  Phononic crystals exhibit Bragg bandgaps, frequency regions within which wave
propagation is forbidden. In solid continua, bandgaps are the outcome of
destructive interferences resulting from periodically alternating layers. Under
certain conditions, natural frequencies emerge within these bandgaps in the
form of high-amplitude localized vibrations near a structural boundary,
referred to as truncation resonances. In this paper, we investigate the
vibrational spectrum of finite phononic crystals which take the form of a
one-dimensional rod, and explain the factors that contribute to the origination
of truncation resonances. By identifying a unit cell symmetry parameter, we
define a family of finite phononic rods which share the same dispersion
relation, yet distinct truncated forms. A transfer matrix method is utilized to
derive closed-form expressions of the characteristic equations governing the
natural frequencies of the finite system and decipher the truncation resonances
emerging across different boundary conditions. The analysis establishes
concrete connections between the localized vibrations associated with a
truncation resonance, boundary conditions, and the overall configuration of the
truncated chain as dictated by unit cell choice. The study provides tools to
predict, tune, and selectively design truncation resonances, to meet the
demands of various applications that require and uniquely benefit from such
truncation resonances.

Applied Physics;Materials Science