On propagation of plane symmetric waves in a periodically corrugated straight elastic layer

The classical Bernoulli−Euler model is employed for analysis of the plane wave propagation in an infinitely long elastic layer with a periodically varying thickness. The Floquet theory is applied to derive asymptotic formulas defining location and broadness of frequency stop bands for several corrugation shapes with different levels of discontinuity. For a layer with the perfectly smooth periodic corrugation, the equation of the axial wave motion is solved by the method of multiple scales in vicinity of critical frequencies. Then it is transformed to the canonical Mathieu equation, and its stability diagram is compared with predictions of the Floquet theory. The qualitative differences between the shapes of stop bands are discussed. The Wentzel−Kramers−Brillouin approximation is employed to solve the Bernoulli−Euler equation for the layer with perfectly smooth periodic corrugation, and the results are discussed in view of existence of pass and stop bands and in view of the levels of approximation used in alternative asymptotic methods.