Surface and normal electroelastic shear waves in even-layered structures

We construct the dispersion equations for surface and normal shear waves propagating in layered periodic structures consisting of alternating layers of piezoelectric and metal. We carry out a numerical analysis of the equations obtained in a wide range of variation of frequency. We describe the distinctive characteristics of dispersion spectra of surface and normal waves and their interrelation. We give the characteristic distributions of the amplitude walues of the mechanical displacements and stresses and the electric potential. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 50–58, 1991.     

Dispersion of normal waves in a linearly orthotropic circular cylinder with a clamped lateral surface

We propose a method of obtaining the dispersion equation for normal waves in an orthotropic cylinder from the boundary conditions on the rigidly clamped boundary using a system of exponential particular solutions of the three-dimensional equations of its stationary wave motions. We compute the real and imaginary branches of the dispersion spectrum for a waveguide made of monocrystalline strontium sulfate. On figure. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 96–99.     

Bending normal waves in a crystal layer surrounded by a liquid

We obtain the dispersion relations that describe the spectrum of the “effluent” harmonic antisymmetric normal waves for an arbitrary direction in the plane of an orthotropic layer surrounded by a viscous or ideally compressible fluid. We present the results of computation of the lower branches of the dispersion spectrum for elastically equivalent directions of propagation in a layer of monocrystal Seignette salt in water. Four figures. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 123–131.     

Unique continuation of some dispersive waves

We demonstrate that a complex-valued wave on space-time Rⁿ⁺¹ , obtained from square-integrable but not necessarily smooth nonzero initial data and having a suitable Hamiltonian generator or dispersion relation, cannot vanish on a measurable rectangle in Rⁿ⁺¹ . Nor can the product of two such waves vanish in such a rectangle - even waves arising from distinct Hamiltonians or dispersion relations. Examples include the solutions of the free particle Schrödinger equation, the positive energy solutions of the free particle Klein-Gordon and Dirac equations, and the positive frequency solutions of the wave equation. Nonlocalization results of this type were obtained by the physicist G. C. Hegerfeldt.     

Wave propagation in an elastic layer with voids located in a liquid

We obtain the dispersion equations that describe the propagation of waves in an elastic layer with voids locted between two liquid half-spaces. We study certain limiting cases corresponding to the absence of voids or liquid. We obtain the roots of the dispersion equations for both dissipative and nondissipative systems. It is shown that the relation of the real part of the phase velocity to the wave number in a dissipative system is qualitatively similar to the corresponding relation for the real value of the phase velocity in the case when dissipation is absent. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 90–96.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder whose physical properties have a longitudinal axis of symmetry

We develop a numerical method of analyzing the dispersion of electroelastic waves in a hollow circular cylinder. The method is based on the reduction of the wave problem to a spectral problem for a system of ordinary differential equations. This system is solved numerically. The roots of the dispersion equation are determined by the method of bisection. The practical convergence of the algorithm is shown using a specific example. One figure. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 151–156.     

Attenuating Resonance Modes in a Cylindrical Waveguide Placed in an Elastic Medium

Interference attenuating waves traveling in a cylindrical elastic waveguide, placed in an elastic medium, are considered. The group velocity of these waves is intermediate between that of the P wave and that of the S wave; the phase velocity equals that of the P wave. The frequency of the waves is almost constant and is determined by the requirement of constructive interference. The dispersion and attenuation of these waves are described. Bibliography: 3 titles.     

Some new solutions of the problems of wave propagation in a two-layer fluid

We present new analytic solutions of the problem of wave propagation in a continuously stratified fluid in the Boussinesq approximation. We study the propagation of internal waves in an ideal fluid in systems of homogeneous-layer/continuously stratified layer and homogeneous-layer/continuously stratified half-space type. We obtain the dispersion equations and study several limiting cases. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pp. 132–137.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

Wave propagation in an isolated porous Biot layer with closed pores on the boundaries

On the boundaries of such an isolated porous Biot layer, the total stresses and normal relative displacement are equal to zero. For this layer, the symmetric and antisymmetric dispersion equations are established and investigated. The wave field consists of normal waves. In this layer, one bending wave, two plate waves, and infinitely many normal waves propagate. For all these waves, we determine dispersion curves by analytical methods. The velocities of the bending wave and the second plate wave for the infinite frequency are equal to the Rayleigh velocity. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 173–189.     

Global Dispersion Relation for Density Waves In a Certain Simplified Model of a Disk Shaped Galaxy

For density waves in a certain simplified model of a disk shaped galaxy, the dominant term of the basic equation (governing density waves) may be represented by a cubic polynomial, in which the stability parameter Q is allowed to be somewhat less than unity near corotation. For such a differential equation, an asymptotic form of the global dispersion relation is presented. It is shown that there exist discrete complex roots of the dispersion relation with small negative imaginary parts. The real parts and the imaginary parts of these roots represent approximately the angular speeds and the growth rate of the amplitudes of the density waves, respectively. It is proved that there exist only a finite number of unstable normal modes of density waves.     

Tube Waves in a Well Filled with a Solid-Fluid Mixture

Kinematic and dynamic parameters of low-frequency waves propagating in a two-phase mixture of solid and fluid particles filling a well are investigated. It is shown that in this model, two tube waves propagate, whereas only one tube wave propagates in a fluidfilled well. The dispersion and attenuation of tube waves are studied. Bibliography: 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 154–161.     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.     

Dispersion relations for normal waves in a cylinder made of orthorhombic monocrystal with pyro- and piezoelectric properties

By integrating the system of differential equations of coupled thermoelectroelastic vibrations of a pyroactive crystal medium of the orthorbombic system in special nonclassical vector-valued functions of ordinary and generalized complex variables we construct the dispersion equation for normal waves in a cylindrical waveguide of circular cross section. The dispersion function is obtained in the form of a determinant of infinite order subject to reduction in numerical investigations. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 118–123.     

Existence and modulation of traveling waves in particles chains

We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where q_n(t) is the displacement of the n^th particle at time t along the chain axis and denotes differentiation with respect to time. We assume that all particles have unit mass and that the interaction potential V between adjacent particles is a convex C∞ function. For this system, we prove the existence of C∞, time‐periodic, traveling‐wave solutions of the form q_n(t) = q(wt kn + where q is a periodic function q(z) = q(z+1) (the period is normalized to equal 1), ω and k are, respectively, the frequency and the wave number, is the mean particle spacing, and can be chosen to be an arbitrary parameter. We present two proofs, one based on a variational principle and the other on topological methods, in particular degree theory. For small‐amplitude waves, based on perturbation techniques, we describe the form of the traveling waves, and we derive the weakly nonlinear dispersion relation. For the fully nonlinear case, when the amplitude of the waves is high, we use numerical methods to compute the traveling‐wave solution and the non‐linear dispersion relation. We finally apply Whitham's method of averaged Lagrangian to derive the modulation equations for the wave parameters α, β, k, and ω. © 1999 John Wiley & Sons, Inc.     

Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations

Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations (NLS) are studied. We start with spatially uniform and temporally periodic solutions (the so-called Stokes waves). We find that the spectra of the linear NLS at the Stokes waves often have surprising limits as dispersion or viscosity tends to zero. When dispersion (or viscosity) is set to zero, the size of invariant manifolds and/or Fenichel fibers approaches zero as viscosity (or dispersion) tends to zero. When dispersion (or viscosity) is nonzero, the size of invariant manifolds and/or Fenichel fibers approaches a nonzero limit as viscosity (or dispersion) tends to zero. When dispersion is nonzero, the center-stable manifold, as a function of viscosity, is not smooth at zero viscosity. A subset of the center-stable manifold is smooth at zero viscosity. The unstable Fenichel fiber is smooth at zero viscosity. When viscosity is nonzero, the stable Fenichel fiber is smooth at zero dispersion.     

Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 217–232.     

Acoustics of gas-saturated foams: Modelling comparisons

We show results obtained by Biot's equations and a simple Theory of Porous Media-based hybrid model describing acoustical waves in a gas-saturated high-porous foam. The resulting dispersion relations for the longitudinal mode and phase velocities vs. porosity show the extraordinary effect of waves in gas-saturated porous media. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Propagation of three-dimensional electroelastic shear waves in a regularly layered medium of metal-piezoelectric type

We construct and study the dispersion equations for electroelastic three-dimensional shear waves propagating in a periodic medium formed by alternating layers of conducting and piezoelectric materials. We determine the structure of the transmission and blocking zones (the spectral regions of stability and instability) for various relations among the parameters of the layers with waves propagating along the interfaces between the layers. Two figures. Bibliography: 8 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 4–8.     

Creeping waves on a highly elongated body of revolution

Creeping waves play an important role in diffraction by a smooth convex body and give an asymptotic of the diffracted field in the shadow. Known results obtained by the boundary-layer method do not allow us to explain some of the properties of creeping waves on highly elongated bodies. In this paper, creeping waves on highly elongated bodies are studied in the case where the binormal curvature of the surface is asymptotically large. The asymptotics derived contains solutions of a differential equation of the Heun type. The analysis of the dispersion equation for the surface waves is carried out numerically. It is discovered that the magnetic creeping wave travels along the surface of a highly elongated body with much less attenuation than predicated by the usual theory. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 22–34. Translated by I. V. Andronov.