On the Group Front and Group Velocity in a Dispersive Medium Upon Refraction From a Nondispersive Medium

2003 ◽  
Author(s):  
Z. M. Zhang ◽  
Keunhan Park
Keyword(s):  
Wave Packet ◽  
Phase Velocity ◽  
Wave Front ◽  
Group Velocity ◽  
Dispersive Medium ◽  
Front Velocity ◽  
Normal Group ◽  
Nondispersive Medium ◽  
Positive Index ◽  
Front Movement

Conventional definitions of the velocities associated with the propagation of the modulated wave are both confusing and insufficient to describe the behavior of the wave packet clearly in a multi-dimensional dispersive medium. There exist infinite solutions to the general equation of wave-front movement, suggesting that there are infinite phase velocities. Therefore, the introduction of “normal phase velocity” becomes necessary to unambiguously define the phase velocity in the direction perpendicular to the wave front. Similarly, there exist infinite solutions to the equation describing the group-front movement in the case of a wave packet, resulting in infinite “group-front velocities.” The “normal group-front velocity” is defined as the smallest speed at which the group-front travels and is in the direction perpendicular to the group front. We show that the group velocity (i.e., the velocity of energy flow) is one of the group-front velocities and, in general, is not the same as the normal group-front velocity. Hence, the direction in which the wave packet travels is not necessarily normal to the group front. Examples are used to demonstrate the behavior of a wave packet that is refracted from vacuum to a positive index medium (PIM) or a negative index medium (NIM).

On the Group Front and Group Velocity in a Dispersive Medium Upon Refraction From a Nondispersive Medium

2004 ◽  
Vol 126 (2) ◽  
pp. 244-249 ◽  
Author(s):  
Z. M. Zhang ◽  
Keunhan Park
Keyword(s):  
Wave Packet ◽  
Phase Velocity ◽  
Wave Front ◽  
Group Velocity ◽  
Dispersive Medium ◽  
Front Velocity ◽  
Nondispersive Medium ◽  
Definition Of ◽  
Positive Index ◽  
Special Case

Conventional definitions of velocities associated with the propagation of modulated waves cannot clearly describe the behavior of the wave packet in a multidimensional dispersive medium. The conventional definition of the phase velocity, which is perpendicular to the wave front, is a special case of the generalized phase velocity defined in this work, since there exist an infinite number of solutions to the equation describing the wave-front movement. Similarly, the generalized group-front velocity is defined for the movement of a wave packet in an arbitrary direction. The group-front velocity is the smallest speed at which the group-front travels in the direction normal to the group front. The group velocity, which is the velocity of energy flow in a nondissipative medium, also satisfies the group-front equation. Because the group-front velocity and the group velocity are not always the same, the direction in which the wave packet travels is not necessarily normal to the group front. In this work, two examples are used to demonstrate this behavior by considering the refraction of a wave packet from vacuum to either a positive-index material (PIM) or a negative-index material (NIM).

Developmental changes in flow-wave propagation velocity in embryonic chick vascular system

1997 ◽  
Vol 273 (3) ◽  
pp. H1523-H1529 ◽  
Author(s):  
M. Yoshigi ◽  
J. M. Ettel ◽  
B. B. Keller
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Wave Front ◽  
Blood Volume ◽  
Propagation Velocity ◽  
Vascular System ◽  
Front Velocity ◽  
Developmental Changes ◽  
Wave Propagation Velocity ◽  
Circulating Blood Volume

We analyzed flow-wave propagation velocity in the early embryonic vascular system and its responses to acute alterations in circulating blood volume. Two 20-MHz pulsed Doppler velocimeters were positioned along the arterial system in stage 18 (n = 12), 21 (n = 10), and 24 (n = 11) chick embryos. Distance between the two measurement sites was measured by video-microscopy. Phase velocity was calculated using Fourier transform up to the fourth harmonics. Wave-front velocity was also calculated by threshold technique. In a subset of embryos at stage 24 (n = 10), circulating blood volume was acutely altered to change stroke volume. Mean phase velocity increased from 42.9 +/- 3.3 to 95.8 +/- 7.5 cm/s from stage 18 to 24 (P < 0.05 by analysis of variance), whereas wave-front velocity increased from 52.8 +/- 2.4 to 72.2 +/- 5.2 cm/s. Stroke volume and mean aortic pressure paralleled the changes in mean phase velocity and wave-front velocity in normal development and in response to changes in circulating blood volume. Thus developmental changes in wave-propagation velocity were consistent with changes in the size of the vascular system, pressure range, and elastic properties of the arterial wall during systemic vasculogenesis in the embryo.

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.

The phase velocity, ray velocity, and group velocity surfaces for a magneto-ionic medium

1977 ◽  
Vol 17 (3) ◽  
pp. 467-486 ◽  
Author(s):  
A. D. M. Walker
Keyword(s):  
Wave Propagation ◽  
Phase Velocity ◽  
Wave Front ◽  
Group Velocity ◽  
Cold Plasma ◽  
Ionic Medium ◽  
Velocity Surface

The phase velocity surface for waves propagating in a uniform cold plasma is sometimes misinterpreted as having the shape of a wave-front. A summary is presented of the correct interpretations of the phase velocity, ray velocity, and group velocity surfaces. A full set of computer generated plots of such surfaces are presented. These are intended as an aid to visualization of wave propagation in such a medium.

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.

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.

Attenuation of dispersed wave trains

1962 ◽  
Vol 52 (1) ◽  
pp. 109-112
Author(s):  
James N. Brune
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Seismic Waves ◽  
Free Oscillations ◽  
Wave Trains

Abstract It is shown that groups of seismic waves are attenuated by the factor exp −exp⁡−πXQUT where X is the distance, U the group velocity, T the period and Q−1 is a measure of the damping of free oscillations. Accordingly, observations of Q given by Ewing and Press (1954 a, b) and Sato (1958) are revised by the ratio of the phase velocity to the group velocity.

scholarly journals Sensitivities of surface wave velocities to the medium parameters in a radially anisotropic spherical Earth and inversion strategies

2015 ◽  
Vol 58 (5) ◽  
Author(s):  
Sankar N. Bhattacharya
Keyword(s):  
Surface Wave ◽  
Phase Velocity ◽  
Group Velocity ◽  
Wave Velocity ◽  
Love Waves ◽  
P Wave ◽  
Model Parameters ◽  
Nonlinear Inversion ◽  
Wave Velocities ◽  
Spherical Earth

<p>Sensitivity kernels or partial derivatives of phase velocity (<em>c</em>) and group velocity (<em>U</em>) with respect to medium parameters are useful to interpret a given set of observed surface wave velocity data. In addition to phase velocities, group velocities are also being observed to find the radial anisotropy of the crust and mantle. However, sensitivities of group velocity for a radially anisotropic Earth have rarely been studied. Here we show sensitivities of group velocity along with those of phase velocity to the medium parameters <em>V<sub>SV</sub>, V<sub>SH </sub>, V<sub>PV</sub>, V<sub>PH , </sub></em><em>h</em><em> </em>and density in a radially anisotropic spherical Earth. The peak sensitivities for <em>U</em> are generally twice of those for <em>c</em>; thus <em>U</em> is more efficient than <em>c</em> to explore anisotropic nature of the medium. Love waves mainly depends on <em>V<sub>SH</sub></em> while Rayleigh waves is nearly independent of <em>V<sub>SH</sub></em> . The sensitivities show that there are trade-offs among these parameters during inversion and there is a need to reduce the number of parameters to be evaluated independently. It is suggested to use a nonlinear inversion jointly for Rayleigh and Love waves; in such a nonlinear inversion best solutions are obtained among the model parameters within prescribed limits for each parameter. We first choose <em>V<sub>SH</sub></em>, <em>V<sub>SV </sub></em>and <em>V<sub>PH</sub></em> within their corresponding limits; <em>V<sub>PV</sub></em> and <em>h</em> can be evaluated from empirical relations among the parameters. The density has small effect on surface wave velocities and it can be considered from other studies or from empirical relation of density to average P-wave velocity.</p>

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