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.

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.

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.

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.

A Dispersive Nonlocal Model for In-Plane Wave Propagation in Laminated Composites With Periodic Structures

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
H. Brito-Santana ◽  
Yue-Sheng Wang ◽  
R. Rodríguez-Ramos ◽  
J. Bravo-Castillero ◽  
R. Guinovart-Díaz ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Phase Velocity ◽  
Incident Angle ◽  
Unit Cell ◽  
Wave Number ◽  
Periodic Distribution ◽  
Phase And Group Velocities ◽  
Group Velocities ◽  
Plane Wave Propagation

In this paper, the problem of in-plane wave propagation with oblique incidence of the wave in an isotropic bilaminated composite under perfect contact between the layers and periodic distribution between them is studied. Based on an asymptotic dispersive method for the description of the dynamic processes, the dispersion equations were derived analytically from the average model. Numerical examples show that the dispersion curves obtained from the present model agree with the exact solutions for a range of wavelengths. Detailed numerical simulations are provided to illustrate graphically the phase and group velocities. Such illustrations allow the identification and comparison of the effects of the unit cell size, wave number and incident angle. It was observed that, as the incident angle increases, the dimensionless quasi-longitudinal phase velocity increases, and the dimensionless quasi-shear phase velocity decreases. In addition, the phase and group velocities decrease as the size of the unit cell increases. The frequency band structure, as a function of the wave-vector components is calculated.

scholarly journals VELOCITIES AND TRAJECTORIES OF WAVE MOTION IN A TWO-LAYER HYDRODYNAMIC SYSTEM

2020 ◽  
pp. 30-37
Author(s):  
Y. Hurtovyi ◽  
O. Kuharenko
Keyword(s):  
Surface Tension ◽  
Phase Velocity ◽  
Internal Waves ◽  
Lower Layer ◽  
World Ocean ◽  
Capillary Waves ◽  
Wave Number ◽  
Hydrodynamic System ◽  
Phase And Group Velocities ◽  
Group Velocities

The paper deals with studying trajectories of motion of individual liquid particles in a two-layer hydrodynamic system with a finite layer thickness as well as analyzing phase and group velocities of internal waves in the system. The problem is modeled for an inviscid incompressible fluid under action of the gravity and surface tension forces in a dimensionless form. Solutions of the problem are sought in the form of progressive waves using the multi-scale method. The solutions are expanded in terms of the nonlinearity coefficient. Dependence of the dispersion ratio of the wavenumber is investigated for different values of the surface tension coefficient and the ratio of the layer densities. Formulas are obtained for the group and phase velocities for internal gravity-capillary waves as well as in the limiting case for capillary waves. A comparison of the values of the phase and group velocities of internal waves for different values of the wave number is carried out. It is proved that with an increase in the wave number, the group velocity begins to outstrip the phase velocity, and their equality occurs at the minimum phase velocity. It is shown that the trajectories are ellipses in which the horizontal semi axes are larger than the vertical ones. Formulas are obtained for the semi axes of elliptic trajectories for each of the layers. The character of the change in the semi axes of elliptical trajectories is analyzed depending on the distance from the interface between two liquid layers as well as on the values of the wave number. It is proved that the semi axes of ellipses decrease unevenly with increasing distance from the boundary. The asymmetry of the particle trajectories of each of the layers is shown for the case when the thickness of the lower layer differs from the thickness of the lower layer. The study of the kinematic characteristics of the particle motion makes it possible to simulate real physical wave processes in the World Ocean. The results are also relevant for creating a theoretical basis for experiments.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

Compressional‐wave velocities in attenuating media: A laboratory physical model study

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1162-1167 ◽  
Author(s):  
Joseph B. Molyneux ◽  
Douglas R. Schmitt
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Wave Velocity ◽  
Propagation Distance ◽  
Compressional Wave ◽  
Arrival Times ◽  
Wave Velocities ◽  
Amplitude Maximum ◽  
Phase And Group Velocities ◽  
Group Velocities

Elastic‐wave velocities are often determined by picking the time of a certain feature of a propagating pulse, such as the first amplitude maximum. However, attenuation and dispersion conspire to change the shape of a propagating wave, making determination of a physically meaningful velocity problematic. As a consequence, the velocities so determined are not necessarily representative of the material’s intrinsic wave phase and group velocities. These phase and group velocities are found experimentally in a highly attenuating medium consisting of glycerol‐saturated, unconsolidated, random packs of glass beads and quartz sand. Our results show that the quality factor Q varies between 2 and 6 over the useful frequency band in these experiments from ∼200 to 600 kHz. The fundamental velocities are compared to more common and simple velocity estimates. In general, the simpler methods estimate the group velocity at the predominant frequency with a 3% discrepancy but are in poor agreement with the corresponding phase velocity. Wave velocities determined from the time at which the pulse is first detected (signal velocity) differ from the predominant group velocity by up to 12%. At best, the onset wave velocity arguably provides a lower bound for the high‐frequency limit of the phase velocity in a material where wave velocity increases with frequency. Each method of time picking, however, is self‐consistent, as indicated by the high quality of linear regressions of observed arrival times versus propagation distance.

Errors Introduced in Estimation of Surface Wave Phase and Group Velocities Due to Wrong Assumptions: An Assessment Using a Simple Model For Love Wave

2021 ◽  
Author(s):  
Akash Kharita ◽  
Sagarika Mukhopadhyay
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
Love Wave ◽  
Arrival Time ◽  
Epicentral Distance ◽  
Horizontal Distance ◽  
Wave Analysis ◽  
Time Period ◽  
Phase And Group Velocities ◽  
Group Velocities

<p>The surface wave phase and group velocities are estimated by dividing the epicentral distance by phase and group travel times respectively in all the available methods, this is based on the assumptions that (1) surface waves originate at the epicentre and (2) the travel time of the particular group or phase of the surface wave is equal to its arrival time to the station minus the origin time of the causative earthquake; However, both assumptions are wrong since surface waves generate at some horizontal distance away from the epicentre. We calculated the actual horizontal distance from the focus at which they generate and assessed the errors caused in the estimation of group and phase velocities by the aforementioned assumptions in a simple isotropic single layered homogeneous half space crustal model using the example of the fundamental mode Love wave. We took the receiver locations in the epicentral distance range of 100-1000 km, as used in the regional surface wave analysis, varied the source depth from 0 to 35 Km with a step size of 5 km and did the forward modelling to calculate the arrival time of Love wave phases at each receiver location. The phase and group velocities are then estimated using the above assumptions and are compared with the actual values of the velocities given by Love wave dispersion equation. We observed that the velocities are underestimated and the errors are found to be; decreasing linearly with focal depth, decreasing inversely with the epicentral distance and increasing parabolically with the time period. We also derived empirical formulas using MATLAB curve fitting toolbox that will give percentage errors for any realistic combination of epicentral distance, time period and depths of earthquake and thickness of layer in this model. The errors are found to be more than 5% for all epicentral distances lesser than 500 km, for all focal depths and time periods indicating that it is not safe to do regional surface wave analysis for epicentral distances lesser than 500 km without incurring significant errors. To the best of our knowledge, the study is first of its kind in assessing such errors.</p>

Free spheroidal vibrations of the earth at very long periods, Part I— Calculation of periods for several earth models

1963 ◽  
Vol 53 (3) ◽  
pp. 483-501 ◽  
Author(s):  
Leonard E. Alsop
Keyword(s):  
Inner Core ◽  
Computer Programs ◽  
Free Vibrations ◽  
Stoneley Waves ◽  
The Core ◽  
The Earth ◽  
Earth Models ◽  
Shear Modes ◽  
Phase And Group Velocities ◽  
Group Velocities

Abstract Periods of free vibrations of the spheroidal type have been calculated numerically on an IBM 7090 for the fundamental and first two shear modes for periods greater than about two hundred seconds. Calculations were made for four different earth models. Phase and group velocities were also computed and are tabulated herein for the first two shear modes. The behavior of particle motions for different modes is discussed. In particular, particle motions for the two shear modes indicate that they behave in some period ranges like Stoneley waves tied to the core-mantle interface. Calculations have been made also for a model which presumes a solid inner core and will be discussed in Part II. The two computer programs which were made for these calculations are described briefly.

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