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

2005 ◽  
Vol 297-300 ◽  
pp. 2213-2218 ◽  
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.

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.

Research on Elastic Scattered Wave of Submerged Elastic Plate

2013 ◽  
Vol 303-306 ◽  
pp. 2779-2783
Author(s):  
Wen Jian Chen ◽  
Hui Sun ◽  
Tie Lin Sun
Keyword(s):  
Elastic Scattering ◽  
Phase Velocity ◽  
Group Velocity ◽  
Elastic Plate ◽  
Lamb Wave ◽  
Arrival Time ◽  
Group Velocity Dispersion ◽  
Scattered Wave ◽  
Energy Propagation ◽  
Scattering Wave

It is proved by theory and experiment that the arrival time of elastic scattering wave is determined by group velocity of Lamb wave in plate and the speed of elastic scattering wave in water. The frequency dispersion equation of Lamb wave is derived for submerged elastic plate, and the phase velocity and group velocity dispersion curves are obtained by numerical calculation method. It is found that the phase velocity is greater or less than the group velocity at different frequency-thickness products. The energy propagation speed of wave is group velocity, so the arrival time of elastic scattering wave is determined by group velocity of Lamb wave in plate and the speed of sound in water. Experimental results show that elastic scattering wave is ahead of or behind the edge wave in echoes of elastic steel plate. The experiment results confirm validity of the theoretical analysis results.

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

2019 ◽  
pp. 6-18
Author(s):  
V I Erofeev ◽  
D A Kolesov ◽  
V L Krupenin
Keyword(s):  
Phase Velocity ◽  
Longitudinal Wave ◽  
Group Velocity ◽  
Mass Density ◽  
Deformable Body ◽  
Mass Model ◽  
Positive Direction ◽  
Normal Dispersion ◽  
Phase And Group Velocities ◽  
Group Velocities

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.

Modification of Planck Black Body Emissive Power and Intensity in Particulate Media Due to Multiple and Dependent Scattering

2005 ◽  
Author(s):  
Ravi Prasher
Keyword(s):  
Heat Flux ◽  
Phase Velocity ◽  
Group Velocity ◽  
Density Of States ◽  
Volume Fraction ◽  
Black Body ◽  
Field Approximation ◽  
Effective Field ◽  
Spherical Particles ◽  
Particulate Media

Dispersion relation for electromagnetic wave is obtained in particulate media using effective field approximation (EFA) and quasi crystalline approximation (QCA). Due to multiple and dependent scattering the density of states, phase velocity and group velocity of photons are modified. Modification of these parameters modifies the Planck black body equilibrium radiation intensity. This modification affects the temperature and the heat flux predictions in multiple and dependent scattering particulate media. Results show that EFA can accurately capture the dependence of density of states, phase velocity and the group velocity on volume fraction of scatterers whereas QCA can capture the dependence of effective attenuation as well as density of states, phase velocity and the group velocity. Comparisons of the temperature, heat flux, and effective attenuation are made between EFA, QCA and work done by C.L Tien and coworkers. Results show that heat flux and temperature predictions made by models in the literature for multiple and dependent scattering are not correct as these models do not take the modification of the equilibrium intensity into account. Finally we introduce a new model called Dependent Effective Field Approximation (DEFA) which accurately captures the effect of volume fraction on the equilibrium intensity, and effective attenuation. All relations derived in the paper are for spherical particles.

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

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.

scholarly journals Studies of gravity wave propagation in the mesosphere observed by MU radar

2013 ◽  
Vol 31 (5) ◽  
pp. 845-858 ◽  
Author(s):  
H. Y. Lue ◽  
F. S. Kuo ◽  
S. Fukao ◽  
T. Nakamura
Keyword(s):  
Wave Packet ◽  
Phase Velocity ◽  
Group Velocity ◽  
Gravity Waves ◽  
Wave Packets ◽  
Wave Period ◽  
Parameter Analysis ◽  
Mu Radar ◽  
Horizontal Wavelength ◽  
Group Velocities

Abstract. Mesospheric data were analyzed by a composite method combining phase and group velocity tracing technique and the spectra method of Stokes parameter analysis to obtain the propagation parameters of atmospheric gravity waves (AGW) in the height ranges between 63.6 and 99.3 km, observed using the MU radar at Shigaraki in Japan in the months of November and July in the years 1986, 1988 and 1989. The data of waves with downward phase velocity and the data of waves with upward phase velocity were independently treated. First, the vertical phase velocity and vertical group velocity as well as the characteristic wave period for each wave packet were obtained by phase and group velocity tracing technique. Then its horizontal wavelength, intrinsic wave period and horizontal group velocity were obtained by the dispersion relation. The intrinsic frequency and azimuth of wave vector of each wave packet were checked by Stokes parameters analysis. The results showed that the waves with intrinsic periods in the range 30 min–4.5 h had horizontal wavelength ranging from 25 to 240 km, vertical wavelength from 2.5 to 12 km, and horizontal group velocities from 15 to 60 m s−1. Both upward moving wave packets and downward moving wave packets had horizontal group velocities mostly directed in the sector between directions NNE (north-north-east) and SEE in the month of November, and mostly in the sector between directions NW and SWS in the month of July. Comparing with mean wind directions, the gravity waves appeared to be more likely to propagate along with mean wind than against it. This apparent prevalence for downstream wave packets was found to be caused by a systematic filtering effect existing in the process of phase and group velocity tracing analysis: A significant portion of upstream wave packets might have been Doppler shifted out of the vertical range in phase and group velocity tracing analysis.

Modification of Planck Blackbody Emissive Power and Intensity in Particulate Media Due to Multiple and Dependent Scattering

2005 ◽  
Vol 127 (8) ◽  
pp. 903-910 ◽  
Author(s):  
Ravi Prasher
Keyword(s):  
Heat Flux ◽  
Phase Velocity ◽  
Group Velocity ◽  
Density Of States ◽  
Volume Fraction ◽  
Field Approximation ◽  
Effective Field ◽  
Emissive Power ◽  
Particulate Media ◽  
Work Done

The dispersion relation for an electromagnetic wave is obtained in particulate media using effective field approximation (EFA) and quasi-crystalline approximation (QCA). Due to multiple and dependent scattering the density of states, phase velocity and group velocity of photons are modified. Modification of these parameters modifies the Planck blackbody equilibrium radiation intensity and emissive power. Results show that EFA can accurately capture the dependence of density of states, phase velocity, and the group velocity on volume fraction of scatterers whereas QCA can capture the dependence of effective attenuation as well as density of states, phase velocity, and the group velocity. Comparisons of the temperature, heat flux, and effective attenuation are made between EFA, QCA, and work done by C. L. Tien and co-workers. Results show that heat flux and temperature predictions made by models in the literature for multiple and dependent scattering are not correct as these models do not take the modification of the equilibrium intensity into account. Finally we introduce a new model called dependent effective field approximation (DEFA) which accurately captures the effect of volume fraction on the equilibrium intensity and effective attenuation.

Transmural heterogeneity of diffusion anisotropy in the sheep myocardium characterized by MR diffusion tensor imaging

2007 ◽  
Vol 293 (4) ◽  
pp. H2377-H2384 ◽  
Author(s):  
Yi Jiang ◽  
Julius M. Guccione ◽  
Mark B. Ratcliffe ◽  
Edward W. Hsu
Keyword(s):  
Diffusion Tensor Imaging ◽  
Volume Fraction ◽  
Diffusion Tensor ◽  
Extracellular Volume ◽  
Extracellular Volume Fraction ◽  
Left Ventricular ◽  
Fiber Direction ◽  
Significant Difference ◽  
Mr Diffusion ◽  
Mr Diffusion Tensor Imaging

The orientation of MRI-measured diffusion tensor in the myocardium has been directly correlated to the tissue fiber direction and widely characterized. However, the scalar anisotropy indexes have mostly been assumed to be uniform throughout the myocardial wall. The present study examines the fractional anisotropy (FA) as a function of transmural depth and circumferential and longitudinal locations in the normal sheep cardiac left ventricle. Results indicate that FA remains relatively constant from the epicardium to the midwall and then decreases (25.7%) steadily toward the endocardium. The decrease of FA corresponds to 7.9% and 12.9% increases in the secondary and tertiary diffusion tensor diffusivities, respectively. The transmural location of the FA transition coincides with the location where myocardial fibers run exactly circumferentially. There is also a significant difference in the midwall-endocardium FA slope between the septum and the posterior or lateral left ventricular free wall. These findings are consistent with the cellular microstructure from histological studies of the myocardium and suggest a role for MR diffusion tensor imaging in characterization of not only fiber orientation but, also, other tissue parameters, such as the extracellular volume fraction.

The relationship between group velocity and phase velocity for finite, discrete observations

1977 ◽  
Vol 67 (5) ◽  
pp. 1249-1258
Author(s):  
Douglas C. Nyman ◽  
Harsh K. Gupta ◽  
Mark Landisman
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
The Other ◽  
Frequency Range ◽  
Finite Frequency ◽  
Discrete Observations ◽  
Practical Situation ◽  
Order Polynomial ◽  
Observational Errors ◽  
The Relationship

abstract The well-known relationship between group velocity and phase velocity, 1/u = d/dω (ω/c), is adapted to the practical situation of discrete observations over a finite frequency range. The transformation of one quantity into the other is achieved in two steps: a low-order polynomial accounts for the dominant trends; the derivative/integral of the residual is evaluated by Fourier analysis. For observations of both group velocity and phase velocity, the requirement that they be mutually consistent can reduce observational errors. The method is also applicable to observations of eigenfrequency and group velocity as functions of normal-mode angular order.

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