A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the essential 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. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trapped modes in a cylindrical elastic waveguide with a damping gasket

An infinite cylindrical body containing a three-dimensional heavy rigid inclusion with a sharp edge is considered. Under certain constraints on the symmetry of the body, it is shown that any prescribed number of eigenvalues of the elasticity operator can be placed on an arbitrary real interval (0, l) by choosing suitable physical properties of the inclusion. In the continuous spectrum, these points correspond to trapped modes, i.e., to exponentially decaying solutions to the homogeneous problem. The results can be used to design filters and dampers of elastic waves in a cylinder.     

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.     

About embedded eigenvalues for a spectral problem arising in the study of elastic surface waves in a topographical waveguide

In this paper, we are interested with the spectral study of an operator given by an elastic topographical waveguide, a deformed half‐space, of which the cross‐section is a local perturbation of a homogeneous half‐plane. We look for guided waves propagating more rapidly than Rayleigh waves (which mathematically would correspond to embedded eigenvalues) and prove that there are no guided waves propagating more rapidly than S‐waves. Thanks to the boundary of the deformed half‐plane and some reduced equations, these eventual eigenmodes must locally vanish. Adapting to our case a unique continuation principle for the elasticity system, we conclude that these eigenmodes vanish everywhere. Copyright © 2002 John Wiley & Sons, Ltd.     

On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Artificial boundary conditions for finding surface waves in the problem of diffraction by a periodic boundary

Transparent artificial boundary conditions and an algorithm for computing the augmented scattering matrix are proposed for finding surface waves in a prescribed range of decay rates. An infinite-dimensional fictitious scattering operator is constructed that determines all waves decaying exponentially with distance from a periodic obstacle.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

Analyticity of the streamlines and of the free surface for periodic equatorial gravity water flows with vorticity

We prove analyticity of the streamlines beneath the surface and the smoothness of the free surface for geophysical equatorial water flows with a general Hölder continuously differentiable underlying vorticity distribution under the assumption of no stagnation points in the flow. Moreover, we prove that the real-analyticity of the vorticity function implies the real-analyticity of the free surface.     

Vibrations of the surface of an elastic double-layered half-space with a periodic system of cracks

The two-dimensional problem of the normal incidence of a plane transverse wave from the far field on to the free surface of an elastic double-layered half-space, comprising a homogeneous layer attached to a semi-infinite base of a different elastic material, is considered. At the boundary between the two media there is a system of plane cracks, arranged periodically along the separation line, which models the fracture zone at the interface between dense solid rock and soft sedimentary rock. The effect of the fractures on the transmission of a transverse seismic wave generated by a deep-focus earthquake, and of the type of vibrations of the free surface of the ground that result, is studied. It is difficult to predict whether the seismic wave is strengthened or weakened by the fracture zone. The effect of the system of cracks on vibrations of the free surface largely depends on the physical and geometrical parameters and, primarily, on the vibration frequencies.     

Gap in the essential spectrum of the Neumann problem for an elliptic system in a periodic domain

We establish the existence of a gap in the essential spectrum of the Neumann problem for an elliptic formally self-adjoint system of second-order differential equations on a quasi-cylinder (a domain with periodically varying cross-section).     

On the structure of the spectrum for the elasticity problem in a body with a supersharp spike

We establish that the continuous spectrum of the Neumann problem for the system of elasticity equations occupies the entire closed positive real semiaxis in the case that a three-dimensional body with a sharp-spiked cusp whose cross-section contracts to a point with the velocity O(r ^1+γ), where r is the distance to the vertex of the spike and γ > 1 is the sharpness exponent.     

Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain Ω^ϵ depending on a small parameter ϵ > 0. The domain has the form , where S^ϵ is an -periodic family of trap-like screens. We prove that, for an arbitrarily large L, the spectrum has precisely one gap in [0,L] when ϵ is small enough; moreover, when ϵ → 0, this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D photonic crystals is discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle

Perturbations of an eigenvalue in the continuous spectrum of the Neumann problem for the Laplacian in a strip waveguide with an obstacle symmetric about the midline are studied. Such an eigenvalue is known to be unstable, and an arbitrarily small perturbation can cause it to leave the spectrum to become a complex resonance point. Conditions on the perturbation of the obstacle boundary are found under which the eigenvalue persists in the continuous spectrum. The result is obtained via the asymptotic analysis of an auxiliary object, namely, an augmented scattering matrix.     

Perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle

Doklady Mathematics -     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

Algebras of almost periodic functions with Bohr-Fourier spectrum in a semigroup: Hermite property and its applications

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener-Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.