The combined effects of transverse isotropy and inhomogeneity on Love waves

1973 ◽  
Vol 63 (1) ◽  
pp. 49-57
Author(s):  
V. Thapliyal
Keyword(s):  
Half Space ◽  
Transverse Isotropy ◽  
Frequency Equation ◽  
Love Waves ◽  
Anisotropy Factor ◽  
Wave Number ◽  
Elastic Parameters ◽  
Short Period ◽  
Phase And Group Velocities ◽  
Group Velocities

abstract The characteristic frequency equation for Love waves propagating in a finite layer overlying an anisotropic and inhomogeneous half-space is derived. This frequency equation takes into account the arbitrary variation of density, elastic parameters, and degree of anisotropy factor in the half-space. In fact, the problem of deriving the frequency equation has been reduced to finding the solution of the equation of motion subject to the appropriate boundary conditions. To illustrate the method, the author has derived the frequency equation for a generalized power law variation of density and elastic parameters with the depth, in the halfspace. As a step toward the systematic investigation of the effects of anisotropy and inhomogeneity, the relationship between the wave number and phase and group velocities has been worked out for increasing, uniform and decreasing anisotropy factor. The pronounced effects of anisotropy have been noticed in the long-period range compared to the short-period one. The numerical analysis shows that for a given phase velocity (or group velocity), the period of propagation depends on the sign and magnitude of power of variation of the density and anisotropy factor in the half-space. For the increased positive rate of variation of the anisotropy factor, the values of phase and group velocities have been found higher whereas the reverse is found true for an increasing negative rate of variation of the anisotropy factor.

Body waves as normal and leaking modes—leaking modes of Love waves

1974 ◽  
Vol 64 (2) ◽  
pp. 301-306
Author(s):  
B. J. Radovich ◽  
J. Cl. De Bremaecker
Keyword(s):  
Group Velocity ◽  
Half Space ◽  
Normal Modes ◽  
Brewster Angle ◽  
Body Waves ◽  
Love Waves ◽  
Wave Number ◽  
Phase Velocities ◽  
Real Group ◽  
Group Velocities

abstract Dispersion curves of leaking modes of Love waves are computed for a one-layer crust-mantle model. The frequency and wave number are complex, but the group velocity is purely real. Group velocities well below the Airy phase of normal modes are found; waves with group velocities between 1.0 and 3.8 km/sec attenuate the least, but even these decay too much ever to be detected. There are no phase velocities between the velocity of the half-space and that corresponding to the Brewster angle.

Period equation for Rayleigh waves in a layer overlying a half space with a sinusoidal interface

1962 ◽  
Vol 52 (4) ◽  
pp. 807-822 ◽  
Author(s):  
John T. Kuo ◽  
John E. Nafe
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Half Space ◽  
Rayleigh Waves ◽  
Wave Length ◽  
Linear Equations ◽  
Wave Number ◽  
Interface Plane ◽  
Period Equation ◽  
Phase And Group Velocities

abstract The problem of the Rayleigh wave propagation in a solid layer overlying a solid half space separated by a sinusoidal interface is investigated. The amplitude of the interface is assumed to be small in comparison to the average thickness of the layer or the wave length of the interface. Either by applying Rayleigh's approximate method or by perturbating the boundary conditions at the sinusoidal interface, plane wave solutions for the equations which satisfy the given boundary conditions are found to form a system of linear equations. These equations may be expressed in a determinant form. The period (or characteristic) equations for the first and second approximation of the wave number k are obtained. The phase and group velocities of Rayleigh waves in the present case depend upon both frequency and distance. At a given point on the surface, there is a local phase and local group velocity of Rayleigh waves that is independent of the direction of wave propagation.

The Frequency Equations for Shear Horizontal Waves in Semiconductor/Piezoelectric Structures Under the Influence of Initial Stress

2016 ◽  
Vol 13 (10) ◽  
pp. 6475-6481 ◽  
Author(s):  
Abo-el-nour N Abd-alla ◽  
N. F Hasbullah ◽  
Hala M Hossen
Keyword(s):  
Initial Stress ◽  
Half Space ◽  
Electrical Potential ◽  
Frequency Equation ◽  
Wave Number ◽  
Semiconductor Layer ◽  
Semiconductor Film ◽  
Stress Effect ◽  
Field Equations ◽  
Shear Horizontal

In this paper, we investigated analytically the frequency equations for shear horizontal wave propagation in a piezoelectric half space covered by a semiconductor film with initial stress effect. The semiconducting layer is influenced by initial stress and the interface between the piezoelectric substrate and the semiconductor layer. The governing equations of the mechanical displacement and electrical potential function under the effect of initial stress are obtained by solving the coupled electromechanical field equations of the piezoelectric half-space and the semiconductor film. Next, the numerical examples are presented to illustrate the influence of initial stress and electromagnetic boundary conditions for the different values of the film thickness and wave number. Furthermore, we studied in more details the effect of initial stress on the frequency equation for piezoelectric Barium Titanate (BaTiO3) and semiconductor silicon. The obtained results provide a predictable and theoretical basis for applications of piezoelectric and semiconductor composites to acoustic wave devices.

Propagation of Impact-Induced Longitudinal Waves in Mechanical Systems With Variable Kinematic Structure

1993 ◽  
Vol 115 (1) ◽  
pp. 1-8 ◽  
Author(s):  
A. A. Shabana ◽  
W. H. Gau
Keyword(s):  
Mechanical Systems ◽  
Wave Number ◽  
Jump Discontinuity ◽  
Harmonic Waves ◽  
Kinematic Structure ◽  
System Configuration ◽  
Longitudinal Waves ◽  
Phase And Group Velocities ◽  
Group Velocities ◽  
System Topology

In previous publications by the authors of this paper it was shown that elastic media become dispersive as the result of the coupling between the finite rotation and the elastic deformation. Impact-induced harmonic waves no longer travel, in a rotating rod, with the same phase velocity and consequently the group velocity becomes dependent on the wave number. In this investigation, the propagation of impact-induced longitudinal waves in mechanical systems with variable kinematic structure is examined. The configuration of the mechanical system is identified using two different sets of modes. The first set describes the system configuration before the change in the system topology, while the second set describes the configuration of the system after the topology changes. In the analysis presented in this investigation, it is assumed that collision between the system components occurs first, followed by a change in the system topology. Both events are assumed to occur in a very short-lived interval of time such that the system configuration does not appreciably change. By using the first set of modes, the jump discontinuity in the system velocities is predicted using the algebraic generalized impulse momentum equations. The propagation of the impact-induced wave motion after the change in the system topology is described using the Fourier method. The series solution obtained is used to examine the effect of the topology change on the propagation of longitudinal elastic waves in constrained mechanical systems. It is shown that, while, for a nonrotating rod, mass capture or mass release has no effect on the phase and group velocities, in rotating rods the phase and group velocities depend on the change in the system topology. In particular the phase velocities of low harmonic longitudinal waves are more affected by the change in the system topology as compared to high frequency harmonic waves.

Experimental investigation of PL modes in a single layer

1962 ◽  
Vol 52 (1) ◽  
pp. 59-66
Author(s):  
Freeman Gilbert ◽  
Stanley J. Laster
Keyword(s):  
Half Space ◽  
Arrival Time ◽  
Single Layer ◽  
Good Resolution ◽  
P Waves ◽  
Small Phase ◽  
Shear Modes ◽  
Phase And Group Velocities ◽  
Set Up ◽  
Group Velocities

Abstract A two dimensional seismic model has been set up to simulate the problem of elastic wave propagation in a single layer overlying a uniform half space. Both the source and the receiver are mounted on the free surface of the layer. Seismograms are presented as a funciton of range. In addition to the Rayleigh and shear modes, PL modes are observed. Experimentally determined phase and group velocities compare fairly well with theoretical curves. The decay factor for PL is maximum at the arrival time of P waves in the half-space. There is also a secondary maximum at the arrival time of P waves in the layer. Although the decay of PL is small, phase equalization of PL does not yield the initial pulse shape because the mode embraces an insufficient frequency band to permit good resolution.

Effects of Ions on the Propagation of Langmuir Oscillations in Cold Quantum Electron-Ion Plasmas

2008 ◽  
Vol 63 (7-8) ◽  
pp. 400-404 ◽  
Author(s):  
Hyun-Jae Rhee ◽  
Young-Dae Jung
Keyword(s):  
Phase Velocity ◽  
Quantum Effect ◽  
Frequency Mode ◽  
Wave Number ◽  
Propagation Mode ◽  
Quantum Electron ◽  
Quantum Plasmas ◽  
Phase And Group Velocities ◽  
Lower Frequency Mode ◽  
Group Velocities

The effects of ions on the propagation of Langmuir oscillations are investigated in cold quantum electron-ion plasmas. It is shown that the higher and lower frequency modes of the Langmuir oscillations would propagate in cold quantum plasmas according to the effects of ions. It is also shown that these two propagation modes merge into one single propagation mode if the contribution of ions is neglected. It is found that the quantum effect enhances the phase and group velocities of the higher frequency mode of the propagation. In addition, it is shown that the phase velocity of the lower frequency mode is saturated with increasing the quantum wavelength and further that the group velocity of the lower frequency mode has a maximum position in the domains of the wave number and quantum wavelength.

Shear-wave group-velocity surfaces in low-symmetry anisotropic media

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Vladimir Grechka
Keyword(s):  
Group Velocity ◽  
Body Wave ◽  
Transverse Isotropy ◽  
Shear Waves ◽  
Anisotropic Media ◽  
The Body ◽  
Velocity Models ◽  
Phase And Group Velocities ◽  
Low Symmetry ◽  
Group Velocities

Shear waves excited by natural sources constitute a significant part of useful energy recorded in downhole microseismic surveys. In rocks, such as fractured shales, exhibiting symmetries lower than transverse isotropy (TI), the shear wavefronts are always multivalued in certain directions, potentially complicating the data processing and analysis. This paper discusses a basic tool — the computation of the phase and group velocities of all waves propagating along a given ray — that intends to facilitate the understanding of geometries of the shear wavefronts in homogeneous anisotropic media. With this tool, arbitrarily complex group-velocity surfaces can be conveniently analyzed, providing insights into possible challenges to be faced when processing shear waves in anisotropic velocity models that have symmetries lower than TI. Among those challenges are complicated multipathing and the presence of cones of directions, known as internal refraction cones, in which no fast shear waves propagate and the entire shear portion of the body-wave seismic data consists of several branches of the slow shear wavefronts.

Graphical Study of the Dispersion of Electro-Magneto-Ionic Waves

1949 ◽  
Vol 2 (2) ◽  
pp. 307
Author(s):  
VA Bailey ◽  
JA Roberts
Keyword(s):  
Graphical Method ◽  
Wave Number ◽  
Refractive Indices ◽  
General Effect ◽  
Wave Groups ◽  
Positive Ions ◽  
Phase And Group Velocities ◽  
The Waves ◽  
Special Case ◽  
Group Velocities

A graphical method for approximating to all the eight roots of the equation of dispersion, corresponding to any numerically specified case, is described. This method uses curves drawn with ω and l as coordinates to give readily the following information about the waves which can exist in the medium : (i) the frequency- bands in which undamped waves or wave-groups can grow as they progress, (ii) the wave-number bands in which unattenuated waves can grow in time, (iii) the phase- and group-velocities, refractive indices, and coefficients of positive and negative attenuation and damping, (iv) the general effect of collisions between electrons and other particles on the attenuation or damping of a wave or wave-group. Several illustrative examples are given. The same method is also applied to a special case of the more comprehensive equation of dispersion which includes the effects due to the motions of the positive ions, and it is shown that there can then exist unattenuated waves which grow with the lapse of time.

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