THE RAMAN EFFECT AND SOLITONS IN AN ELLIPTICAL OPTICAL FIBER

1996 ◽  
Vol 05 (03) ◽  
pp. 559-574 ◽  
Author(s):  
BORIS A. MALOMED ◽  
RICHARD S. TASGAL
Keyword(s):  
Optical Fiber ◽  
Phase Velocity ◽  
Group Velocity ◽  
Optical Activity ◽  
Raman Effect ◽  
Complex Form ◽  
General System ◽  
Ellipticity Angle ◽  
Constant Polarization ◽  
Form Stability

We derive a general system of coupled nonlinear Schrödinger equations, describing light in a bimodal optical fiber, taking into account the quasi-instantaneous Raman effect, group-velocity birefringence, phase-velocity birefringence, and optical activity, and for light with general ellipticity. The Raman coefficients prove to obey a relation which depends on the ellipticity. When group-velocity birefringence and optical activity are both non-zero, polarization couples to the changing frequency, so soliton polarization cannot be held constant in this case. Without optical activity, there are solitons with constant polarization either entirely in one polarization (“simple”) or equally divided between the two polarizations (“vector”). At ellipticity angle 0° to 35.3°, the simple solitons are stable and the vector solitons are unstable, and vice versa for ellipticity angle 35.3° to 90°. With optical activity, the polarization can be constant, but with a rather more complex form. Stability is also examined, and bistability is found in some circumstances.

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>

Do traveltimes in pulse‐transmission experiments yield anisotropic group or phase velocities?

Geophysics ◽  
1994 ◽  
Vol 59 (11) ◽  
pp. 1774-1779 ◽  
Author(s):  
Joe Dellinger ◽  
Lev Vernik
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Elastic Constants ◽  
Near Field ◽  
Frequency Dispersion ◽  
Transversely Isotropic ◽  
Sh Waves ◽  
Phase Velocities ◽  
Pulse Transmission ◽  
Transmission Technique

The elastic properties of layered rocks are often measured using the pulse through‐transmission technique on sets of cylindrical cores cut at angles of 0, 90, and 45 degrees to the layering normal (e.g., Vernik and Nur, 1992; Lo et al., 1986; Jones and Wang, 1981). In this method transducers are attached to the flat ends of the three cores (see Figure 1), the first‐break traveltimes of P, SV, and SH‐waves down the axes are measured, and a set of transversely isotropic elastic constants are fit to the results. The usual assumption is that frequency dispersion, boundary reflections, and near‐field effects can all be safely ignored, and that the traveltimes measure either vertical anisotropic group velocity (if the transducers are very small compared to their separation) or phase velocity (if the transducers are relatively wide compared to their separation) (Auld, 1973).

Theory of 2-D complex seismic trace analysis

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 264-272 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Hilbert Transform ◽  
Instantaneous Frequency ◽  
Trace Analysis ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Instantaneous Amplitude ◽  
Dip Angle

The ideas of 1-D complex seismic trace analysis extend readily to two dimensions. Two‐dimensional instantaneous amplitude and phase are scalars, and 2-D instantaneous frequency and bandwidth are vectors perpendicular to local wavefronts, each defined by a magnitude and a dip angle. The two independent measures of instantaneous dip correspond to instantaneous apparent phase velocity and group velocity. Instantaneous phase dips are aliased for steep reflection dips following the same rule that governs the aliasing of 2-D sinusoids in f-k space. Two‐dimensional frequency and bandwidth are appropriate for migrated data, whereas 1-D frequency and bandwidth are appropriate for unmigrated data. The 2-D Hilbert transform and 2-D complex trace attributes can be efficiently computed with little more effort than their 1-D counterparts. In three dimensions, amplitude and phase remain scalars, but frequency and bandwidth are 3-D vectors with magnitude, dip angle, and azimuth.

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