Umov-Poynting-Mandelstam radiation conditions in periodic composite piezoelectric waveguides

We develop and investigate radiation conditions at infinity for composite piezo-elastic waveguides. The approach is based on the Mandelstam radiation principle according to which the energy flux at infinity is directed away from the source and which implies constraints on the (sign of the) group velocities. On the other side, the Sommerfeld radiation condition implies limitations on the wave phase velocity and is, in fact, not applicable in the context of piezo-elastic wave guides. We analyze the passage to the limit when the piezo-electric moduli tend to zero in certain regions yielding purely elastic inclusions there. We provide a number of examples, e.g. elastic and acoustic waveguides as well as purely elastic insulating and conducting inclusions.