In this paper, we consider the generalized approximate boundary
synchronization for a coupled system of wave equations with Dirichlet
boundary controls. We analyse the relationship between the generalized
approximate boundary synchronization and the generalized exact boundary
synchronization, give a sufficient condition to realize the generalized
approximate boundary synchronization and a necessary condition in terms
of Kalman's matrix, and show the meaning of the number of total
controls. Besides, by the generalized synchronization decomposition, we
define the generalized approximately synchronizable state, and obtain
its properties and a sufficient condition for it to be independent of
applied boundary controls.