Multi‐parametric analysis of strongly inhomogeneous periodic waveguideswith internal cutoff frequencies

In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by a uniform elastic Winkler foundation. A multi‐parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non‐typical quasi‐static uniform or, possibly, linear microscale displacement variations over the high‐frequency domain. Macroscale governing equations are derived within the framework of the Floquet–Bloch theory as well as using a high‐frequency‐type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macroscale behavior is governed by the same equation as for a beam supported by a two‐parameter Pasternak foundation.