Free transverse vibrations of uniform circular plates and membranes with eccentric holes

An analytical method, based on the transformation of cylindrical wave functions, is presented for calculating the free transverse vibrations of uniform circular plates and membranes with eccentric holes. The wave equations governing the small transverse motion of the plates and membranes are solved exactly to satisfy the boundary conditions at both inner and outer edges, and the associated frequency equations for the plates and membranes are derived. The analysis also includes the vibrations of uniform circular plates and membranes with concentric holes as special cases. Numerical examples are presented to show the dependence of the eigenfrequency on the eccentricity and on the ratio of the inner radius to outer radius. The main purpose of the paper is to illustrate that an exact, analytical solution of the vibrations of eccentric annular plates and membranes can be obtained.     

Permittivity of a randomly inhomogeneous medium

A theory of permittivity of suspension-type systems is developed that allows one to calculate such optical parameters of inhomogeneous systems as the length of scattering and the transport length. It is shown that, in the Born approximation, which takes into account two-particle correlations in the arrangement of scattering particles, the theoretical and experimental data are in agreement only to within tens of percent. The contribution of three-particle correlations to the permittivity of a system of solid spheres is determined. It is shown that, in describing the optical properties of suspensions with a large difference between the refractive indices of the medium and the particles, it does not suffice to replace the Rayleigh-Gans form factor by the Mie form factor, even under a restriction to two-particle correlations.     

Beam propagation in a randomly inhomogeneous medium

An integrodifferential equation describing the angular distribution of beams is analyzed for a medium with random inhomogeneities. Beams are trapped because inhomogeneities give rise to wave localization at random locations and random times. The expressions obtained for the mean square deviation from the initial direction of beam propagation generalize the “3/2 law.”     

Average pedersen-ray intensity in randomly inhomogeneous ionosphere

A formula is derived for the average field intensity of high-angle rays in a randomly inhomogeneous ionosphere. The Huygens-Kirchhoff method is employed. Numerical studies show that the average intensity in a randomly inhomogeneous ionosphere is greater than the high-angle-ray intensity in an unperturbed medium. The dependence of this effect on propagation distance is shown.Institute of Applied Physics, Irkutsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 5, pp. 398–403, May, 1993.     

Theory of transition radiation in randomly inhomogeneous medium

A theory of the radiation emitted by a charged particle moving in a randomly inhomogeneous correlated medium is constructed. It is shown that the radiation is isotropic and unpolarized. The frequency dependence of the radiation intensity is investigated and a peak is predicted in the wavelength region of e₂a (where e₂ is the dielectricconstant fluctuation and a is the correlation radius of the random field).Institute of Radio Physics and Electronics, Academy of Sciences of Armenia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 1, pp. 51–64, January, 1993.     

Electromagnetic waves in a randomly inhomogeneous Josephson junction

Spectrum modification and damping of Josephson plasma waves induced by random inhomogeneities of the critical current through the superconductor contact and the averaged Green function of such excitations are analyzed. In the self-consistent approximation that makes it possible to take into account multiple wave scattering on the inhomogeneities, the frequency and damping of averaged waves, as well as position ν _m and peak width Δν of the Fourier transform imaginary part of the averaged Green function, are determined as functions of wavevector k. The evolution of such functions with the variation of the correlation radius and the relative r.m.s. fluctuations of inhomogeneities is studied. The inhomogeneity-induced wave frequency decrease observed in the long wavelength spectral region qualitatively agrees with the ν _m behavior. It is established that in the case of “long-range” inhomogeneities, the linear dependence of damping on k changes to the inversely proportional one, and damping tends to zero as k → 0, while Δν at small k attains its maximal values due to nonuniform broadening. In the presence of “short-range” inhomogeneities, the wave damping and Δν are found to be similar functions of k. The results are compared to the numerical calculation data.     

Bulk plasma waves in a randomly inhomogeneous conductor

The modification of the spectrum and damping of bulk plasma waves due to three-dimensional random inhomogeneities of the density of a degenerate electron gas in a conductor have been investigated using the averaged Green??s function method. The dependences of the frequency and damping of the averaged plasma waves, as well as the position ?? _m and width ???? of the peak of the imaginary part of the Fourier trans-form of the averaged Green??s function, on the wave vector k have been determined in the self-consistent approximation, which makes it possible to take into account multiple scattering of plasma waves by inhomogeneities. It has been found that, in the long-wavelength region of the spectrum, the decrease revealed in the frequency of the plasma waves is caused by the inhomogeneities, which agrees qualitatively with the behavior of the position of the peak ?? _m . In the range of large values of the correlation length of inhomogeneities and small values of k, the damping of the plasma waves tends to zero, whereas the width of the peak ???? remains finite, which is due to the nonuniform broadening. A comparison with the data of numerical calculations has been performed.     

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.     

Piezoelastic vibrations of composite Reissner-Mindlin-type plates

Linear vibrations of Reissner-Mindlin-type composite plates in the presence of piezoelectric eigenstrains are studied. Piezoelectric eigenstrains are produced by applying electrical loads to piezoelectric layers embedded in or attached to substrate layers. The influence of the mechanical field upon the electric field is taken into account in the modelling, ending up with electro-mechanically coupled field equations and boundary conditions, which describe the mechanical and the electrical dynamic response of the plate.The mechanical displacements are approximated by means of the kinematic hypothesis of Hencky. The electric potential distribution is assumed to be composed of a superposition of a linear and a parabolic distribution in the thickness direction. The linear part accounts for the electric potential difference between the electrodes of the totally electroded piezoelectric layers. The parabolic part is considered in order to take into account the influence of the mechanical field upon the electric potential by means of the direct piezoelectric effect. A weak two-dimensional formulation of the three-dimensional field equations is obtained by utilizing mechanical and electrical variational principles. This formulation is characterized by resultants of stress and electric displacement. The electro-mechanically coupled behaviour comes into play by means of the constitutive relations. In case the electric potential difference is not prescribed, it can be calculated from a relation, which connects the total electric charge and the electric potential difference to each other. Because this relation is obtained from the Gauss law of electrostatics, requiring integration with respect to the area of the electrode, non-local constitutive relations for the plate are found. The non-local constitutive relations bring a new aspect into the theory of plates. An analysis for the practically interesting one-dimensional case of composite, piezoelectric plates in cylindrical motion completes the paper.     

Transverse vibrations of hybrid laminated plates

In this paper, an attempt is made to obtain the free vibration response of hybrid, laminated rectangular and skew plates. The Galerkin technique is employed to obtain an approximate solution of the governing differential equations. It is found that this technique is well suited for the study of such problems. Results are presented in a graphical form for plates with one pair of opposite edges simply supported and the other two edges clamped. The method is quite general and can be applied to any other boundary conditions.