Dispersion and attenuation of a longitudinal wave propagating in a metamaterial defined as a mass-to-mass chain

We study the features of propagation of a longitudinal wave in an acoustic (mechanical) metamaterial, modeled as a one-dimensional chain, containing equal masses, connected by elastic elements (springs), and having the same rigidity. Each mass contains within itself a series connection of another mass and viscous element (damper). The mass-to-mass model is free from the drawbacks of a number of other mechanical models of metamaterials: i.e. it eliminates the need to have the property of a deformable body to possess a negative mass, density, and (or) a negative elastic modulus. It is shown that the model under consideration makes it possible to describe the dispersion and frequency-dependent attenuation of a longitudinal wave, the character of which essentially depends on the ratio of the external and internal mass of the metamaterial. The behavior of the phase and group velocities of the wave is studied, as well as the evolution of its profile, both in the low-frequency and high-frequency ranges. The mass ratios were found at which the phase velocity exceeds the group velocity (normal dispersion) in magnitude and those at which the group velocity exceeds the phase velocity (anomalous dispersion) in a wide frequency range. Having the same asymptotic values when the frequency tends to infinity, the phase and group velocities have significant differences in behavior, namely, that the phase velocity is a monotonic function of frequency, and the group velocity has a maximum. In addition, in the region of normal dispersion, the group velocity may be negative, i.e. the so-called “reverse wave” effect is true, when, despite the fact that the phase velocity is directed in the positive direction of the spatial axis, the energy in such a wave is transferred in the negative direction.

Download Full-text

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

Geophysics ◽

10.1190/1.1444809 ◽

2000 ◽

Vol 65 (4) ◽

pp. 1162-1167 ◽

Cited By ~ 33

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.

Download Full-text

Frequency derivative of Rayleigh wave phase velocity for fundamental mode dispersion inversion: parametric study and experimental application

Geophysical Journal International ◽

10.1093/gji/ggaa417 ◽

2020 ◽

Vol 224 (1) ◽

pp. 649-668

Author(s):

A Wang ◽

D Leparoux ◽

O Abraham ◽

M Le Feuvre

Keyword(s):

Phase Velocity ◽

Rayleigh Wave ◽

Group Velocity ◽

Fundamental Mode ◽

Deep Layer ◽

Wave Phase Velocity ◽

Phase And Group Velocities ◽

Velocity Derivative ◽

Rayleigh Wave Phase Velocity ◽

Group Velocities

SUMMARY Monitoring the small variations of a medium is increasingly important in subsurface geophysics due to climate change. Classical seismic surface wave dispersion methods are limited to quantitative estimations of these small variations when the variation ratio is smaller than 10 per cent, especially in the case of variations in deep media. Based on these findings, we propose to study the contributions of the Rayleigh wave phase velocity derivative with respect to frequency. More precisely, in the first step of assessing its feasibility, we analyse the effects of the phase velocity derivative on the inversion of the fundamental mode in the simple case of a two-layer model. The behaviour of the phase velocity derivative is first analysed qualitatively: the dispersion curves of phase velocity, group velocity and the phase velocity derivative are calculated theoretically for several series of media with small variations. It is shown that the phase velocity derivatives are more sensitive to variations of a medium. The sensitivity curves are then calculated for the phase velocity, the group velocity and the phase velocity derivative to perform quantitative analyses. Compared to the phase and group velocities, the phase velocity derivative is sensitive to variations of the shallow layer and the deep layer shear wave velocity in the same wavelength (frequency) range. Numerical data are used and processed to obtain dispersion curves to test the feasibility of the phase velocity derivative in the inversion. The inversion results of the phase velocity derivative are compared with those of phase and group velocities and show improved estimations for small variations (variation ratio less than 5 per cent) of deep layer shear wave velocities. The study is focused on laboratory experiments using two reduced-scale resin-epoxy models. The differences of these two-layer models are in the deep layer in which the variation ratio is estimated as 16.4 ± 1.1 per cent for the phase velocity inversion and 17.1 ± 0.3 per cent for the phase velocity derivative. The latter is closer to the reference value 17 per cent, with a smaller error.

Download Full-text

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

Geophysics ◽

10.1190/geo2013-0203.1 ◽

2014 ◽

Vol 79 (1) ◽

pp. R27-R39 ◽

Cited By ~ 16

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.

Download Full-text

Group Velocity of Lamb Wave S0 Mode in Laminated Unidirectional CFRP Plates

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.297-300.2213 ◽

2005 ◽

Vol 297-300 ◽

pp. 2213-2218 ◽

Cited By ~ 3

Author(s):

Jeong Ki Lee ◽

Young H. Kim ◽

Ho Chul Kim

Keyword(s):

Phase Velocity ◽

Group Velocity ◽

Lamb Wave ◽

Volume Fraction ◽

Flow Direction ◽

Frequency Region ◽

Fiber Direction ◽

Symmetric Mode ◽

Group Velocities ◽

Propagation Velocities

The elastic waves in the isotropic plate are dispersive waves with the characteristics of Lamb wave, however, S0 symmetric mode is less dispersive in the frequency region less than the first cut-off frequency. In the anisotropic plates such as CFRP plates, the propagation velocities vary with the directions as well as the dispersion of the Lamb wave, and the phase velocity direction does not accord with the group velocity direction. The phase velocity direction is equivalent the wave vector direction, while the group velocity direction is equivalent the energy flow direction. In this work, the group velocity dispersion curves were obtained by the dispersion relation of the Lamb wave in unidirectional CFRP plate with an orthotropic structure. The group velocities of the S0 symmetric mode in the frequency region less than the first cut-off frequency were corrected by applying the slowness surface. The propagation velocities of Lamb wave were decided by measuring the arrival time of the Lamb wave signals received with the two pinducers varying the propagating direction in the laminated unidirectional CFRP plates of 8, 16 and 24 plies having a volume fraction of 67%. The measured velocities are better agreement with corrected group velocity curve, except near the fiber direction at the cusp region. When the propagating direction is not accorded with the principal axis, the direction of the group velocities inclines toward the fiber direction in the unidirectional CFRP plates, suggesting that the energy propagates preferentially toward fiber direction.

Download Full-text

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

Zeitschrift für Naturforschung A ◽

10.1515/zna-2008-7-803 ◽

2008 ◽

Vol 63 (7-8) ◽

pp. 400-404 ◽

Cited By ~ 1

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.

Download Full-text

Дисперсия объемных волн в структуре "графен--диэлектрик--графен"

Журнал технической физики ◽

10.21883/os.2019.02.47209.239-18 ◽

2019 ◽

Vol 126 (2) ◽

pp. 224

Author(s):

А.С. Абрамов ◽

Д.А. Евсеев ◽

И.О. Золотовский ◽

Д.И. Семенцов

Keyword(s):

Electric Field ◽

Phase Velocity ◽

Waveguide Mode ◽

Chemical Potential ◽

Dielectric Film ◽

Dispersion Characteristics ◽

Graphene Layers ◽

Waveguide Modes ◽

Phase And Group Velocities ◽

Group Velocities

AbstractWe investigate the dispersion properties of the first waveguide modes in a dielectric film that is coated with graphene layers having different chemical potential values. The control over the phase and group velocities of the first waveguide mode is considered. Spectral intervals in which the phase velocity of the waveguide modes is small, while their group velocity is negative, are revealed. We show that the dispersion characteristics of the waveguide modes can be rearranged using an external electric field.

Download Full-text

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

Geophysics ◽

10.1190/geo2014-0156.1 ◽

2015 ◽

Vol 80 (1) ◽

pp. C1-C7 ◽

Cited By ~ 19

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.

Download Full-text

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

10.5194/egusphere-egu2020-21956 ◽

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

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&#8217;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.Following the completion of the GeMEX* 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&#8217;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.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.&#160;&#160;&#160;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 Comisi&#243;n 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&#8217;s Horizon 2020 research and innovation programme under grant agreement No. 727550 and the Mexican Energy Sustainability Fund CONACYT-SENER, project 2015-04-68074.&#160;* http://www.gemex-h2020.eu/index.php?option=com_content&view=featured&Itemid=101&lang=en

Download Full-text

IDENTIFICATION OF MICROSTRUCTURED MATERIALS BY PHASE AND GROUP VELOCITIES

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2009.14.57-68 ◽

2009 ◽

Vol 14 (1) ◽

pp. 57-68 ◽

Cited By ~ 2

Author(s):

Jaan Janno ◽

Jüri Engelbrecht

Keyword(s):

Inverse Problem ◽

Wave Packets ◽

Anomalous Dispersion ◽

Normal Dispersion ◽

Physical Solution ◽

Numerical Tests ◽

Phase Velocities ◽

Phase And Group Velocities ◽

Group And Phase Velocities ◽

Group Velocities

An inverse problem to determine parameters of microstructured solids by means of group and phase velocities of wave packets is studied. It is proved that in the case of normal dispersion the physical solution is unique and in the case of anomalous dispersion two physical solutions occur. Numerical tests are presented.

Download Full-text

Generalized velocity approximation

Geophysics ◽

10.1190/geo2018-0401.1 ◽

2019 ◽

Vol 84 (1) ◽

pp. C27-C40 ◽

Cited By ~ 7

Author(s):

Alexey Stovas ◽

Sergey Fomel

Keyword(s):

Phase Velocity ◽

Functional Form ◽

Transversely Isotropic ◽

P Wave ◽

Transversely Isotropic Medium ◽

Numerical Examples ◽

P Wave Velocity ◽

Phase And Group Velocities ◽

Group Velocities ◽

Velocity Approximation

We have developed a new approximation for P-wave velocity that has the same functional form for phase and group velocities. We call it the generalized velocity approximation (GVA) because it is similar to the generalized moveout approximation. The 2D GVA has five parameters, and the 3D GVA has 12 parameters. Our approximation is exact for the phase velocity in a transversely isotropic medium. The parameters of the 3D version of the proposed approximation are defined in all of the symmetry planes in the same fashion. Numerical examples indicate that our approximation has the same accuracy as recently proposed anelliptic approximation, but it has fewer parameters.

Download Full-text