Dispersion and attenuation of torsional wave in a viscoelastic layer bonded between a layer and a half-space of dry sandy media

Propagation of a torsional wave in a doubly-layered half-space structure of an initially stressed heterogeneous viscoelastic layer sandwiched between a layer and a half-space of heterogeneous dry sandy media is studied. A closed form complex expression for the velocity profile is obtained under effective boundary conditions. The real part of the complex expression provides a dispersion equation, and the imaginary part yields a damping equation. The derived dispersion and damped equations are in well agreement with the classical Love wave condition. In addition, to study the effect of the dissipation factor, the attenuation coefficient, the sandy parameters, the initial stress, the heterogeneity parameters, and the thickness ratio parameter, some noteworthy contemplations are made by numerical calculations and graphical visuals. The results of this paper may present a deeper insight into the behaviour of propagation phenomena in heterogeneous viscoelastic and heterogeneous dry sandy materials that can provide a theoretical guide for the design and optimization in the field of earthquake engineering. The study also reveals that the presence of a damping part due to viscoelasticity affects the torsional wave propagation significantly.     

Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half-space

The present paper is concerned with the study of propagation of torsional waves in an inhomogeneous isotropic layer whose material properties vary harmonically with a space variable, lying over a semi-infinite inhomogeneous isotropic half-space. The closed form solutions for the displacement in the layer and half-space are obtained separately. The dimensionless phase velocity has been plotted against dimensionless wave number and scaled wave number for different values of inhomogeneity parameters. The effects of inhomogeneity have been shown in the dispersion curves using 2D and 3D plot.     

Torsional wave frequency in heterogeneous earth crust lying over dry sandy semi-infinite substratum

The present study is carried out to investigate the transference of torsional surface waves in a heterogeneous anisotropic crust lying over a dry sandy half-space. The rigidities and densities as well as the initial stress are assumed varying as a function of depth in both the media. These variations are the product of the polynomial function of depth in degree n(n ∈ R) and the exponential function of depth. Following the theory of elastic waves, the mathematical model is established. Separation of variables is used to obtain the displacement in the layer and the half-space. Intrinsic boundary conditions are imposed to derive the dispersion equation. The inhomogeneity parameters associated with the rigidity, the density, and the initial stress of the medium are found to have substantial influence on the phase velocity of the torsional surface wave. The graphical presentations are drawn to exhibit the findings. The results thus obtained are significant for the investigation and characterization of torsional surface wave in the heterogeneous anisotropic layer.     

Propagation of SH waves in an irregular monoclinic crustal layer

The present paper discusses the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity. In the absence of the irregularity, the dispersion equation reduces to standard dispersion equation for SH waves in a monoclinic layer over an isotropic semi-infinite medium. The dispersion curves for different size of the irregularity are computed and compared for the half-space without any irregularity. It can be seen that the phase velocity is strongly influenced by the wave number and the depth of the irregularity.     

Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space

The paper studies the propagation of Love waves in a non-homogeneous substratum over an initially stressed heterogeneous half-space. The dispersion equation of phase velocity is derived. The velocities of Love waves are calculated numerically as a function of kH and presented in a number of graphs, where k is the wave number, and H is the thickness of the layer. The case of Gibson’s half-space is also considered. It is observed that the speed of Love waves is finite in the vicinity of the surface of the half-space and vanishes as the depth increases for a particular wave number. It is also observed that an increase in compressive initial stresses causes decreases of Love waves velocity for the same frequency, and the tensile initial stress of small magnitude in the half-space causes increase of the velocity.     

Diffraction of torsional waves by a penny-shaped crack in a non-homogeneous thick fibre bonded to a non-homogeneous matrix

In this paper the equation of motion is solved when the shear modulus and density are functions of r and z and the latter part of this paper contains an analysis of the interaction of torsional waves normally with penny-shaped crack located in a thick infinite elastic fibre. The infinite elastic fibre is bonded to an infinite elastic matrix. The matrix and the thick elastic fibre are non-homogeneous and are of dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack. Finally the results of the stress intensity factors are displayed graphically.     

Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress

The present work deals with the possibility of propagation of torsional surface wave in fluid saturated poroelastic layer lying over nonhomogeneous elastic half space. Both the media are assumed to be under compressive initial stress. The half space has two types of inhomogeneity, viz; hyperbolic and quadratic. The dispersion equation for torsional wave in porous layer has been derived and observed that the presence of fluid in pores increases the velocity of the torsional surface wave but the phase velocity diminishes due to the presence of compressive initial stress in the porous layer. It is also observed that the velocity of the torsional surface wave increases due to the increase of initial stress in inhomogeneous half space. The inhomogeneity factor due to quadratic and hyperbolic variations in rigidity, density and initial stress of the medium decreases the phase velocity as it increases.     

Influence of linearly varying density and rigidity on torsional surface waves in inhomogeneous crustal layer

The present work deals with the possibility of propagation of torsional surface wave in an inhomogeneous crustal layer over an inhomogeneous half space. The layer has inhomogeneity which varies linearly with depth whereas the inhomogeneous half space exhibits inhomogeneity of three types, namely, exponential, quadratic, and hyperbolic discussed separately. The dispersion equation is deduced for each case in a closed form. For a layer over a homogeneous half space, the dispersion equation agrees with the equation of the classical case. It is observed that the inhomogeneity factor due to linear variation in density in the inhomogeneous crustal layer decreases as the phase velocity increases, while the inhomogeneity factor in rigidity has the reverse effect on the phase velocity.     

Propagation of wave packets in the boundary layer on a curved surface

The development of wave packets excited in a boundary layer by means of a local deformation of the surface in the longitudinal-transverse interaction regime is considered. A solution of the linearized system of equations of interaction theory is constructed using a Laplace transformation with respect to time and a Fourier transformation with respect to the space variables. Two problems are separately examined. In the first, the disturbances are induced by a surface deformation sinusoidal in the transverse direction. It is shown that the center of the wave packet with the greatest oscillation amplitude moves in a direction opposite to that of the flow in the boundary layer. At the same time the wave packet expands, so that in the course of time any fixed point will enter the region of growing oscillations. In the second problem the source of the disturbances is isolated. In this case the wave packet acquires a horseshoe shape. Expanding, it carries the disturbances away from the source in all directions, including upstream relative to the flow in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 59–68, March–April, 1990.     

Propagation of an electromagnetic wave across a magnetic field in a parabolic plasma layer

The reflection and transmission coefficients are obtained, as well as the coefficient of transformation of an electromagnetic wave into a plasma wave. The problem of choosing the physical path of analytic continuation of the solutions is considered in the case of a wave equation with two poles.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 28–34, May–June, 1971.     

Flow of a thin layer of a viscous liquid over a dry surface

The problem of flow of a thin layer of a viscous liquid over a dry surface is formulated (ignoring surface tension). Examples of calculations are given. The results are compared with a solution using an appropriate one-dimensional model that admits an exact solution. Singularities of the solution are analyzed. Deceased. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 47–51, March–April, 1998.     

Effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a heterogeneous half-space

In the present paper we study the effect of rigid boundary on the propagation of torsional waves in a homogeneous layer over a semi-infinite heterogeneous half-space, where the heterogeneity is both in rigidity and density. The present study demonstrates that torsional waves can propagate in the layer. The velocities of torsional waves have been calculated numerically as a functions of KH, (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs. It is also observed that, for a layer over a homogeneous half-space, the velocity of torsional waves does not coincide with that of Love waves in the presence of the rigid boundary whereas it does at the free boundary.     

Propagation of torsional waves in a radially layered cylindrical waveguide

It was shown in [1–3] that the spectrum of homogeneous solutions for layered bodies with alternating rigid and soft layers splits into the “lower” and “higher” parts. Moreover, the “lower” part is always associated with some applied theory. In [4], the method developed in [1–3] is generalized to problems of steady torsional vibrations of a radially inhomogeneous cylinder and to the dynamic case of the applied theory constructed in [2]. In the present paper, we use analytic and numerical methods to study the propagation of torsional waves in a radially layered cylindrical waveguide.     

Propagation of a Tollmien-Schlichting wave over the junction between rigid and compliant surfaces

The propagation of an instability wave over the junction region between rigid and compliant panels is studied theoretically. The problem is investigated using three different methods with reference to flow in a plane channel containing sections with elastic walls. Within the framework of the first approach, using the solution of the problem of flow receptivity to local wall vibration, the problem considered is reduced to the solution of an integro-differential equation for the complex wall oscillation amplitude. It is shown that at the junction of rigid and elastic channel walls the instability-wave amplitude changes stepwise. For calculating the step value, another, analytical, method of investigating the perturbation propagation process, based on representing the solution as a superposition of modes of the locally homogeneous problem, is proposed. This method is also applied to calculating the flow stability characteristics in channels containing one or more elastic sections or consisting of periodically alternating rigid and compliant sections. The third method represents the unknown solution as the sum of a local forced solution and a superposition of orthogonal modes of flow in a channel with rigid walls. The latter method can be used for calculating the eigenvalues and eigenfunctions of the stability problem for flow in a channel with uniformly compliant walls.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 31–48. Original Russian Text Copyright © 2004 by Manuilovich.     

Transient torsional wave propagation in a transversely isotropic tube

Summary By use of the separation of variables method and the Laplace transformation, the two-dimensional transient torsional wave propagation problem in a transversely isotropic tube is studied when a torque is suddenly applied to its end surface. The results show that, for the discontinuous distribution of the impact shear stress, the region of the 2D stress distribution in a transversely isotropic tube becomes large with the increase of the anisotropy of the material. Received 13 June 1997; accepted for publication 17 June 1998