Intrinsic transfer matrix method and split quaternion formalism for multilayer media

•Split quaternion formalism for 1D wave propagation in a multilayer medium is obtained based on the transfer matrix method.•Periodic media with a defect are considered and several particular cases are analysed.•A class of commutative split quaternions is identified, corresponding to defects that can be placed anywhere in the structure with the same effect.•A medium composed of commutative split quaternion elements is described.

The intrinsic transfer matrix method for 1D longitudinal elastic wave propagation through multilayer media is used to obtain an equivalent split quaternion formalism. Periodic media are analysed in this framework and the presence of a defect is considered. A simple one layer defect and an inversion defect are analysed. A commutative type of split quaternion is identified, which corresponds to defects of the periodic structure that can be placed in any position, the overall acoustic properties of the medium being conserved. Also, a nonperiodic medium composed of such commutative elements has a behaviour independent of the order of elements. Several possible applications in sound wave measurement and processing are outlined. The proposed split quaternion formalism is compact and can make analytical and numerical computations easier.