Showing papers on "Group velocity published in 1989"

Achromatic phase matching for second harmonic generation of femtosecond pulses

Abstract: A scheme is proposed by which the phase matching bandwidth can be increased by adjusting higher-order terms in frequency shift in the phase-matching condition. This is achieved by introducing a spectral angular dispersion so that the different spectral components propagate at their phase-matching angles. This is equivalent to canceling the group velocity mismatch. The equations describing the requirements on the angular dispersion are discussed and applied to the particular case of type I phase matching. A possible experimental setup is shown that would meet all the conditions required. >

Recovering the full velocity and density fields from large-scale redshift-distance samples

Abstract: A new method for extracting the large-scale three-dimensional velocity and mass density fields from measurements of the radial peculiar velocities is presented. Galaxies are assumed to trace the velocity field rather than the mass. The key assumption made is that the Lagrangian velocity field has negligible vorticity, as might be expected from perturbations that grew by gravitational instability. By applying the method to cosmological N-body simulations, it is demonstrated that it accurately reconstructs the velocity field. This technique promises a direct determination of the mass density field and the initial conditions for the formation of large-scale structure from galaxy peculiar velocity surveys.

Sensitivity of Near-Surface Shear-Wave Velocity Determination From Rayleigh And Love Waves

Abstract: Rayleigh and Love waves recorded on seismic-shot gathers can be used to determine the thickness and shear-wave velocity of shallow subsurface layers. After the data are transformed into the k-f domain, the dispersion curve for each of the phases can be picked from maxima on the contour plot. This dispersion curve is then inverted for the velocities and depths. Different frequencies in the dispersion curve yield information about different depths. The fundamental mode has proven to be of greater use than higher modes. Both Rayleigh and Love waves are easily inverted. However, the Love waves seem to yield information in a lower portion of the spectrum than the Rayleigh modes. Three examples are given from field experiments conducted near Canton, Texas.

Lg-wave propagation in heterogeneous media

Abstract: The Lg -wave phase, which is of considerable interest for nuclear discrimination problems, is normally observed after propagation through a few hundred kilometres. This phase is dominantly guided in the crustal waveguide, which is known to be a region with a very significant horizontal variability in properties. The effect of heterogeneous crustal structures on Lg waves has been determined by using a “coupled-mode” technique in which the local seismic wavefield in the real medium is expressed as a horizontally varying combination of the modal eigenfunctions of a stratified reference structure. Departures of the seismic properties in the medium from those of the reference medium lead to coupling between the various amplitude coefficients in the modal expansion. The evolution of these modal weighting factors with horizontal position are described by a coupled set of ordinary differential equations. This approach provides a calculation scheme for studying guided wave propagation over extended distances, at frequencies of 1 Hz and above. The heterogeneity models that have been used are two-dimensional and calculations are carried out for one frequency at a time. A sequence of models with varying levels of heterogeneity have been considered in order to determine the merits and limitations of the computation scheme. The coupled mode technique works well with heterogeneous models in which the local seismic velocities differ from the stratified reference model by up to two per cent and there are no significant distortions of the main discontinuities (e.g., the crust-mantle boundary). The approach can be used for higher levels of heterogeneity and with distorted interfaces but a large number of modes needs to be considered with consequent high computation costs. If the level of heterogeneity is not too large, the interaction between modes can be restricted, rather than extending over the whole mode set, with consequent reduction in computation cost. One of the major effects of crustal heterogeneity is to introduce the possibility of smearing out the main amplitude peak in the Lg wave train over a band of group velocities. As a result, an effective measure of the energy content of the Lg waves will be to consider the integrated amplitude along the trace between group velocities of 3.6 and 3.3 km/sec. The effects of heterogeneity vary between different parts of the Lg wave train and the representation of the wavefield in terms of modal contributions allow a detailed analysis in terms of the group velocity components, which can be illustrated by constructing theoretical seismograms (with a narrow bandwidth in frequency) for the heterogeneous models.

Effect of velocity ratio on plane mixing layer development

Abstract: An experimental study has been conducted to investigate the effect of velocity ratio on the development of a plane mixing layer to self-similarity. Plane mixing layers with five different velocity ratios (0.5, 0.6, 0.7, 0.8 and 0.9) were generated in a newly designed mixing layer wind tunnel with both initial boundary layers tripped. For each velocity ratio, mean flow and turbulence measurements were obtained at eight streamwise locations with a single cross-wire probe. The results indicate that the splittter plate wake plays a very dominant and, in some cases, a lasting role in the development of the mixing layer. For velocity ratios between 0.5 and 0.7, self-similarity of the mixing layer was observed with the asymptotic states comparable. Mixing layers with the higher velocity ratios failed to achieve a self-similar state within the measurement domain. The development distance decreased with increasing velocity ratio up to 0.7, after which it appeared to increase. Almost all of the observed effects may be attributed to the presence and subsequent behavior of the splitter plate wake.

Insights into the hydrodynamics of free falling wavy films

Abstract: On selectionne trois ondes isolees de differentes amplitudes et formes a partir de mesures sur un film liquide tombant. On calcule les champs de vitesse et de pression ainsi que la vitesse de l'onde

A comparison of finite-difference and fourier method calculations of synthetic seismograms

Abstract: The Fourier method, the second-order finite-difference method, and a fourth-order implicit finite-difference method have been tested using analytical phase and group velocity calculations, homogeneous velocity model calculations for disperson analysis, two-dimensional layered-interface calculations, comparisons with the Cagniard-de Hoop method, and calculations for a laterally heterogeneous model. Group velocity rather than phase velocity dispersion calculations are shown to be a more useful aid in predicting the frequency-dependent travel-time errors resulting from grid dispersion, and in establishing criteria for estimating equivalent accuracy between discrete grid methods. Comparison of the Fourier method with the Cagniard-de Hoop method showed that the Fourier method produced accurate seismic traces for a planar interface model even when a relatively coarse grid calculation was used. Computations using an IBM 3083 showed that Fourier method calculations using fourth-order time derivatives can be performed using as little as one-fourth the CPU time of an equivalent second-order finite-difference calculation. The Fourier method required a factor of 20 less computer storage than the equivalent second-order finite-difference calculation. The fourth-order finite-difference method required two-thirds the CPU time and a factor of 4 less computer storage than the second-order calculation. For comparison purposes, equivalent runs were determined by allowing a group velocity error tolerance of 2.5 per cent numerical dispersion for the maximum seismic frequency in each calculation. The Fourier method was also applied to a laterally heterogeneous model consisting of random velocity variations in the lower half-space. Seismograms for the random velocity model resulted in anticipated variations in amplitude with distance, particularly for refracted phases.

Mode competition and suppression in free electron laser oscillators

Abstract: The stability of single‐mode operation of free electron laser (FEL) oscillators is investigated. Two models of an untapered FEL oscillator are considered. The first model is called the klystron model. In the klystron model the FEL interaction occurs at two points: a prebunching point and an energy extraction point. In this model the nonlinear electron dynamics are solvable exactly, leading to a complex delay equation for the wave fields. The stability of single‐mode operation can then be determined easily as a function of a single‐pass gain, energy mismatch–frequency, and the difference between the group velocity of the radiation and the beam velocity. The second, more realistic model has a distributed interaction region of finite length: Stability of single‐mode operation in this device must be determined numerically. The results of the models in the low gain regime are compared and the parameter regimes where stable single‐mode operation is possible are determined. It is found that the time to reach a single‐mode state can be extremely long.

Wave Propagation in a Vertical Transversely Isotropic Medium: Field Experiment and Model Study

Abstract: Summary A down-hole experiment was carried out in the transversely isotropic Oxford Clay outcropping in the south of England. Different moveout curves for the two shear wave types and anomalous amplitude features for the SV-wave were found in the field data. Based on velocity measurements carried out formerly at the site a model study was performed to explain the results. Phase velocity and group velocity curves computed analytically with the method of characteristics, and synthetic seismograms computed with the Alekseev-Mikhailenko method, are presented. the field experiment and the model studies demonstrate that the occurrence of cuspidal triangles in the qSV-wavefront is an essential feature of wave propagation in transversely isotropic media. Even for weak transversely isotropic media there is a focusing effect into the direction of the cusp which leads to prominent shear wave amplitudes in this direction. Furthermore, we examined the effect of numerical anisotropy which can contaminate the synthetic seismograms. Velocity errors are one order of magnitude higher for shear waves than for compressional waves and increase with increasing Poisson's ratio. It was found that the error can be restricted to less than 1 percent only if using a spatial sampling of three times higher than a value that would generally be regarded as sufficient in finite difference computations.

Observations of motional electromagnetic fields during EMSLAB

Abstract: The long-period (>1 day) behavior of the seafloor electromagnetic fields during EMSLAB is considered in detail with an emphasis on interpretation in terms of oceanic motions. The study begins with a summary of the physics of motional electromagnetic induction, in which the seawater conductivity-weighted, vertically integrated velocity measured by the horizontal electric field is emphasized. Using frequency-domain methods, it is shown that seafloor and terrestrial magnetic variations have similar spectral shapes, indicating a common origin, but the seafloor electric field is not consistent with either at periods longer than 4 days, suggesting an oceanic source. The magnetic field variations are highly coherent across the EMSLAB array at periods shorter than 9 days, but the similarity decreases at longer periods, probably due to long-term instrumental drift. The seafloor horizontal electric field data exhibit the broadband coherence characteristic of ionospheric sources only at periods shorter than 1–2 days and are essentially incoherent between 4 days and a week except in association with certain propagating wave phenomena. The distinction between the depth-averaged velocity inferred from the horizontal electric field and its point measurement by the vertical electric field is demonstrated with data. Some specific properties of the seafloor electric fields are then considered in detail. The first of these is a 4-day wave observed to propagate from north-to-south along the east flank of the Juan de Fuca Ridge. The wavelength, propagation sense, and association with the ridge are all consistent with topographically trapped Rossby wave behavior at a zero in the group velocity. A second north-to-south propagating wave is seen at periods of 8–12 days in the middle of the EMSLAB area with a probable wavelength of order 1000 km. Unresolved low-frequency structures in the electric field are also mentioned. These observations clearly demonstrate the power of electromagnetic array methods for the study of long-period oceanic behavior.

Wave growth and spreading of a turbulent spot in plane Poiseuille flow

Abstract: Previous numerical simulations have shown that waves at the wing tips of plane Poiseuille spots grow rapidly as they are overtaken by the turbulent region. Kinematic wave theory is used to investigate the growth mechanism by calculating the amplitude of waves along rays defined by the group velocity. A dispersion relation is found by solving an extended form of the Orr–Sommerfeld equation using spatially averaged mean velocity profiles. The results indicate that the waves grow because of a combination of wave energy focusing and ‘‘cross‐flow’’ instability. The exponential growth resulting from the ‘‘cross‐flow’’ inflectional profiles is the dominating factor. Good agreement is obtained between the characteristics of the waves calculated by the theory and those found in the numerical simulation.

Seismic migration in elliptically anisotropic media1

Abstract: The study of wave propagation in media with elliptical velocity anisotropy shows that seismic energy is focused according to the horizontal component of the velocity field while the vertical component controls the time-to-depth relation. This implies that the vertical component cannot be determined from surface seismic velocity analysis but must be obtained using borehole or regional geological information. Both components of the velocity field are required to produce a correctly focused depth image. A paraxial wave equation is developed for elliptical anisotropic wave propagation which can be used for modelling or migration. This equation is then transformed by a change of variable to a second paraxial equation which only depends on one effective velocity field. A complete anisotropic depth migration using this transformed equation involves an imaging step followed by a depth stretching operation. This allows an approximate separation or splitting of the focusing and depth conversion steps of depth migration allowing a different velocity model to be used for each step. This split anisotropic depth migration produces a more accurate result than that obtained by a time migration using the horizontal velocity field followed by an image-ray depth conversion using the vertical velocity field. The results are also more more » accurate than isotropic depth migration and yield accurate imaging in depth as long as the lateral variations in the anisotropy are slow. « less

Compensation of dispersion in optical fibers for the 1.3-1.6 mu m region with a grating and telescope

Abstract: The transmission of ultrashort optical pulses over long distances in optical fibers is limited by pulse broadening due to group velocity dispersion. A grating and telescope dispersion compensator with group velocity dispersion of equal magnitude and opposite sign can compensate for the fiber dispersion. The possible benefits of such dispersion compensation in the 1.3-1.6- mu m wavelength region are investigated. The results show that compensation of first-order dispersion at 1.55 mu m in a fiber with zero dispersion near 1.3 mu m is primarily limited by the second-order dispersion of the grating and the telescope compensator. For a wavelength slightly greater than the zero dispersion wavelength, both the first- and second-order group velocity dispersion can be canceled by the grating and telescope dispersion compensator, allowing transmission exceeding 100 Gb/s over 100 km. >

Doppler flow velocity distribution measuring apparatus

Abstract: An apparatus for measuring the velocity distribution in a fluid by a pulse-Doppler method using an ultrasonic waver or an electromagnetic wave. The observation wave is transmitted to and received from the fluid and the reflected echo from the fluid is received so as to obtain the Doppler frequency. A predetermined transverse section is scanned with the observation wave so as to obtain the Doppler velocity distribution in the transverse section. Since the Doppler velocity distribution only contains the component in the direction of the observation wave, the Doppler velocity distribution is cumulatively integrated in the direction orthogonal to the direction of the observation wave and the integrated values are differentiated again in the direction of the observation wave, thereby obtaining the component in the orthogonal direction by calculation. If the observation wave is an ultrasonic beam, it is possible to observe the velocity distribution of the bloodstream in the heart.

Solitons and related periodic pulse evolutions in a femtosecond ring dye laser

Abstract: The generation of solitons and other related periodic pulse evolutions in a passively mode-locked dye laser is controlled by adjustment of group velocity dispersion, self-phase modulation, and spectral filtering. Without spectral filtering, periodic pulse evolutions reminiscent of higher-order solitons are observed. The pulses differ from the classic solitons because of additional shaping mechanisms. With spectral filtering, pulses are generated that can be described analytically as true asymmetric N=2 solitons. The introduction of the filter appears to simplify the intracavity shaping mechanisms in such a way that the nonlinear Schrodinger equation becomes a good approximation for pulse generation in the laser. The remarkable stability achieved allows for accurate characterization and control. >

Full waveform inversion of reflection data

Abstract: The theory of a direct inversion method to obtain the P and S wave seismic velocities and the density of a horizontally, finely layered elastic medium from vertical and horizontal component plane wave seismograms in the tau-p (slant stacked) domain has been developed. The method is based on the downward continuation method of Bube and Burridge (1983), but plane wave seismograms with a range of ray parameters are used simultaneously to obtain the velocity and density profiles. A corrected acoustic approximation of the elastic case has also been developed with which one may obtain a model of the P wave velocity and density from vertical component seismic recordings. However, it requires the S wave velocity to be estimated a priori, for instance as a fixed fraction of the P wave velocity. Many direct inverse methods, including the Bube and Burridge procedure followed here, may lack stability when implemented numerically, specially when they are applied to real data which necessarily are band limited at both low and high frequencies. The possibility of a stable implementation of our methods has been investigated, firstly by using synthetic data for layered acoustic media containing a full range of low frequencies. In this case, the inverted velocity and density profiles are in good agreement with the test models. Secondly, the indeterminacy associated with the lack, in practice, of low frequencies has been studied, and a method has been devised to estimate the missing low-frequency data from travel time information. Finally, the inversion algorithm has then been applied to high-quality marine seismic reflection data obtained by the North Atlantic Transect (NAT) group in the deep ocean using an expanding spread profile (ESP). In this test, only the P wave velocity profile was estimated independently, because the move-out data cannot provide low-frequency information about density. Also, the “source wavelet” of the seismic system had to be estimated from the seismic recordings. A stable and plausible estimate of the velocity variation was obtained. Unfortunately, no independent measurement of this velocity structure is available with which to verify the method's accuracy.

Group-velocity-dispersion compensation of a passively mode-locked ring LiF:F 2 + color-center laser

Abstract: By using an intracavity prism sequence to control group-velocity dispersion, pulses as short as 180 fsec have been generated near 850 nm from a passively mode-locked, traveling-wave LiF:F2+ color-center laser.

Wave packets, group velocities, and rays in lossy media revisited

Abstract: A study prompted by discussions of the group velocity concept in absorptive media and the correctness of Maxwell's equation is presented. A simple model for a wave packet train is obtained by beating two waves with adjacent frequencies and wavenumbers. By extending frequencies and propagation vectors into the complex domain, it is shown that a consistent definition of real group velocity in absorptive media can be stated. Numerical examples are displayed, and theoretical and experimental aspects are briefly discussed. An extension of Hamilton's ray equations for absorptive media is given. This formalism is used for numerical ray tracing in an absorptive ionosphere. The arguments show that a physically meaningful definition of the group velocity in absorptive media is consistent with the Fermat principle and the special relativistic limitation on the speed of light. >

Determining angular velocity from two quadrature signals by squaring the derivative of each signal and taking the square root of the sum

Abstract: A method and an apparatus for determining angular velocity from two voltage signals which are produced by an angle transmitter (1) have a sinusoidal function with respect to the angle of rotation of a rotatable member and are shifted in phase by 90 angular degrees. The amount and sign of the angular velocity (ω) are determined by the division of the derivative of a first voltage signal with respect to time by the second voltage signal. In that signal range in which the second voltage signal has passages through zero, the derivative of the second voltage signal with respect to time is divided by the first voltage signal. Alternatively, the amount of the angular velocity (ω) is determined from the vector sum of the derivatives of the two voltage signals with respect to time. The sign of the angular velocity is determined from the two voltage signals and their derivatives.

Group velocity dispersion in asymmetric slab guides

Abstract: Calcul de la dispersion de la vitesse de groupe dans les guides d'onde multicouches asymetriques a fort guidage

Radar target velocity estimator

Abstract: A radar target velocity estimator apparatus and method for computing a radial velocity of radar targets from differences in Doppler frequency shift between pulse-returns of multi-pulse waveforms The velocity estimator uses Doppler frequency shift which is obtained from a finite impulse response (FIR) filter in combination with logarithm tables stored in read-only-memory (ROMs) to calculate the target's velocity The estimation process requires the calculation of each complex FIR filter value twice during a pulse repetition interval; once for returns of a leading set of radar pulses and then for a trailing set The estimated velocity is proportional to the phase difference between each corresponding pair of filter values The estimate is a function of the arctangent of the quotient of the in-phase component of the complex value divided by the quadrature component

Phase velocity estimation of diffusely scattered waves

Abstract: Two methods are investigated for estimating the phase velocity of diffusely scattered seismic waves simultaneously arriving from different azimuths and recorded by a two-dimensional array of seismometers. The Hankel spectrum is the average of the frequency-wavenumber (FK) spectrum over all azimuths, while the wavenumber spectrum is derived by integrating the FK spectrum around a contour of constant phase velocity, i.e., a circle centered on the origin in the wavenumber plane. If the conventional estimate of the FK spectrum using the covariance matrix of the seismometer signals is integrated, a closed form for both the Hankel spectrum and the wavenumber spectrum may be found; the two spectra are very similar, the wavenumber spectrum being equal to the Hankel spectrum times the wavenumber. In spite of this similarity, however, we find that the two formulations have significantly different behavior for small wavenumbers, i.e., high phase velocities. In both cases there is a highest (true) velocity that can be estimated from the spectral maximum for a given array aperture (“velocity cut-off”). The Hankel spectrum estimates too high a velocity; for true wavenumbers below a certain limit, infinite velocity is estimated. The wavenumber spectrum, on the other hand, estimates too low a velocity, and there is an upper limit on the estimated velocity. An example illustrating these difficulties for the two methods is given for teleseismic P coda of an event recorded at the NORESS array in southern Norway: in spite of the problems, the analysis is able to demonstrate that the coda consists of two components; a coherent P -wave component with a high phase velocity and a diffuse S -wave component of low phase velocity. The cut-off and bias problem are investigated by numerical simulation for the NORESS array using azimuthal averaging and synthetic signals. The results confirm and quantify the cut-off problem at low wavenumbers and indicate that wavenumbers estimated from the Hankel and wavenumber spectra maxima bracket the true wavenumber, with the Hankel spectrum estimate being low (phase velocity too high) and the wavenumber spectrum estimate high. The bias of both methods decreases with increasing wavenumber (decreasing phase velocity) and they are both asymptotically unbiased. The wavenumber spectrum has a superior performance at low wavenumbers (high phase velocity), but the Hankel spectrum gives superior results at high wavenumbers (low phase velocity). The product of the linear wavenumber (= 1/wavelength) and the array aperture define “high” and “low” wavenumbers; for low wavenumbers, the product is 1 or less. In an Appendix, we find absolute lower bounds on the cut-offs analytically. The problems could be mitigated by using high resolution methods.

Effects of optical guiding on sideband instabilities in a free-electron laser.

Abstract: The effects of optical guiding on sideband instabilities in a Raman free-electron laser (FEL) are studied numerically and experimentally. An axisymmetric two-dimensional (2D) computer code that includes the effects of space charge and diffraction in an overmoded waveguide is developed to simulate sideband growth in the Columbia University FEL, which generates radiation of millimeter wavelength. It is found in both the simulation and the experiment that the effect of refractive optical guiding, which slows down the radiation group velocity, shifts the sidebands away from the signal carrier. We also find numerically that refractive optical guiding enhances the filling factor of the electron beam and perturbs the electron distribution, and thereby increases the sideband growth rate. We show that the sideband growth rate can be depressed by tuning the FEL so that the real part of the effective index of refraction associated with the electron beam decreases. The effect of wiggler tapering on the sideband growth is also studied with the 2D code. A significant reduction in the sideband growth rate in an efficiency-enhanced wiggler has been demonstrated and is qualitatively consistent with experimental measurements.

Energy propagation in linear hyperbolic systems

Abstract: The concept of energy velocity for linear dispersive waves is usually given for a normal mode solution of the system as the ratio between the mean energy flux and the mean energy density. In the absence of dissipation this velocity is known to coincide with the corresponding group velocity. When dispersion is accompanied by dissipation, this interpretation is not correct since the group velocity loses its original meaning and can assume nonphysical values. In this note the relation between energy velocity and group velocity is derived for dissipative, uniaxial waves, governed by a linear hyperbolic system. An example is provided where the energy velocity is compared with the phase and group velocities.

Relativistic phase velocity transformation

Abstract: By utilizing the appropriate Doppler formulas, it is shown that phase velocities in both Galilean and relativistic kinematics transform according to rules different from the ordinary velocity addition laws. As corollaries, it is demonstrated that (i) phase velocities parallel to the inertial frames’ relative velocity transform according to the Lorentz velocity addition rule; (ii) regardless of its direction, the velocity of light in vacuum transforms as a relativistic particle velocity; and (iii) the relativistic phase velocity transformation rule reduces to the Galilean rule in the nonrelativistic limit.