In this paper the authors obtain necessary conditions, presented as criteria of Kalman's type, to the approximate null controllability, the approximate synchronization, and the approximate synchronization by groups, respectively, for a coupled system of wave equations with Dirichlet boundary controls. The sufficiency of these conditions is also established for N x N cascade systems, for some 2 x 2 systems, as well as for diagonalizable systems in the one-space-dimensional case.