  A complete and systematic approach to compute the causal boundary of
wave-type spacetimes is carried out. The case of a 1-dimensional boundary is
specially analyzed and its critical appearance in pp-wave type spacetimes is
emphasized. In particular, the corresponding results obtained in the framework
of the AdS/CFT correspondence for holography on the boundary, are reinterpreted
and very widely generalized. Technically, a recent new definition of causal
boundary is used and stressed. Moreover, a set of mathematical tools is
introduced (analytical functional approach, Sturm-Liouville theory, Fermat-type
arrival time, Busemann-type functions).
