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.

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.

Analytic calculation of phase and group velocities of P-waves in orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C79-C97 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Transversely Isotropic ◽  
P Wave ◽  
P Waves ◽  
Type Approximation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Potential Applications ◽  
Phase And Group Velocities ◽  
And Migration ◽  
Group Velocities

We have developed an approximate method to calculate the P-wave phase and group velocities for orthorhombic media. Two forms of analytic approximations for P-wave velocities in orthorhombic media were built by analogy with the five-parameter moveout approximation and the four-parameter velocity approximation for transversely isotropic media, respectively. They are called the generalized moveout approximation (GMA)-type approximation and the Fomel approximation, respectively. We have developed approximations for elastic and acoustic orthorhombic media. We have characterized the elastic orthorhombic media in Voigt notation, and we can describe the acoustic orthorhombic media by introducing the modified Alkhalifah’s notation. Our numerical evaluations indicate that the GMA-type and Fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy, and the GMA-type approximation is comparable with the approximation recently proposed by Sripanich and Fomel. Potential applications of the proposed approximations include forward modeling and migration based on the dispersion relation and the forward traveltime calculation for seismic tomography.

scholarly journals Rayleigh Wave Dispersion for a Single Layer on an Elastic Half Space

1960 ◽  
Vol 13 (3) ◽  
pp. 498 ◽  
Author(s):  
BA Bolt ◽  
JC Butcher
Keyword(s):  
Numerical Solutions ◽  
Single Layer ◽  
Wave Dispersion ◽  
Elastic Half Space ◽  
Seismic Parameters ◽  
Period Equation ◽  
Phase And Group Velocities ◽  
The University ◽  
Rayleigh Wave Dispersion ◽  
Group Velocities

Numerical solutions of the period equation for Rayleigh waves in a single surface layer were calculated using the SILLIAC computer at the University of Sydney. Values of the phase and group velocities for both the fundamental and first higher mode are tabulated against period for eleven models. These related models allow a sensitivity analysis of the effect of variation in the seismic parameters.

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.

scholarly journals The Effect of Viscosity and Heterogeneity on Propagation of G–Type Waves

2017 ◽  
Vol 27 (2) ◽  
pp. 253-260
Author(s):  
Shishir Gupta ◽  
Keyword(s):  
Wave Propagation ◽  
Closed Form ◽  
Dispersion Equation ◽  
Initial Stress ◽  
Half Space ◽  
Plane Surface ◽  
Lower Half Space ◽  
Type Wave ◽  
Phase And Group Velocities ◽  
Group Velocities

AbstractEarthquakes yield motions of massive rock layers accompanied by vibrations which travel in waves. This paper analyses the possibility of G-type wave propagation along the plane surface at the interface of two different media which is assumed to be heterogeneous and viscoelastic. The upper layer is considered to be viscoelastic and the lower half space is considered to be an initially stressed heterogeneous half space. The dispersion equation, as well as the phase and group velocities, is obtained in closed form. The dispersion equation agrees with the classical Love type wave. The effects of the nonhomogeneity of the parameters and the initial stress on the phase and group velocities are expressed by means of a graph.

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.

Diffraction of elastic waves by a fluid-filled crack in a fluid-saturated poroelastic half-space

2021 ◽  
Author(s):  
Zhongxian Liu ◽  
Jiaqiao Liu ◽  
Sibo Meng ◽  
Xiaojian Sun
Keyword(s):  
Half Space ◽  
Seismic Waves ◽  
Incident Angle ◽  
Excitation Frequency ◽  
Vertical Displacement ◽  
Crack Model ◽  
P Waves ◽  
Amplification Effect ◽  
Resonance Frequencies ◽  
Angle Crack

Summary An indirect boundary element method (IBEM) is developed to model the two-dimensional (2D) diffraction of seismic waves by a fluid-filled crack in a fluid-saturated poroelastic half-space, using Green's functions computed considering the distributed loads, flow, and fluid characteristics. The influence of the fluid-filled crack on the diffraction characteristics is investigated by analyzing key parameters, such as the excitation frequency, incident angle, crack width and depth, and medium porosity. The results for the fluid-filled crack model are compared to those for the fluid-free crack model under the same conditions. The numerical results demonstrate that the fluid-filled crack has a significant amplification effect on the surface displacements, and that the effect of the depth of the fluid-filled crack is more complex compared to the influence of other parameters. The resonance diffraction generates an amplification effect in the case of normally incident P waves. Furthermore, the horizontal and vertical displacement amplitudes reach 4.2 and 14.1, respectively. In the corresponding case of the fluid-free crack, the vertical displacement amplitude is only equal to 4.1, indicating the amplification effect of the fluid in the crack. Conversely, for normally incident SV waves at certain resonance frequencies, the displacement amplitudes above a fluid-filled crack may be lower than the displacement amplitudes observed in the corresponding case of a fluid-free crack.

Sign in / Sign up

Export Citation Format

Share Document