Showing papers on "Group velocity published in 1988"

Front propagation into unstable states: Marginal stability as a dynamical mechanism for velocity selection

Abstract: In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ``wave number'' that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -\ensuremath{\gamma} ${\ensuremath{\partial}}^{4}$\ensuremath{\varphi}/\ensuremath{\partial}${x}^{4}$ to it. I predict that for \ensuremath{\gamma}(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form \ensuremath{\varphi}(x-vt) that propagates with the marginal stability velocity. For \ensuremath{\gamma}g(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts \ensuremath{\varphi}(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

Entropy and 2-D velocity distribution in open channels

Abstract: Equations based on the entropy concept have been derived for describing the two‐dimensional velocity distribution in an openchannel cross section. The velocity equation derived is capable of describing the variation of velocity in both the vertical and transverse directions, with the maximum velocity occurring on or below the water surface. Equations for determining the location of mean velocity have also been derived along with those, such as the entropy function, that can be used as measures of the homogeneity of velocity distribution in a channel cross section. A dimensionless parameter of the entropy function named the M number has been found useful as an index for characterizing and comparing various patterns of velocity distribution and states of open‐channel flow systems. The definition and demonstrated usefulness of this parameter indicate the importance and value of the information given by the location and magnitude of maximum velocity in a cross section, and suggest the need for future experime...

Distortion of femtosecond laser pulses in lenses and lens systems.

Abstract: Owing to the difference between the phase and group velocity, the pulse front may be delayed with respect to the phase front by several picoseconds when traversing a lens or a lens system. The delay is a parabolic function of the input radius. The delay is calculated for singlet lenses, achromats, compound lenses, telescopes. The effect may be two to three orders of magnitude larger than the broadening due to group velocity dispersion in the materials of the lenses.

Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber.

Abstract: The temporal response of a polarization-dispersive medium is derived for the case of a highly coherent source using principal states of polarization Experimental results in dispersion-shifted fiber at 155 μm confirm that the first-order effect of polarization dispersion is a difference in the time of flight for pulses launched in the two principal states, while the second-order effect on pulse propagation is an effective chromatic dispersion of opposite sign for these two states The latter effect is demonstrated by the compression of frequency-chirped pulses in one principal state and pulse broadening in the orthogonal state

Synthetic Aperture Radar imaging of ocean waves from an airborne platform: Focus and tracking issues

Abstract: There has been a controversial issue of many years standing in airborne Synthetic Aperture Radar (SAR) ocean imaging which this paper addresses and resolves. Investigators have been strongly divided on the reasons for apparent improvement in wave contrast in response to processor focus adjustment. The dispute has centered on two parameters of wave dynamics: orbital velocity and phase velocity. This paper shows that both orbital velocity and phase velocity are of fundamental importance in the SAR wave imaging problem. The first affects the phase of the received signal, leads to velocity bunching, and is scaled by the ratio of sensor altitude to sensor velocity. The second affects the magnitude of the received signal, leads to translation of wave features during image formation (observed as blurring in the image), and is scaled by the ratio of wave phase velocity to sensor velocity, thus becoming significant for airborne radars. This treatment of the phase velocity parameter is new. It is shown that focus adjustment, as a side effect, shifts image position. This explains why experiments have appeared to “prove” that focus adjustment may be optimised for wave movement. The paper shows that better performance is obtained by compensating individual looks for wave movement before look summation, while using nominal perfect focus just as for static scatterers. The work is applicable to a full ocean wave spectrum and does not depend on the details of the scattering mechanism itself.

Acoustic plasmons in a conducting double layer

Abstract: We have investigated the acoustic plasma branch present in the longitudinal spectrum of two spatially separated parallel quasi-two-dimensional conducting layers. Our approach is based on the dielectric theory and is completely analytical within the random-phase approximation. By means of a systematic analysis we have obtained several exact results concerning the plasma dispersion relation. In particular, we have derived an exact expression for ${c}_{p}$ the acoustic plasmon group velocity in terms of the effective masses, densities and geometrical parameters of the heterostructure. We find that when the two layers are identical the system always admits a branch of acoustic plasmons as undamped modes for any finite value of the distance between the layers.

A new technique for ultrasonic-nondestructive evaluation of thin specimens

Abstract: Combining standard FFT methods with conventional ultrasonics, a method has been developed for measuring the phase velocity, the group velocity and the attenuation in ultrathin specimens (submillimeter or subwavelength in thickness). A detailed description of this technique is given. The technique was used on four disparate materials: aluminum, an epoxy, a particulate composite and a graphite-fiber/epoxy composite. The method works equally well for thin or thick specimens, and for dispersive as well as nondispersive media.

Modeling Slow Phase Velocity Generation during Off‐Vertical Axis Rotationa

Abstract: 1. Rotation about an off-vertical axis (OVAR) causes continuous unidirectional nystagmus in darkness. An analysis of the dynamics of the nystagmus suggests that the continuous slow-phase velocity is generated by a signal that is an estimate of the velocity of a traveling wave pattern associated with the excitation and inhibition of the cells of the otolith maculae. The estimated velocity signal then excites the velocity storage integrator. 2. A mathematical model has been developed that shows how the velocity of the traveling wave might be estimated from patterns of otolith activation related to head position. The estimation of velocity is based on a "template matching" algorithm. It is assumed that the signal arising in each cell of the macula is delayed by a certain time (T). As the head rotates in the gravitational field, a delayed pattern representing a previous position of the head is available as a "template" that can be compared to the pattern associated with the present position of the head. 3. The delayed signal level for each cell is approximated from the present pattern by a spatial extrapolation in pattern space using information from the given cell and an adjacent one. The value of the displacement that minimizes the mean square error between the extrapolated and the delayed signal values over all cells gives a best estimate of head rotation (d) in time T. The estimated head velocity is proportional to the estimated head displacement (d) and inversely proportional to the delay time (T). 4. By using a linear spatial extrapolation function and assuming a uniformly spaced distribution of polarization vectors over 360 degrees, sinusoidal spatial patterns are obtained. The formula for the estimated head velocity (ŵ) reduces to a sinusoidal function of angular head velocity (w) and delay time (T). For T = 0.85 seconds, the model predicts that the steady state estimate of head velocity will rise as a function of stimulus velocity (w) to a peak value at w = 50 deg/sec. The estimate then declines for larger values of stimulus velocity (w). This type of behavior is observed in the slow-phase velocity characteristics of OVAR in monkeys. 5. The model predicts that when animals are tilted after prolonged rotation about a vertical axis, the estimate of head velocity is delayed relative to actual head velocity. This accounts for the delay in the buildup of slow-phase velocity during the initial second.(ABSTRACT TRUNCATED AT 400 WORDS)

Investigation of Lamb waves having a negative group velocity

Abstract: The propagation characteristics of Lamb waves in a solid plate are typically represented by a set of dispersion curves, which describe the Lamb‐wave phase velocity as a function of the product fd, where f is the acoustic frequency and d is the plate thickness. For certain modes, within a range of phase velocity and fd, it has been theoretically predicted that the associated group velocity could be negative, i.e., the energy transport is in the opposite direction to the phase velocity. In the present study, Lamb waves are generated via mode conversion from a water‐borne sound beam incident onto a flat brass plate. Measurement of the phase and group velocities of the Lamb waves of the S1 mode is performed for the fd range of 2.0–2.3 MHz‐mm. Comparison of the measured and computed values of phase and group velocities shows good agreement and clearly demonstrates that S1‐mode Lamb waves have a negative group velocity for fd=2.08–2.24 MHz‐mm.

Gravity–capillary rings generated by water drops

Abstract: A theory for water waves created by the impact of small objects such as raindrops on an initially quiescent body of water is established. Capillary and dissipative viscous effects are taken into account in addition to gravity. It is shown that the prevailing waves are in a mixed capillary–gravity regime around a wavenumber km which corresponds to the minimum value of the group velocity. The waves are described as function of time and distance by the linear superposition of two transient wave components, a ‘sub-km’ (k km) component. The super-km components prevail at a short distance from the drop, whereas only the sub-km ones remain at a larger distance. The relative time history of the wavetrain is independent of the size of the drop, and its amplitude is proportional to the drop momentum when it hits the free surface. The wave pattern is composed of a multiplicity of rings of amplitude increasing towards the drop location and is terminated by a trailing wave with an exponential decay. The number of rings increases with time and distance.

Transient second-harmonic generation: influence of effective group-velocity dispersion

Abstract: Second-harmonic generation using high-intensity ultrashort pulses is studied numerically. In a highly dispersive medium and for ultrashort (subpicosecond to femtosecond) fundamental pulses the effective group-velocity dispersion (GVD) as well as the group-velocity mismatch should be taken into account. The numerical results indicate that in this case the conversion efficiency is reduced, compared with the case when only the group-velocity mismatch is taken into account, and for somewhat larger values of effective GVD parameters it exhibits an oscillatory behavior. Fundamental and harmonic pulse shapes inside the medium are also calculated. The cases with the effective GVD taken into account show that inside the medium both of the pulses split into two or more pulses.

Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium

Abstract: The velocity of propagation of electromagnetic energy for a monochromatic plane-wave field in a causally dispersive dielectric, medium with absorption as described by the Lorentz model, is considered within the framework of Poynting’s theorem. A general, rigorous expression for the energy-transport velocity in a Lorentz medium with multiple-resonance frequencies is derived. From this rigorous result, an approximate expression for the energy velocity is obtained that is in a form that is independent of the medium model and so is likely to be applicable to general dispersive media.

Response of a floating sea ice sheet to a steadily moving load

Abstract: An extended analysis of previous experimental work (Takizawa, 1985) is made. Using the dispersion relation for flexural free waves in a floating ice sheet, a diagram that predicts the critical speed, at which the ice deflection is amplified markedly, is presented. An understanding of the group velocity provides one explanation for the question why two waves appear in front of and at the rear of the load. The two-dimensional deflection patterns obtained in the experiment are compared with the existing theoretical results. At high speeds the ice beneath the load is found not to be depressed but somewhat elevated. It is shown that viscous damping has substantial effects on the ice deflection profile. The wave number curves, which give the two-dimensional wave patterns, are also examined, and the results verify that a one-dimensional model quite adequately describes the wave characteristics in Takizawa's experiment.

Optical generation and detection of surface acoustic waves on a sphere

Abstract: Surface acoustic waves (SAW’s) have been optically generated and detected on metallic spheres The source was a focused yttrium aluminum garnet (YAG) laser operated in the Q‐switched mode and the probe was a sensitive optical heterodyne interferometer The curvature of the surface gives rise to dispersive effects The variation of the SAW group velocity is deduced from the evolving waveforms of pulses detected after propagating a few turns around the sphere The observation of SAW pulse waveforms at positions away from the pole (a point diametrically opposite to the source) reveals that the spectral components undergo a π‐phase shift when passing through the pole

Solitons in the region of the minimum group-velocity dispersion of single-mode optical fibers.

Abstract: Pulses launched with their central wavelength in the region of the minimum-dispersion wavelength of a single-mode optical fiber exhibit a distinct spectral splitting due to the nonlinearity. As a solitary wave evolves, the corresponding central wavelength of this component frequency downshifts while the dispersive wave is upshifted, in qualitative agreement with theoretical prediction.

The scattering of ultrasonic waves by multiphase polycrystals

Abstract: The theory of ultrasonic propagation in single‐phase polycrystals presented in previous articles [J. Acoust. Soc. Am. 72, 1021–1031 (1982); 73, 1160–1163 (1983)] is generalized to calculate the scattering coefficients and phase and group velocities of plane compressional and shear waves in multiphase polycrystals. The analytical calculation was done for macroscopically isotropic materials with phases of cubic, hexagonal, or orthorhombic single‐crystal symmetry in second‐order perturbation theory using the assumption that the relative mean quadratic deviation of the wavenumbers in the polycrystal considered from its homogeneous isotropic approximation is small. Numerical evaluation is carried out for a two‐phase material of α‐ and γ‐Fe.

Group Velocity and the Linear Response of Stratified Fluids to Internal Heat or Mass Sources

Abstract: A steadily maintained line heat or mass source turned on in an unbounded, steadily moving, uniformly stratified flow will in general create ever-increasing vertical displacements of the fluid. Lin and Smith viewed a maintained heat source as a train of heat pulses. A pulse occurring a time T before the observation time creates a negative displacement proportional to T−1 at the heat source position when T is large. They pointed out that superposing the pulse responses leads to a displacement that grows logarithmically with time. This paper uses group velocity arguments to recreate the gravity wave field a time T after a heat pulse. The T−1 decay of the displacement is shown to be a geometrical consequence of dispersion in two dimensions. The growing response to a maintained source can be understood as the result of energy being pumped into the gravity wave modes, whose group velocity is near zero, faster than it can spread in physical space due to dispersion. A steady response is shown to be possi...

Wave propagation in a fluid‐filled fracture — An experimental study

Abstract: A laboratory experimental study has been carried out to investigate the mode trapping characteristics of a fluid-filled fracture between two elastic solids. Using a small circular cylindrical receiver of 2.7 mm diameter, we were able to measure the wave motion directly inside a 2.8 mm thick fracture and to obtain array data for the propagating waves. The data was processed using Prony's method to give velocity of the wave modes as a function of frequency. The experimental results agree with the theoretical predictions quite well. Specifically, in a “hard” (aluminum) fracture where the shear velocity of the solid is greater than the fluid velocity, four normal modes were detected in the frequency range up to 2.4 MHz. Whereas in a “soft” (lucite) fracture where the shear velocity is smaller than the fluid velocity, four leaky-P modes were detected in the same frequency range. In both cases, a fundamental mode analogous to Stoneley waves in a borehole was detected. In particular, the velocity of this mode approaches zero in the low frequency limit, as indicated by the theory and confirmed by the experiment in a low frequency range down to 25 kHz.

Pulse Asymmetry in the Colliding-pulse Mode-locked Dye Laser

Abstract: The design and performance characteristics of a group velocity dispersion-compensated colliding-pulse passively mode-locked ring dye laser are discussed. Pulses as short as 19 fs have been observed and analyses of experimental autocorrelations—both intensity and interferometric—together with spectral data that show evidence of pulse asymmetry are presented.

Dispersion of waves in piano strings

Abstract: A recent survey of piano acoustics literature revealed an apparent lack of attention to various aspects of dispersion in piano strings, apart from some information on its effect on inharmonicity of piano tone partials. In this article, it will be shown how group velocity of transverse waves in piano strings can be measured as a function of frequency with the aid of a short‐time spectral analysis method. Examples of group velocity measurements appear to be essentially in agreement with the theoretical predictions based on a model of a flexurally stiff string. In addition, the relationship between the group and phase velocity, as a function of frequency, is also illustrated, indicating correspondence between theoretical predictions and experimental results. A short‐time spectral analysis of transverse string displacement as a function of time, monitored near the bridge, has revealed a quasiperiodic succession of frequency glides. This effect is due to dispersion and is particularly prominent in the lowest bass strings. However, the same type of analysis applied to the sound‐pressure signal of the corresponding radiated sound yielded somewhat different results. While frequency glides similar to those found in the string displacement spectra were partly evident in the sound‐pressure spectra, strong precursive components of longitudinal string vibration origin were found to dominate the higher frequency portion of the attack transient of the radiated sound.

Time-of-flight measurement of excitonic polaritons in a GaAs/AlGaAs quantum well

Abstract: We report the first experimental observation of excitonic polaritons in a GaAs/AlGaAs quantum well with incident light propagating parallel to the quantum well layer. We have built a time‐of‐flight measurement system which allows us to investigate low‐temperature optical properties of semiconductor waveguides with picosecond time resolution. This system has been used to measure propagation delay time of an incident light pulse transmitted through the quantum well. The delay time increases drastically near the photon energies resonant to the optical absorption lines of the quantum well excitons. This behavior shows that the group velocity of the light pulse decreases as a result of the formation of quantum well excitonic polaritons. The group velocity drops to 7×104 m/s at a heavy‐hole exciton absorption line at 6.0 K.

Phase velocity dispersion of a generalized metal-segment-loaded helix as used in broad-band traveling-wave tubes

Abstract: An analytical model for predicting phase velocity dispersion of tape helix circuits loaded with solid-type or vane-type metal segments is described. An accurate design formula for dispersion is derived from the concept of the developed tape helix, taking account of the thickness effects of the tape and the edge effects of the metal segments. Comparisons with measurements of four solid-type and one vane-type helices show about +or-1% accuracy, which is seen to be sufficient for practical applications, The model can also be used to calculate a wide variety of asymmetry effects in loading elements around a tape helix. >

Optical fibers with negative group-velocity dispersion in the visible.

Abstract: Dispersion measurements have been performed in optical fibers having a photoinduced refractive-index grating in the core. The results show that negative group-velocity dispersion can be obtained in these fibers over a frequency region of 500 MHz for wavelengths shorter than 550 nm.