Possible flexural wave propagation regimes in periodic beams

Flexural wave propagation in periodic beams has been the object of studies for decades and it remains an active field of research, particularly because of the possibility of generating bandgaps – frequency intervals at which propagation is not allowed – and the new possibilities of acting on structures to precisely control their stiffness over time (modulation). For Euler–Bernoulli, Rayleigh and Timoshenko beam models, there exist two wave modes, and each one of them can be either propagating, evanescent (exponentially decaying) or decaying with oscillations. The present work establishes which combinations in modes can emerge when the beam is assumed as infinitely periodic and the unit cell is piece-wise constant. Regardless of number of layers in the unit cell and material and cross-section properties, it is shown that Euler–Bernoulli beams always feature an evanescent mode, while any combination of modes is possible for Rayleigh and Timoshenko, unless the design parameter space is constrained. Bloch’s theorem and spectral analysis of transfer matrices (specifically, the notions of symplecticity and dominant eigenvalue) are the mathematical levers we use to derive the results, once wave propagation is related to the eigenvalues of the transfer matrix. We show why these results do not carry over when sharp temporal interfaces replace spatial interfaces. Other practical consequences and the possible extension of the results towards continuous beam profiles are also discussed. This work offers foundational results to guide bandgap optimization for flexural waves.