Quaternion Formalism for the Intrinsic Transfer Matrix

A quaternion formulation is applied to the intrinsic transfer matrix for longitudinal elastic wave propagation through a multilayer medium in order to find the spectral response of a sonic crystal. Resonance conditions and the band structure of the crystal are obtained. The presence of a defect is also analysed. The analysis is carried out theoretically and through simulations. A coupled oscillators model is used to validate the obtained results from a phenomenological point of view. Experimental measurements are carried out for some periodic multilayer arrangements and they are correlated with theory. The obtained spectral response and band structure are essential in characterising the sonic crystal and also in optimising its structure in order to obtain specific passbands and stopbands. The adaptedness of the quaternion formulation to periodic structures and to the inclusion of defects is considered.