CLASSICAL BAND-STRUCTURE OF PERIODIC ELASTIC COMPOSITES

The rich diversity and the fundamental character of the essential theo retical problems associated with it have given band theory a width of interest which contrasts strongly with the apparent narrowness of its subject matter. This review, dealing mainly with the classical band st ructures of periodic elastic and acoustic binary systems, offers brief ly a systematic survey of the historical development of the principles , tools, and applications of band theory for electrons, phonons, photo ns, and vibrations giving what may be called the ''background'' to the more recent developments in the fields of photonic and phononic band- gap crystals. Attention is given to survey the physical conditions req uired to achieve the complete spectral gaps within which the respectiv e propagating modes are utterly forbidden irrespective of the directio n of propagation. The existence of complete spectral gaps for cleverly synthesized photonic crystals guarantees the observability of classic al Anderson localization of photons and the influence on the spontaneo us emission which was, until the 1980's, often regarded as a natural a nd uncontrollable phenomenon. The phononic band-gap crystals, on the o ther hand, offer the feasibility of constructing the ultrasound filter s, polarization filters, and improvements in designing the transducers , as well as the observability of classical elastic or acoustic wave l ocalization. Abiding by the central theme of the review, numerous theo retical results on the band structure related problems for periodic el astic and acoustic binary systems have been gathered and reviewed. Thi s survey is preceded by a detailed mathematical machinery that provide s the reader with numerous useful analytical results applicable to a w ide range of systems of varying interest. Finally, the report conclude s with a summary of anticipated implications of photonic and phononic band-gap crystals and proposes some interesting relevant problems conc erned with the spectral gaps and the classical wave localization. Our satisfaction in writing this review, like any other review which cover s a considerably longer period, was to reach a reasonably self-contain ed unity by wanting to ''leave nothing unexplained''. The background p rovided is believed to make less formidable the task of future writers of reviews in this rather general field and hence enable them to deal more readily with particular aspects of the subject, or with recent a dvances in those directions in which notable progress may have been ma de.