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.

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 Group and Phase Velocity of Love Waves Propagating in Elastic Functionally Graded Materials

2015 ◽  
Vol 40 (2) ◽  
pp. 273-281 ◽  
Author(s):  
Piotr Kiełczyński ◽  
Marek Szalewski ◽  
Andrzej Balcerzak ◽  
Krzysztof Wieja
Keyword(s):  
Group Velocity ◽  
Half Space ◽  
Functionally Graded Materials ◽  
Integral Formula ◽  
Liouville Problem ◽  
Elastic Half Space ◽  
Functionally Graded ◽  
Love Waves ◽  
Graded Materials ◽  
Sturm Liouville

AbstractThis paper presents a theoretical study of the propagation behaviour of surface Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in acoustics. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). Two Love wave waveguide structures are analyzed: 1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and 2) a semi-infinite nonhomogeneous elastic half-space. The Direct Sturm-Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved 1) analytically in the case of the step profile, exponential profile and 1cosh2type profile, and 2) numerically in the case of the power type profiles (i.e. linear and quadratic), by using two numerical methods: i.e. a) Finite Difference Method, and b) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The results obtained in this paper can give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials.

scholarly journals Velocities of mantle Love and Rayleigh waves over multiple paths

1963 ◽  
Vol 53 (4) ◽  
pp. 741-764 ◽  
Author(s):  
M. Nafi Toksöz ◽  
Ari Ben-Menahem
Keyword(s):  
Rayleigh Waves ◽  
Great Circle ◽  
Love Waves ◽  
Period Range ◽  
Phase Velocities ◽  
Multiple Paths ◽  
Lateral Variations ◽  
Phase Spectra ◽  
Group Velocities ◽  
Good Agreement

Abstract Phase velocities of Love waves from five major earthquakes are measured over six great circle paths in the period range of 50 to 400 seconds. For two of the great circle paths the phase velocities of Rayleigh waves are also obtained. The digitized seismograph traces are Fourier analyzed, and the phase spectra are used in determining the phase velocities. Where the great circle paths are close, the phase velocities over these paths are found to be in very good agreement with each other indicating that the measured velocities are accurate and reliable. Phase velocities of Love waves over paths that lie far from each other are different, and this difference is consistent and much greater than the experimental error. From this it is concluded that there are lateral variations in the structure of the earth's mantle. One interpretation of this variation is that the mantle under the continents is different from that under the oceans, since the path with the highest phase velocities is almost completely oceanic. This interpretation, however, is not unique and variations under the oceans and continents are also possible. Group velocities are computed from the phase velocities and are also directly measured from the seismograms. The group-velocity curve of Love waves has a plateau between periods of 100 and 300 seconds with a shallow minimum at about 290 seconds. The sources of error in both Fourier analysis and direct time domain methods of phase velocity measurement are discussed.

Estimation of elastic stiffness coefficients of an orthorhombic physical model using group velocity analysis on transmission data

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. R27-R39 ◽  
Author(s):  
Faranak Mahmoudian ◽  
G. F. Margrave ◽  
P. F. Daley ◽  
J. Wong ◽  
D. C. Henley
Keyword(s):  
Physical Model ◽  
Group Velocity ◽  
Body Wave ◽  
Elastic Stiffness ◽  
Laboratory Model ◽  
Theoretical Dependence ◽  
Stiffness Coefficients ◽  
Phase Velocities ◽  
Phase And Group Velocities ◽  
Group Velocities

We acquired 3C ultrasonic transmission seismograms, measured group velocities associated with the three quasi-body wave types, and determined density-normalized stiffness coefficients ([Formula: see text]) over an orthorhombic physical (laboratory) model. The estimation of [Formula: see text] is based on an approximate relationship between the nine orthorhombic [Formula: see text] and group velocities. Estimation of the anisotropic [Formula: see text] is usually done using phase velocities with well-known formulas expressing their theoretical dependence on [Formula: see text]. However, on time-domain seismograms, arrivals are observed traveling with group velocities. Group velocity measurements are found to be straightforward, reasonably accurate, and independent of the size of the transducers used. In contrast, the accuracy of phase velocities derived from the [Formula: see text] transform analysis was found to be very sensitive to small differences in picked arrival times and to transducer size. Theoretical phase and group velocities, calculated in a forward manner from the [Formula: see text] estimates, agreed with the originally measured phase and group velocities, respectively. This agreement confirms that it is valid to use easily measured group velocities with their approximate theoretical relationship to the [Formula: see text] to determine the full stiffness matrix. Compared to the phase-velocity procedure, the technique involving group velocities is much less prone to error due to time-picking uncertainties, and therefore is more suitable for analyzing physical model seismic data.

3D-S wave velocity model of the Los Humeros geothermal field, Mexico, by ambient-noise tomography

2020 ◽  
Author(s):  
Joana Martins ◽  
Anne Obermann ◽  
Arie Verdel ◽  
Philippe Jousset
Keyword(s):  
Group Velocity ◽  
Ambient Noise ◽  
Cross Correlation ◽  
Geothermal Field ◽  
Seismic Network ◽  
Ambient Noise Tomography ◽  
Phase Velocities ◽  
Phase And Group Velocities ◽  
Velocity Anomalies ◽  
Group Velocities

<p>Since the successful retrieval of surface-wave responses from the ambient seismic field via cross-correlation, noise-based interferometry has been widely used for high-resolution imaging of the Earth’s lithosphere from all around the globe. Further applications on geothermal fields reveal the potential of ambient noise techniques to either characterize the subsurface velocity field or to understand the temporal evolution of the velocity models due to field operations.</p><p>Following the completion of the GeMEX<sup>*</sup> project, a European-Mexican collaboration to improve our understanding of two geothermal sites in Mexico, we present the results of ambient noise tomography (ANT) techniques over the Los Humeros geothermal field. We used the vertical component of the data recorded by the seismic network active from September 2017 to September 2018. The total network is composed of 45 seismometers from which 25 are Broadband (BB) and the remaining ones short-period stations. From the ambient noise recorded at the deployed seismic network, we extract surface-waves after the computation of the empirical Green’s functions (EGF) by cross-correlation and consecutive stacking. After the cross-correlations, we pick both phase and group velocity arrival times of the ballistic surface-waves for which we derive independent tomographic maps. Finally, using both the retrieved phase and group velocities, we jointly invert the tomographic results from frequency to depth.</p><p>We identify positive and negative velocity variations from an average velocity between -15% and 15% for group and between -10% and 10% for phase velocities in the frequency domain. While the velocity variations are consistent for both the phase and group velocities (with expected group velocities lower than the phase velocities), the group velocity anomalies are more pronounced than the phase velocity anomalies. Low-velocity anomalies fall mostly within the inner volcano caldera, the area of highest interest for geothermal energy. This is consistent with the surface temperatures measured at the Los Humeros caldera, indicating the presence of a heat source. Finally, we compare our results with other geophysical studies (e.g geodesy, gravity, earthquake tomography and magnetotelluric) performed during the GeMEX project within the same area.</p><p> </p><p> </p><p> </p><p>We thank the European and Mexican GEMex team for setting up the seismic network and station maintenance as well as data retrieval (amongst which Tania Toledo, Emmanuel Gaucher, Angel Figueroa and Marco Calo). We thank the Comisión Federal de Electricidad (CFE) who kindly provided us with access to their geothermal field and permission to install the seismic stations. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 727550 and the Mexican Energy Sustainability Fund CONACYT-SENER, project 2015-04-68074.</p><p> </p><p>* http://www.gemex-h2020.eu/index.php?option=com_content&view=featured&Itemid=101&lang=en</p>

scholarly journals Love wave in porous layer under initial stress over heterogeneous elastic half-space under gravity and initial stress

2021 ◽  
Vol 60 (3) ◽  
pp. 193-210
Author(s):  
Asit Kumar Gupta ◽  
Anup Kumar Mukhopadhyay ◽  
Santimoy Kundu ◽  
Pulak Patra
Keyword(s):  
Free Boundary ◽  
Initial Stress ◽  
Half Space ◽  
Porous Layer ◽  
Elastic Half Space ◽  
Love Waves ◽  
Initial Stresses ◽  
Rigid Boundary ◽  
Free Surfaces ◽  
Phase Velocities

In the present paper, effect of initial stresses and gravity on the propagation of Love waves has been studied in porous layer surface over a heterogeneous half-space. We have considered two types of boundary on free surfaces: (a) rigid boundary and (b) traction free boundary. The propagation of Love waves has been investigated under assumed media in both the cases of boundary and discusses a comparison study of two cases. The dispersion equations and phase velocities have been obtained in both the cases. The numerical calculations have been done and presented graphically. This study of Love waves in the assumed medium reveals that the presence of initial stress in the half-space and absence of initial stress in the layer, the displacement of phase velocity in rigid boundary  is more than the traction free boundary.

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.

Dispersion of Love waves in prestressed double-layered medium over a gravitating half-space

2020 ◽  
Vol 13 (13) ◽  
Author(s):  
Bishwanath Prasad ◽  
Santimoy Kundu ◽  
Prakash Chandra Pal ◽  
Parvez Alam
Keyword(s):  
Half Space ◽  
Layered Medium ◽  
Love Waves
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