Showing papers on "Group velocity published in 1984"

Negative dispersion using pairs of prisms.

Abstract: We show that pairs of prisms can have negative group-velocity dispersion in the absence of any negative material dispersion. A prism arrangement is described that limits losses to Brewster-surface reflections, avoids transverse displacement of the temporally dispersed rays, permits continuous adjustment of the dispersion through zero, and yields a transmitted beam collinear with the incident beam.

Use of ray theory to calculate high-frequency radiation from earthquake sources having spatially variable rupture velocity and stress drop

Abstract: We analyze the problem of calculating high-frequency ground motions (>1 Hz) caused by earthquakes having arbitrary spatial variations of rupture velocity and slip velocity (or stress drop) over the fault. We approximate the elastic wave Green's functions by far-field body waves, which we calculate using geometric ray theory. However, we do not make the traditional Fraunhofer approximation, so our method may be used close to large faults. The method is confined to high frequencies (greater than about 1 Hz) due to the omisson of near-field terms, and must be used at source-observer distances less than a few source depths, due to the omission of surface waves. It is easily used in laterally varying velocity structures. Assuming a simple parameterization of the slip function, the computational problem collapses to the evaluation of a series of line integrals over the fault, with one line integral per each time ti in the observer seismogram. The path of integration corresponding to observation time ti consists of only those points on the fault which radiate body waves arriving at the observer at exactly time ti . This path is an isochron of the arrival time function. An isochron velocity may be defined that depends on rupture velocity and resembles the usual directivity function. Observed ground motions are directly dependent upon this isochron velocity. Ground velocity is proportional to isochron velocity and ground acceleration is proportional to isochron acceleration in dislocation models of rupture. Ground acceleration may also be related to spatial variations of slip velocity on the fault, using the square of isochron velocity as a constant of proportionality. We show two rupture models, one with variable slip velocity and the other with variable rupture velocity, that cause the same ground acceleration at a single observer. The computational method is shown to produce reasonably accurate synthetic seismograms, compared to a method using complete Green's functions, and requires about 0.5 per cent of the computer time. It may be very effective in calculating ground motions in the frequency band 1 to 10 Hz at observers within a few source depths of large earthquakes, where most of the high-frequency motions may be caused by direct P and S waves. We suggest a possible method for inverting ground motions for both slip velocity and rupture velocity over the fault.

Turbulent structure in the edge plasma of the TEXT tokamak

Abstract: A reversal has been observed in the mean phase velocity of the turbulent fluctuations in the edge plasma of the TEXT tokamak. The observations can be described by a model in which the wave velocity in the lab frame is dominated by a nonuniform Er×B velocity and a gradient driven drift. The measurements also exhibit a localized instability which occurs in the region of maximum velocity shear.

Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion

Abstract: Closed-form analytical solutions are obtained for a passively mode-locked laser for the case in which self-phase modulation and group-velocity dispersion, in addition to the more conventional mechanisms of saturable absorption and gain, shape the laser pulses. Provided that the self-phase modulation and group-velocity dispersion are related in a manner similar to that which causes soliton formation in optical fibers, this additional pulse shaping can reduce the pulse duration below the limit otherwise set by the laser bandwidth.

Measurements of mantle wave velocities and inversion for lateral heterogeneity and anisotropy. II - Analysis by the single-station method

Abstract: Phase and group velocities of G_2, G_3, R_2 and R_3 (100-330_s) are measured by the single-station method and are inverted to give a spherical harmonic representation of the velocity lateral variation. Approximately 200 paths have been studied. The results are presented for degrees and orders up to 6. The even harmonics of the phase velocity representation are consistent with those obtained from great circle phase velocities (Paper I). The odd harmonics are less constrained and generally have larger standard deviations than the even harmonics. To suppress the poorly determined harmonics in the velocity contour maps we construct a filter which is derived from an inverse problem formulation. The filter reduces the amplitudes of regional variations, but does not change the overall pattern. The patterns of the regional variations are generally consistent with those obtained by regionalized inversion of great circle data (Paper I). The velocity maps show significant differences within oceans and continents. An analysis is made of correlations of surface wave velocities with heat flow and the non-hydrostatic geoid. The slownesses correlate well with heat flow for l = 1-6. The correlation peaks at l = 2 and 5. The geoid has an anticorrelation with the slownesses at l = 2 and 3, and a positive correlation from l = 4 to 6.

Optical resonator with negative dispersion.

Abstract: Analysis of an optical ring resonator consisting of a prism and two mirrors demonstrates that such a resonator can have adjustable dispersion of either sign. The dispersion is proportional to the second derivative of the optical path length in the resonator with respect to wavelength. Adjustable dispersion may have important application to the production of ultrashort laser pulses.

Measurements of surf beat and set-down beneath wave groups

Abstract: A laboratory study was conducted to measure the amplitudes of long waves In shallow water as induced by wave grouping. In a 55 m long wave channel with a plane beach at the end, two primary waves of nearly equal frequency were generated. Due to a sophisticated control of the wave paddle - including second order wave generation as well as active wave absorption at the paddle face - the wave action at the difference frequency was limited to an incident forced wave, propagating at the group velocity, and a reflected free wave generated in the surf zone. For the incident forced - or bound - wave, also known as set-down, the experimental results show good agreement with the existing theory. Furthermore, the experiments confirm qualitatively a theoretical model by Symonds et al. (198 2) explaining two-dimensional surf beat as a result of the time-varying breakpoint of the incident primary waves.

Stimulated Raman crosstalk in optical transmission: effects of group velocity dispersion

Abstract: The nonlinear crosstalk which can degrade monomode optical transmission systems which use wavelength-division multiplexing is analysed, taking into account the influence of group velocity dispersion.

Cold plasma waves

Abstract: 1 Elementary properties of a plasma- Plasma- Equations of drift motion- Isothermal atmosphere in equilibrium- Types of wave- Effect of collisions- The continuity equations- 2 Maxwell's equations- Equations in terms of current and charge densities- Equations in terms of electric moment per unit volume- The exponential wave function- The concept of a dispersion relation- Calculation of the dispersion relation (electric current method)- Calculation of the dispersion relation (electric moment method)- 3 Isotropic plasma- Mobility and conductivity of an isotropic plasma- Susceptibility and dielectric constant of a collisionless isotropic plasma- The plasma frequency- Refractive index of a collisionless isotropic plasma- Wave dispersion in a collisionless isotropic plasma- Effect of collisions in an isotropic plasma- Importance of ordered kinetic energy in a plasma- Poynting's theorem in a plasma- The energy significance of the complex dielectric constant of an isotropic plasma- 4 Alternating current in a magnetoplasma- Mobility tensor for a magnetoplasma- Conductivity tensor for a magnetoplasma- Low-frequency conduction properties of an infinite homogenous magnetoplasma- Low-frequency conduction properties of a slab of magnetoplasma- Effect of plasma scale on wave propagation- 5 General properties of phase propagation in a magnetoplasma- Susceptibility tensor for a magnetoplasma- Alternative expressions for the susceptibility tensor components in the absence of multiple ion species- Dispersion relation for a magnetoplasma- Elliptic polarization- Alternative derivation of the dispersion relation for a magnetoplasma- The radio and hydromagnetic approximations- Effect of collisions in a magnetoplasma- 6 General properties of group propagation in a magnetoplasma- Frequency and angular spectra- Velocity of a wave packet- Relation between phase and group propagation- Method for calculating group velocity in a magnetoplasma- Formulae for group velocity in a magnetoplasma- Beam radiation in a magnetoplasma- 7 Propagation of phase along the imposed magnetic field- Circular polarization- The dispersion relation for longitudinal propagation- Longitudinal Alfven waves- The violin-string approach to longitudinal Alfven waves- The hydromagnetic approximation for longitudinal propagation- The radio approximation for longitudinal propagation- The Eckersley approximation for longitudinal propagation- Comparison of approximations- Pass and stop bands of frequency for longitudinal propagation- Particle vibration for longitudinal propagation- Plasma motion in a longitudinal Alfven wave- Longitudinal propagation in low-density and high-density magnetoplasmas- Effect of collisions on longitudinal propagation- Effect of an additional ion species on longitudinal propagation- Pass and stop bands of ionization density for longitudinal propagation- 8 Energy flow and group velocity for longitudinal propagation- Electromagnetic energy density for longitudinal propagation- Kinetic energy density for longitudinal propagation- Energy flow and group velocity for longitudinal propagation- Energy in a longitudinal Alfven wave- Faraday rotation for longitudinal Alfven waves- A resonator for longitudinal Alfven waves- The mode of operation of a hydromagnetic violin-string- Freezing of the magnetic field in the plasma (longitudinal Alfven waves)- Energy in a longitudinal whistler wave in the band ?Mi ? ? ? ?Me- A resonator for longitudinal whistler waves in the band ?Mi ? ? ? ?Me- Freezing of the magnetic field in the electron gas (longitudinal whistler wave)- Solid-state plasmas- 9 Propagation of phase transverse to the imposed magnetic field- The O wave- The X wave- Superposition of the O and X waves- Pass and stop bands of frequency for transverse propagation- The hybrid resonant frequencies- Transverse propagation in a low-density magnetoplasma- Pass and stop bands of ionization density for transverse propagation- Effect of collisions on transverse propagation- 10 Elliptic polarization of the X wave for transverse propagation- The electric ellipse for transverse propagation of the X wave- Frequency dependence of the electric ellipse- Particle vibration for transverse propagation of the X wave- Plasma compressions and dilations for transverse propagation of the X wave- Non-reciprocity- 11 Energy behaviour of the X wave for transverse propagation- Electromagnetic energy density for transverse propagation of the X wave- Kinetic energy density for transverse propagation of the X wave- Energy flow and group velocity for transverse propagation of the X wave- A resonator for transverse Alfven waves- The mode of operation of a hydromagnetic organ-pipe- Freezing of the magnetic field in the plasma (transverse Alfven waves)- 12 Propagation at any angle to the imposed magnetic field- The zeros in the frequency dispersion curves- Nomenclature for the characteristic waves- The cross-connection phenomenon for frequency dispersion curves- Frequency dispersion curves for nearly transverse propagation- Frequency dispersion curves for nearly longitudinal propagation- The elliptic polarizations of the O and X waves at the plasma frequency- Effect of an additional ion species on cross-connection phenomena- The infinities in the frequency dispersion curves- Permitted regions for the frequency dispersion curves- The cross-connection phenomenon for ionization dispersion curves- Permitted regions for the ionization dispersion curves- Propagation into a magnetoplasma from free space- 13 The radio approximation- The radio approximation to the dispersion relation- Frequency dispersion curves in the radio band- Frequency dependence of elliptic polarization in the radio band- Frequency dependence of the direction of group propagation in the radio band- Variation in the angle of squint of a rotating broadside antenna in the radio band- Dependence of refractive index on ionization density in the radio band- Dependence of elliptic polarization on ionization density in the radio band- Dependence of the direction of group propagation on ionization density in the radio band- 14 The hydromagnetic approximation- The hydromagnetic approximation to the dispersion relation- Frequency dispersion curves in the hydromagnetic band- Effect of ionic collisions in the hydromagnetic band- The fit between the hydromagnetic and radio approximations- Frequency dependence of elliptic polarization in the hydromagnetic band- Frequency dependence of the tilts of the electronic and current ellipses in the hydromagnetic band- Frequency dependence of the direction of group propagation in the hydromagnetic band- Polar diagrams for group velocity in the hydromagnetic band- Dependence of refractive index on ionization density in the hydromagnetic band- Dependence of elliptic polarization on ionization density in the hydromagnetic band- Dependence of the direction of group propagation on ionization density in the hydromagnetic band- 15 The quasi-longitudinal and quasi-transverse approximations- The transition angle between the quasi-longitudinal and quasi-transverse approximations- The regions of validity for the first- order angular approximations- Importance of avoiding angular approximations that upset an infinity of a refractive index- The regions of validity for angular approximations of practical value- Accuracy of ?2n/??p2 using angular approximations- The quasi-transverse approximation when ? ? ?Mi- The quasi-longitudinal approximation when ? ? Max(?N,?Me)- The quasi-longitudinal approximation when ?Mi ? ? ?C2- Group propagation of the O wave in the pass band ? > ?N- Group propagation in the upper part of the whistler band [??1 ? > ??2- Radiation in the frequency band ?? > ? > ?N- Radiation in the frequency band ?N > ? > ?C1- Radiation in the frequency band ?Me > ? > ??1- Radiation in the frequency band ??1 > ? > ?Mi- Radiation in the frequency band ? < ?Mi- Frequency dependence of radiation resistance- The relation between beaming and guidance in a homogeneous magnetoplasma- The relation between beaming and guidance for the whistler wave when ??1 < ? ? Min (?N, ?Me)- The relation between beaming and guidance for the O wave when ? ? ?Mi- The relation between beaming and guidance for the combined O and X waves when ? ? ?Mi- Effect of energy absorption on Alfven guidance- Symbols- Index of subjects

Method and apparatus for acoustic dipole shear wave well logging

Abstract: The shear velocity of a formation traversed by a fluid-filled borehole is determined in the presence of a significant flexural mode arrival. A plurality of waveforms are obtained from a dipole acoustic investigation of the formation relative to a common location in the borehole. The flexural mode phase velocity as a function of frequency is determined. If a low frequency asymptote is identifiable, it is reported as the shear velocity. If a low frequency asymptote is not identifiable, however, a plurality of additional curves of the flexural mode phase velocity as a function of frequency are theoretically determined, based on respective estimated shear velocities. These theoretically determined curves are respectively fitted to the curve determined from the waveforms, until a satisfactory least error fit is achieved. The last-estimated shear velocity is reported as the shear velocity of the formation.

Long waves induced by short-wave groups over an uneven bottom

Abstract: Unidirectional and periodically modulated short waves on a horizontal or very nearly horizontal bottom are known to be accompanied by long waves which propagate together with the envelope of the short waves at their group velocity. However, for variable depth with a horizontal lengthscale which is not too great compared with the group length, long waves of another kind are further induced. If the variation of depth is only one-dimensional and localized in a finite region, then the additional long waves can radiate away from this region, in directions which differ from those of the short waves and their envelopes. There are also critical depths which define caustics for these new long waves but not for the short waves. Thus, while obliquely incident short waves can pass over a topography, these second-order long waves may be trapped on a ridge or away from a canyon.

Travel time inversion for three-dimensional p-wave velocity anisotropy

Abstract: Inversion method for estimating the three-dimensional isotropic seismic velocity structure developed by Aki and his colleagues is extended to include the effect of velocity anisotropy. Assuming weakly inhomogeneous anisotropic media with some symmetry in anisotropy, we reduce the number of unknown parameters. We present two sets of linearized equations of quasi P-wave travel time residuals suitable for inversion. One set is for a weakly anisotropic medium with a hexagonal symmetry about a horizontal axis, and the other for a medium with a spheroidal P-wave velocity surface. Using these equations, we invert P-wave travel time residual data to estimate the isotropic and anisotropic velocity perturbations in the subdivided three-dimensional blocks within the modeling space. For local earthquakes, the hypocentral parameter perturbations should also be included in the inversion. This three-dimensional anisotropic velocity inversion is a natural extension of the previous isotropic velocity inversion. The three-dimensional anisotropic inversion method is applied to seismic data in Southwest Japan to determine velocity perturbations in the blocks which were well-determined in the previous isotropic study. The isotropic velocity perturbations are well determined, but the anisotropic velocity perturbations are less resolved. In spite of the rather poor resolution, our preliminary analysis shows some interesting features. Within the descending Pacific and Philippine Sea plates, the fast directions of P-wave velocity seem to be perpendicular to the magnetic lineations in the sea-floors near the trenches, suggesting that the descending lithosphere maintains the anisotropic structures which were formed before subducting.

Seismic velocity estimation

Abstract: The estimate of propagation velocities in the reflection or refraction seismic method is essential to the effective imaging of the subsurface. Wave propagation in a fully elastic medium gives rise to several propagation modes, among them are the longitudinal and transverse compressional waves (P-waves and S-waves), which are commonly used in the reflection and refraction seismic methods. With appropriately sorted data, the arrival time of a wave returning from a particular reflecting interface in the subsurface will vary as a function of source-receiver offset. This variation in arrival time with offset is called "moveout" and is controlled by the propagation velocity. The velocity of propagation enters the processing of reflection seismic data first as an essential parameter of a time coordinate transformation required before data of varying source-receiver offsets can be stacked to enhance signal-to-noise ratio. Velocities estimated in this manner are called "moveout velocities." Velocities also enter the processing sequence as a parameter of an imaging operation called "migration." Early efforts at velocity estimation were only accurate enough to provide parameters to process data, but high-quality data collected using present-day technology allow us to make accurate enough estimates of propagation velocity to infer subsurface geology. Complications arise, however, due to the effects of reflector structure and lateral velocity gradients. Current developments in seismic velocity estimation include measurement of shear wave (S-wave) velocities, use of wide-angle arrivals for more accurate P-wave velocity estimates, and methods requiring areal coverage (three-dimensional seismic).

Turbulent current measurements in a wind-wave tank

Abstract: Laboratory experiments were conducted in a wind-wave tank to study the turbulence characteristics in the air and wave boundary layers generated by wind and waves. The turbulent air boundary layer over the water surface was surveyed using two hot-wire probes. It was found to be a good simulation of an atmospheric boundary layer over a body of water. Capacitance probes were used to record the wave height. Spectra, wave phase velocity, dominant wavelength and frequency, and other statistics of the wind waves were computed. The water boundary layer below the water surface was surveyed using an array of 10 cross-element hot-film probes to measure the stream wise and vertical components of velocity. Various turbulence statistics including mean velocity, root-mean-square velocity fluctuations, Reynolds stress, dissipation rate, spectra, and higher-order statistics were computed. Similarity profiles were found for the mean subsurface current and the rms turbulent fluctuations in the streamwise and vertical directions. The velocity spectra indicated the relative importance of the wave-induced motion, which attenuated ex-pontentially with depth.

Wave‐current Models for Design of Marine Structures

Abstract: Modifications to vertical velocity profiles of coastal currents due to surface gravity waves were experimentally and theoretically examined. Results on these modifications were used to develop guidelines for approximating the form of current velocity profiles to be used in predictive models of wave kinematics. The validity of the guidelines was established by comparing the experimental data of wave particle velocities with the theoretical predictions of two wave‐current models which approximate the shear current by: (1) A constant vorticity current; and (2) a uniform velocity over the water depth. The results show that opposing waves decrease the current mean velocity close to the bottom and increase the mean velocity and the current shear near the free surface. Following waves increase mean velocity and velocity gradient of the current, close to the bottom, and might cause the shear at the surface to be negative. Models for constant vorticity and uniform velocity approximations are found to yield accurat...

Seismic to ultrasonic velocity drift: Intrinsic absorption and dispersion in crystalline rock

Abstract: An intrinsic velocity dispersion of 5% has been measured from 2 to 7 kHz in a water saturated crystalline rock. The porosity of the granite is 0.008 and consists of microcracks. Specific attenuation was observed to be weakly dependent on frequency over the same band. We also measured longitudinal and shear specific attenuation as a continuous function of water saturation at 1-2 kHz. The constant-Q model inadequately describes the dispersion. The dispersion mechanism is hydrodynamic relaxation of the microcracks. An effective pressure of 10 MPa reduces the dispersion to less than 1% per decade, or 4% from 1 Hz to 1 MHz. The intrinsic dispersion is insufficient to account for the discrepancies between seismic, sonic, and ultrasonic velocities which are observed in the field. The in-situ velocity drift is a factor of four greater than the maximum intrinsic dispersion. We conclude that such a velocity drift in crystalline rocks is caused by sampling resolution of macroscopic fractures.

Elementary derivation of the wake pattern of a boat

Abstract: An elementary derivation is given for the shape of the wake pattern of a boat traveling at constant velocity in deep water. We find essentially the same results as those found in standard treatments, but our mathematics is easier. All results follow from the fact that the group velocity is half the phase velocity.

Spreading of a relativistic wave packet

Abstract: We present a simple general proof that the spreading velocity of a relativistic free wave packet of de Broglie waves is limited. For a wide class of packets we confirm that the limit is the velocity of light, and we show how this limit is approached when the width Δp of the wave packet in momentum space tends to infinity and the minimum width σ(t=0) in ordinary space tends to zero.

Topographic and inertial waves on the continental rise of the Mid‐Atlantic Bight

Abstract: Analysis of an array of four closely spaced moorings, 2800 m water depth, instrumented by near-bottom current meters, has been performed with a view to isolating coherent wave motions in the low-frequency and inertial bands. The low-frequency motions are dominated by topographic Rossby waves similar to the results of previous studies of deep currents on the continental rise of the Mid-Atlantic Bight. Estimated wave parameters show good agreement with the linear topographic Rossby wave theory of Rhines (1970). Inertial oscillations are shown to be coherent over the array (horizontal separations ∼15–20 km). The peak frequency and phase and group velocity estimates are consistent with surface layer generation at a site north of the array. The relative vorticity of the mean currents and the low-frequency wave motions do not account for the observed shift of the near-inertial peak to frequencies ∼7% above f. An event of strong near-bottom inertial currents (amplitudes ∼10 cm s−1) appears to be attributable to the passage of Hurricane Belle (August 9–10, 1976) 24 days earlier, primarily because the vertical travel times are consistent with the estimates of vertical group velocity. Strong ringing inertial currents north of the array (amplitude ∼40 cm s−1), due to Hurricane Belle were observed at the shelf edge (Mayer et al., 1981) with peak frequencies comparable to the near-inertial frequencies observed at the array on the rise.

Wave-Particle Interactions

Abstract: Note: Magnetospheric studies of the interactions between plasma waves and particles have generally been performed using Gaussian units. To allow the material presented here to be conveniently used in conjunction with other published material on magnetospheric wave-particle interactions, we have not converted equations from the literature into MKS units. Thus all equations in this chapter are in Gaussian units.

On the Structure of the Velocity Field over Progressive Mechanically-Generated Water Waves

Abstract: The structure of the velocity field over a propagating wave of fixed frequency is examined. The vertical and horizontal velocities were measured in a transformed Eulerian wave-following frame of reference in a wind-wave research facility at Stanford University. Experimental results are given for seven different wind speeds in the range 140–402 cm s−1, with 1 Hz, 2.54 cm nominal amplitude, mechanically-generated sinusoidal water waves. The mean velocity profiles have a log-linear form with a wake free-stream characteristic. The constant C which characterizes these profiles decreases with increasing wind speed, as a result of the variation of surface roughness condition between the transition region and the fully rough regime. The wave-associated stresses with their main component at twice the fundamental wave frequency were found to be significant. Therefore, the nonlinear terms encountered in the wave-induced Navier-Stokes equations associated with these stresses cannot be neglected, and lineariz...