MATHEMATICAL THEORY OF NORMAL WAVES IN AN OPEN METAL-DIELECTRIC REGULAR WAVEGUIDE OF ARBITRARY CROSS SECTION

The problem of normal waves in an open metal-dielectric regular waveguide of arbitrary cross-section is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operatorfunction on the complex plane is found.

О неоднородных поляризованных состояниях вблизи точки фазового перехода в тонкой сегнетоэлектрической пленке

It is shown in terms of the phenomenological Landau theory of phase transitions that a phase transition to an inhomogeneous polar phase preceding in temperature a phase transition to a homogeneous polar state is possible. As a result of solving a boundary eigenvalue problem for the polarization equilibrium equation and electrostatics equations, wave vector k _⊥ characterizing the inhomogeneous phase has been determined and the temperature boundaries of its existence in the dependence on the film thickness and its surface properties have been found.

On the neutral stability curve for shallow conical shells subjected to lateral pressure

This work presents a detailed asymptotic description of the neutral stability envelope for the linear bifurcations of a shallow conical shell subjected to lateral pressure. The eighth-order boundary-eigenvalue problem investigated originates in the Donnell shallow-shell theory coupled with a linear membrane pre-bifurcation state, and leads to a neutral stability curve that exhibits two distinct growth rates. By using singular perturbation methods we propose accurate approximations for both regimes and explore a number of other novel features of this problem. Our theoretical results are compared with several direct numerical simulations that shed further light on the problem.

Dynamic Stability of Generalized Beck-Reut’s Beam

The classical non-conservative Beck’s beam, loaded by follower compressive force, is generalized by allowing an arbitrary angle of action of the follower force as well as allowing for excentric positioning of the applied force. For the corresponding boundary eigenvalue problem, the frequency equation is derived. Results of parametric studies are presented with an emphasis laid on the lowest eigenfrequencies. The characteristic shape of the computed curves indicates whether stability loss by divergence or by flutter occurs. A map of stability is presented in terms of parameters describing the excentricity and the angle under which the follower force acts on the beam.