  Bandgaps, or frequency ranges of forbidden wave propagation, are a hallmark
of Phononic Crystals (PnCs). Unlike their lattice counterparts, PnCs taking the
form of continuous structures exhibit an infinite number of bandgaps of varying
location, bandwidth, and distribution along the frequency spectrum. While these
bandgaps are commonly predicted from benchmark tools such as the Bloch-wave
theory, the conditions that dictate the patterns associated with bandgap
symmetry, attenuation, or even closing in multi-bandgap PnCs remain an enigma.
In this work, we establish these patterns in one-dimensional rods undergoing
longitudinal motion via a canonical transfer-matrix-based approach. In doing
so, we connect the conditions governing bandgap formation and closing to their
physical origins in the context of the Bragg condition (for infinite media) and
natural resonances (for finite counterparts). The developed framework uniquely
characterizes individual bandgaps within a larger dispersion spectrum
regardless of their parity (i.e., odd vs even bandgaps) or location (low vs
high-frequency), by exploiting dimensionless constants of the PnC unit cell
which quantify the different contrasts between its constitutive layers. These
developments are detailed for a bi-layered PnC and then generalized for a PnC
of any number of layers by increasing the model complexity. We envision this
mathematical development to be a future standard for the realization of
hierarchically-structured PnCs with prescribed and finely tailored bandgap
profiles.
