Investigation of normal waves in a porous layer surrounded by elastic half-spaces

The wave field and dispersion equations are found for a porous layer surrounded by two elastic half-spaces. The porous layer is described by the effective model of a medium in which elastic and fluid layers alternate. To investigate the normal waves, all real roots of dispersion equations are determined and their movements as the wave number increases are investigated. As a result, the dispersion curves of all normal waves are constructed and the dependence of normal waves on the parameters of the porous layer and elastic half-spaces is analyzed. Bibliography: 6 titles.     

Investigation of low-frequency normal waves in a Biot layer surrounded by an elastic medium

The porous Biot layer surrounded by two elastic half-plates is considered. For this medium, two dispersion equations are established. These dispersion equations correspond to symmetric and antisymmetric parts of the medium. The investigation of these equations is conducted in the region of low frequencies. The imaginary roots of these equations in this region determine low-frequency normal waves. Bibliography: 6 titles.     

Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator H with a periodic potential perturbed by a function ?W that is periodic in two variables and exponentially decreasing in the third variable. Here, ? is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrödinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ?W. We obtain simple formulas for the transmission and reflection probabilities.     

Scattering of emerging normal elastic waves for a crystalline plate in a liquid medium

Frequency dependences are given for the imaginary components of the three lowest energy scattering branches of the dispersion spectra of symmetric and antisymmetric normal emerging waves propagating along the elastically equivalent directions in a plate of single-crystal Rochelle salt immersed in water. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 156–161, 1999.     

On the transition to the normal phase for superconductors surrounded by normal conductors

For a cylindrical superconductor surrounded by a normal material, we discuss transition to the normal phase of stable, locally stable and critical configurations. Associated with those phase transitions, we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide. In particular, when the conductivity ratio of the superconducting and the normal material is large, we show that the aforementioned critical magnetic fields coincide, thereby proving that the transition to the normal phase is sharp. One key-ingredient in the paper is the analysis of an elliptic boundary value problem involving ‘transmission’ boundary conditions. Another key-ingredient involves a monotonicity result (with respect to the magnetic field strength) of the first eigenvalue of a magnetic Schrödinger operator with discontinuous coefficients.     

Phase and group velocities of normal waves in an orthotropic layer

The article analyzes the distribution of the phase and group velocities of normal, transversely symmetric waves in an anisotropic layer of the orthrhombic class. The numerical investigation of a complex variance equation made it possible to establish qualitative peculiarities of the behavior of phase and group velocities for stresses different from elastoequivalent ones.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 101–105, 1987.     

Propagation of waves in a micropolar elastic layer with stretch immersed in an infinite liquid

The effect of liquid on the propagation of waves in a micropolar elastic layer with stretch has been investigated. The frequency and wave velocity equations for symmetric and antisymmetric vibrations are derived. Propagation of monochromatic waves in a micropolar elastic layer with stretch is discussed. Results of this analysis reduce to those without stretch.     

Diffraction of elastic waves by an inclined crack in a layer

For the problem of the diffraction of normal modes by an inclined crack in an elastic layer, an integral equation with explicit representation of the Fourier symbol kernel is derived in the form of the product of matrices. The algorithm for calculating the wave fields, based on the analytical representations obtained, enables a rapid parametric analysis to be carried out of the influence of the size and orientation of the crack on the transmission of travelling waves. The influence of the inclination of the crack on the effects of resonance trapping and localization of wave energy, which were established previously for the case of a horizontal crack, is analysed.     

Attenuation of normal waves in a cavity with a liquid in the presence of an elastic shell

The attenuation of normal waves is investigated in a cylindrical cavity of an unbounded viscoelastic medium, in which there is a thin elastic shell filled with a viscous compressible liquid. It is assumed that the absorption of waves in a liquid or solid is small. The effect of the shell thickness on the value of the absorption coefficient of the normal waves is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 49–52, 1987.     

Subsonic and supersonic disturbances in the generation of surface waves in a liquid layer adjacent to a high-speed gas-stream

In an effort to shed further light upon the nature of “supersonic” disturbances as distinct from that of ‘subsonic’ disturbances in parallel compressible flows, this paper makes an investigation of the stability characteristics of the surface waves generated in a liquid layer adjacent to a high-speed gas-stream. It turns out that the nature of the surface waves generated in the liquid layer depends markedly upon the type of disturbances present in the high-speed gas-stream. For the case of ‘subsonic’ disturbances it is shown that the energy transfer from the gas stream to the surface waves is contributed predominantly by the Fourier component of the normal gas-pressure force-field in phase with the slope of the wavy surface. For the case of ‘supersonic’ disturbances, this energy transfer is shown to be predominantly due to the component of the pressure-field in phase with the surface-wave displacement and is related to the presence of travelling periodic waves in the gas-stream—this energy transfer is shown to promote always the growth of the surface waves.     

Surface and internal waves in a stratified layer of liquid and an analysis of the impedance boundary conditions

The surface and internal waves in a multilayered ideal liquid for specified displacements of the bottom are considered. The upper surface of the liquid is either free or displacements are specified on it. In the long-wave approximation, asymptotically accurate models of the waves propagating along the surface layers of an incompressible liquid are constructed both for intense stratification (the ratio of the densities of neighbouring layers are associated with a small parameter) and for weak stratification. The important case of this problem of the dynamic contact of rigid bodies through a layer of incompressible or compressible liquid is investigated. High-order impedance boundary conditions are constructed and the results of testing them using the exact solutions are presented.     

Propagation of TM waves in a Kerr nonlinear layer

TM electromagnetic waves propagating through a nonlinear homogeneous isotropic unmagnetized dielectric layer located between two homogeneous isotropic half-spaces are studied. The nonlinearity in the layer obeys the Kerr law. The problem is reduced to a system of nonlinear ordinary differential equations. A dispersion relation for the propagation constants is derived. The results are compared with those in the case of a linear layer.     

Propagation of internal waves in a three-layer liquid

The propagation o f internal harmonic waves in a three-layer liquid is examined in a linear approximation for arbitrary wave numbers. A travelling wave sohaion is obtained and a dispersion equation is derived. The roots of the dispersion equation and the dependence of the solution on the ratio of the densities of the liquids are examined in detail. The phase and group velocities are plotted as functions of wavelength and graphs of the various wave modes are presented. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 170–174, 1999.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

Surface waves in a non-newtonian incompressible liquid film

The perturbation techniques of Friedrichs, Keller and Benney ar used to derive the governing equations of long wavelength surface waves in a liquid film of a fluid which obeys a generalized Norton type consititutive relationship. Motions are assumed to be small so that acceleration terms are negligible. It is shown that the form of the surface wave equation depends verystrongly on the stress-strain-rate relationship at very small strain rates. The result is that dispension is a secondary effect, if the material response is soft at low strain rates, but may be dominant, if it is stiff.

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Zusammenfassung Die Methode der singulären Strömungsrechnung von Friedrichs, Keller und Benney wird zur Herleitung der, lange Oberflächenwellen in Flüssigkeitsfilmen beschreibenden Differentialgleichung benutzt und zwar für eine, einem verallgemeinerten Nortonschen Fliessgesetz gehochende viskose Flüssigkeit. Die Bewegung se ein langsames kriechen, so dass Beschleunigungsterme vernachlässigbar sind. Es wird gezeigt, dass die Form der Oberflächenwellen-Gleichung stark von der Spannungs-Dehnungs-Beziehung abhängt, indem Dispersion einen kleinen Effekt darstellt, falls das Material sich für kleine Verzerrungsgeschwindigkeiten weich verhält, aber einen entscheidenden Einfluss ausübt, wenn es hart ist.

     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.