The rich diversity and the fundamental character of the essential theoretical 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 structures of periodic elastic and acoustic binary systems, offers briefly a systematic survey of the historical development of the principles, tools, and applications of band theory for electrons, phonons, photons, 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 required to achieve the complete spectral gaps within which the respective propagating modes are utterly forbidden irrespective of the direction of propagation. The existence of complete spectral gaps for cleverly synthesized photonic crystals guarantees the observability of classical Anderson localization of photons and the influence on the spontaneous emission which was, until the 1980's, often regarded as a natural and uncontrollable phenomenon. The phononic band-gap crystals, on the other hand, offer the feasibility of constructing the ultrasound filters, polarization filters, and improvements in designing the transducers, as well as the observability of classical elastic or acoustic wave localization. Abiding by the central theme of the review, numerous theoretical results on the band structure related problems for periodic elastic and acoustic binary sytems have been gathered and reviewed. This survey is preceded by a detailed mathematical machinery that provides the reader with numerous useful analytical results applicable to a wide range of systems of varying interest. Finally, the report concludes with a summary of anticipated implications of photonic and phononic band-gap crystals and proposes some interesting relevant problems concerned with the spectral gaps and the classical wave localization. Our satisfaction in writing this review, like any other review which covers a considerably longer period, was to reach a reasonably self-contained unity by wanting to "leave nothing unexplained". The background provided 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 advances in those directions in which notable progress may have been made.
Corner states in a second-order mechanical topological insulator
