of the direct problem, especially along the characteristic lines, as well as to prove necessary energy estimates including also stability estimate for auxiliary hyperbolic problem. Based on this analysis we prove the compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator , corresponding to the inverse problem. The last property allows us to prove an existence of a quasi-solution of the inverse problem defined as a minimum of the Tikhonov functional and also Frechet differentiability of this functional. For the case when and , a uniqueness theorem is derived. An explicit formula for the Frechet gradient of the Tikhonov functional and its justification are derived by making use of the unique solution to corresponding adjoint problem. The approach proposed in this paper is expected to lead to very effective computational identification algorithms.