Spectrum of Elastic Honeycombs with a Clamped Surface

In the low-frequency range of the spectrum of an isotropic waveguide in the shape of a thick layer of thin-walled honeycombs with completely clamped surface, we find a family of open wide gaps between narrow spectral segments (the wave-stopping and wave-passing zones, respectively). Elastic waves concentrate near and oscillate along the edges of the honeycomb cells. The results are obtained by constructing asymptotics for eigenpairs of the model problem on the periodicity cell depending on the Floquet parameter. The main role is played by the boundary layer phenomenon, described by solutions to two elasticity problems—the plane vector problem and the out-of-plane scalar problem—in a symmetric two-dimensional tripod composed of unit half-strips. The decisive observation is as follows: the unique eigenvalue in the discrete spectrum of the scalar problem lies strictly below the spectrum of the vector problem. The asymptotics is justified by means of the classical lemma on “almost eigenvalues” and “almost eigenvectors,” as well as by checking the convergence of the attributes of vector-valued eigenfunctions.