Recursive Dispersion Relations in One-Dimensional Periodic Elastic Media

A frequency bandgap is a range of wave frequencies that are prohibited from passing through a medium. The dispersion relation, which links the frequency to the wave number, enables us to illustrate the bandgaps. In [E. H. Lee, “A survey of variational methods for elastic wave propagation analysis in composites with periodic structures,” in Dynamics of Composite Materials, E. H. Lee, ed., ASME, New York, 1972, pp. 122–138] and [E. H. Lee and W. H. Yang, SIAM J. Appl. Math., 25 (1973), pp. 492–499] the dispersion relation was studied theoretically for the one-dimensional periodic structure made of two materials arranged symmetrically with respect to the center of the cell. Their dispersion relation formulas can be similarly extended to a multilayered symmetric cell configuration, but not to a general (nonsymmetric) cell configuration. The general model was considered in [M. Shen and W. Cao, J. Phys. D, 33 (2000), pp. 1150–1154], where each unit cell of the periodic layered structure contains several sublayers of arbitrary lengths and materials. Using the transfer matrix method, the dispersion relation was successfully derived, involving very lengthy explicit formulas. In this paper, we generalize the work of Lee and Yang and develop recursive dispersion relation formulas for a general cell configuration. The recursive formulas are easy to implement and, through several numerical experiments, successfully corroborate the results of Shen and Cao.