A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh's Quotient and Rayleigh's Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.