Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer

This paper is concerned with the Rayleigh wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer under initial stress. Both the layer and half-space have subjected to the incompressible in nature. The particle motion of the Rayleigh type wave is elliptically polarized in the plane, which described by the normal to the surface and the focal point along with wave generation. The dispersion of waves refers typically to frequency dispersion, which means different wavelengths travel at a different velocity of phase. To deal with the analytical solution of displacement components of Rayleigh type waves in a layer over a half-space, we have taken the assistance of different methods like exponential, characteristic polynomial and undetermined coefficients. The dispersion relation has been derived based upon suitable boundary conditions. The finite difference scheme has been introduced to calculate the phase velocity and group velocity of the Rayleigh type waves. We also have derived the stability condition of the finite difference scheme (FDS) for the phase and group velocities. If a wave equation has to travel in the time domain, it is necessary to achieve both accuracy and stability requirements. In such cases, FDS is preferred because of its power, accuracy, reliability, rapidity, and flexibility. The effect of various parameters involved in the model like non-homogeneity, porosity, and internal pre-stress on the propagation of Rayleigh type waves have been studied in detail. Graphical representations for the effects of various parameters on the dispersion equation have been represented. Numerical results demonstrated the accuracy and versatility of the group and phase velocity depending on the stability ratio of the FDS.     

Propagation of Normal Waves in an Isolated Porous Fluid-Saturated Biot Layer

For a porous fluid-saturated Biot layer with boundaries free from stresses and pressure, the wave field is found and dispersion equations are derived. The roots of the dispersion equations and the dependence of the phase velocities of the normal waves on the wave number are investigated by analytic methods. It is shown that the phase velocities of most of the normal waves decrease with increasing wave number. Special investigations are conducted in the case of bend and plate waves and their phase velocities for high and low frequencies. It is also shown that on the boundary of a porous Biot half-space, the Rayleigh wave does not always originate, and conditions for the existence of such a wave are established. Bibliography: 7 titles.     

An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

Estudio de la estabilidad y dispersión del problema de propagación de ondas sísmicas en 2-D utilizando el método de diferencias finitas generalizadas

This paper shows the solution to the problem of seismic wave propagation in 2-D using generalized finite difference (GFD) explicit schemes. Regular and irregular meshes can be used with this method.As we are using an explicit method, it is necessary to obtain the stability condition by using the von Neumann analysis. We also obtained the star dispersion formulas for the phase velocities for the P and S waves, as well as the ones for the group velocities.As the control over the irregularity in the mesh is very important in the application of this method, we have defined an index of irregularity for the star (IIS) and another for the cloud (IIC), analyzing its relationship with the dispersion and time step used in the calculations.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson     

Time-dependent waves of rayleigh type near the surface of an inhomogeneous elastic body

The well studied, high-frequency Rayleigh waves polarized in a plane normal to a cross section of the surface of an inhomogeneous elastic body with phase speed close to the speed of transverse waves are generalized to the case of the time-dependent equations of elasticity. For the wave field uniform asymptotics are obtained in the form of space-time ray expansions of two types: with a real eikonal (for transverse waves diffracted at the surface) and with a complex eikonal (for a longitudinal wave damped away from the surface).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 168–183, 1986.     

表面效应对偏场下介电高弹体表面波传播的影响

采用表面薄层模型考察偏场下介电高弹体的表面效应,针对不同边界情形,建立一阶等效边界条件.基于有限变形电弹性体的线性增量理论,利用Stroh公式和Ting方法,给出等效边界条件的严格推导过程.进一步利用Stroh公式,获得了偏场下具有表面效应的介电高弹体中表面波的频散方程.以可压缩Neo-Hookean介电高弹体为例,分析了表面效应对预变形和电学偏场作用下的介电高弹体表面波传播特性的影响.结果表明,通过施加适当的偏场,可以调控和优化纳米声表器件的性能.     

Waves from a Deep Source of Longitudinal Waves in an Elastic Half Space with a Free Boundary

The nonstationary propagation of waves on the surface of an elastic half space from a deep expansion source (model of an explosion in a half space) is examined. Exact solutions are obtained in the form of integrals with finite limits and the general solution is calculated. Algebraic expressions are obtained for the Rayleigh wave. The transition of Rayleigh waves at the surface of the half space is studied. Calculations of Rayleigh waves from discontinuous pulsed sources are presented.     

Rayleigh wave scattering by a plane strip in a deep ocean

A theoretical formulation to study the problem of scattering of Rayleigh waves due to the presence of a rigid plane strip in a deep ocean is presented. A rigid plane strip (0 ≤z ≤ H, 0 ≤x ≤ l) is fixed in the surface of the ocean occupyingz ≥ 0. Fourier transformation and Wiener-Hopf technique are used to arrive at the solution. The scattered Rayleigh waves behave as cylindrical waves emerging out of the corner of the strip and its image in the free surface of the ocean. The scattered waves are obtained in terms of Bessel functions whose behaviour near and far from the strip is well-known. The numerical calculations for the scattered waves show that their amplitude increases rapidly for a small increase in the value of the wave number. Scattering of Rayleigh waves due to a thin plane vertical barrier and a thin barrier in the free surface of the ocean has been considered as the special cases.     

Nonlinear behavior of electromagnetic waves on surface of antiferromagnet

The properties of the nonlinear TM waves on the interface between a dielectric and an antiferromagnet are studied. The relationship between the field components of TM wave is discussed in detail, and the dispersion characteristics as well as the position of the peak field are exposed. The theoretical analysis shows that for the nonlinear TM waves there exist passband(s) and stopband(s) which can be switched into each other by varying the power. It is revealed that, in the case of , the nonlinear TM waves on the interface are backward surface waves with the group and phase velocities opposite. Project supported by the National Natural Science Foundation of China (Grant No. 69477020).     

Propagation of surface electromagnetic waves similar to Rayleigh waves in the case of Leontovich boundary conditions

If on a surface Σ bounding an electromagnetic field, the Leontovich boundary conditions with a pure imaginary exponent are fulfilled, then surface electromagnetic waves propagating along Σ may exist. These waves have much in common with the Rayleigh waves in elasticity theory. In the paper, the ray theory of this electromagnetic analog of the Rayleigh waves is constructed. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 5–19.     

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.     

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.     

Love waves with a transverse structure

For an arbitrary layered isotropic structure, new exact solutions of the elastodynamic problem for the propagation of surface waves are presented. These solutions describe waves with rectilinear wave fronts propagating at the phase velocities of common SH-polarized Love waves. They linearly depend on a lateral transverse variable and, in addition to being standardly SH-polarized, have a longitudinally polarized anomalous component. The construction uses the assumption of the existence of standard Love waves. It is based on a potential representation of the wavefield and is quite elementary.     

Energy Spectrum Width and Nonequilibrium Energy of Rayleigh Waves Scattered on a Two-Dimensional Near-Surface Electron Layer

We obtain expressions for the energy spectrum widths of Rayleigh waves corresponding to their deformational coupling to Fermi and Boltzmann electrons in a two-dimensional layer near the surface of a semibounded solid. We evaluate the nonequilibrium energy of Rayleigh waves that depends on these widths and is caused by the same coupling to the corresponding hot electrons. We show that this energy is independent of the degeneracy degree of the electrons and is given by the mean energy of free Rayleigh waves heated up to temperature of the electrons. We find conditions under which the thermodynamics is determined by this nonequilibrium energy of Rayleigh waves in films of a certain thickness with Fermi electrons near the surface and by the equilibrium energy of bulk phonons in thicker samples. All the results are obtained using the Keldysh diagram technique applied to the case of semibounded media.     

Kinematic approach to the group velocity of high-frequency space-time Love (sh) and Rayleigh (sv) waves

The kinematic approach to the interpretation of the group velocity of a modulated slowly changing (as a function of the properties of the medium) wave is applied under the assumption that the dispersion equation of the problem is given. Namely, the space-time eikonal equation is such an equation for the Love and Rayleigh space-time waves. The result obtained is consistent with the known formulas for the stationary case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 182–186, 1989.     

Propagation of shear waves in a poroelastic layer constrained between two elastic layers

In this paper, we investigated the propagation of shear waves in a transversely isotropic poroelastic layer constrained between two elastic layers. Following Biot’s theory, the dispersion equation for shear waves in this structure was derived. The numerical values on the dimensionless phase velocities are calculated and presented graphically to illustrate the dependences upon geometry, anisotropy and porosity comparatively. It is observed that the phase velocities increase with the increase of the porosity and the decrease of the anisotropy. In addition, the geometry in this structure has a significant effect on the phase velocity of the shear waves.     

Wave Propagation in a Layer of Magnetic Liquid

A linearized equation for the propagation of surface gravitational waves in a layer of magnetized liquid of finite depth is examined. The liquid is assumed to be inviscid, incompressible, and to possess magnetization properties in the absence of electrical conductivity, while the motion is assumed to be irrotational. Travelling wave solutions are obtained. The dependences of the phase and group velocities of the magnetic liquid on the magnetic parameters are studied. It is shown that for some values of the magnetic parameters there is an interval of short wavelengths for which the group velocity is negative, which indicates that the wave energy propagates in the negative direction.     

Ausbreitungsgeschwindigkeiten magnetoakustischer Wellen in dissipationsfreien Plasmen

Scaled polar plots of the phase velocities and group velocities of plane magnetoacoustic waves are given in the form suggested by K. O. Friedrichs [1]. The first kind of Friedrichs diagram shows the dependence of the propagation speeds of fast and slow waves on the angle between the wave normal and the magnetic field, for several values of the ratio of Alfvén speedb to sound speeda. The second kind represents the disturbance pattern created by a magnetoacoustic pulse at the origin, for several values of the ratiob/a.