Reciprocity theorems and perception of non-reciprocal behavior in vibrating systems, imetamaterials, acoustic devices, and electroacoustic devices

Equations governing physical systems often admit a reciprocity corollary. Such systems include mechanical, acoustical, vibrational, and electroacoustic systems. There is no universally applicable succinct definition of reciprocity, although it is sometimes loosely stated that reciprocity means that, “if you see it, then it can see you.” There have been a considerable number of derivations of reciprocity theorems for different types of acoustical and mechanical systems published over the years --- notable names are Helmholtz, Rayleigh, Lamb, Lyamshev, Schottky, Foldy, Primakoff, Payton, de Hoop, Knopoff, Gangi, Arntsen, Carcione, and Fokkema. There appears to be no universally applicable derivation, and no universally applicable reciprocity principle. The emergence of metamaterial systems poses new challenges and opportunities in regard to reciprocity. Given one such system, one naturally asks if there is some reciprocity theorem that applies. If one perceives that there is no such theorem, then one might naturally ask whether the perception is wrong, and if it is simply the case that no one has yet bothered to try to derive one or that one is not sufficiently clever to find one. It may also be that there is an analogous system for which there is a theorem, and that the theorem applies for the present system in some approximate sense. If so, how well? Present paper examines several metamaterial systems introduced in the recent literature from the viewpoint point of reciprocity theorems.