A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.Comment: 24 pages, 9 figure

A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide

We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.Comment: 26 pages, 6 figure

Asymptotic Theory for Rayleigh and Rayleigh-Type Waves

Explicit asymptotic formulations are derived for Rayleigh and Rayleigh-type interfacial and edge waves. The hyperbolic–elliptic duality of surface and interfacial waves is established, along with the parabolic–elliptic duality of the dispersive edge wave on a Kirchhoff plate. The effects of anisotropy, piezoelectricity, thin elastic coatings, and mixed boundary conditions are taken into consideration. The advantages of the developed approach are illustrated by steady-state and transient problems for a moving load on an elastic half-space