Group Velocity Of Mode Research Articles

Group velocity of Rayleigh waves in the Antarctic region

We analyzed dispersion of intermediate and long period surface waves, recorded at permanent observatories in the Antarctic region, in a tomographic study based on group velocity. We considered Rayleigh waves from available records of all events which occurred during years 1991–1995, with latitude lower than 40°S and magnitude M≥5.0. We performed classic single-station dispersion analysis on the surface wave train to measure group velocity of the Rayleigh fundamental mode in the period range between 30 and 120 s. We iteratively applied two different techniques of filter analysis: a multiple-filter [Dziewonski et al., 1969] and a phase-matched filter algorithm [Herrin, and Goforth, 1977]. We used such dispersion measurements to compute two-dimensional maps of velocity anomalies in the region. We parameterized group velocity with linear splines, and inverted the linear system by means of singular value decomposition. Results are in significant agreement with earlier studies, but reach a considerably higher detail. Our model differentiates well among geologically different regions, such as the old East Antarctica craton, the accreted terranes of West Antarctica, and the oceanic lithosphere surrounded by the ring of mid-ocean ridges. We image slow areas corresponding to hot-spot regions, including a broad anomaly corresponding to recent volcanism in the Ross Sea, where the existence of a mantle plume has been proposed.

Robust diagnosis for the focal stability in time-reversal acoustics

For a time-independent medium, one can use stored probe pulses to focus on a specific location in time-reversal acoustics. However, the temporal variability of the ocean is expected to limit such a procedure. In the earlier low-frequency (445 Hz) experiments, we found that probe pulses up to 1 week old still produced a significant focus at the original probe source location. In the recent high-frequency (3500 Hz) experiments, however, stability of the focus was limited to less than an hour or even a few minutes, depending upon the oceanographic variability since the acoustic wavelength is much smaller than the one for 445 Hz. A quick and robust diagnosis for the focal stability appears to be the modal group velocity versus phase velocity (gp curves) derived from the sound-speed profile, where the phase speed is directly related to the mode number and the slopes of the curves separate different groups of modes. In this paper, we investigate the stability and scintillation of the focus observed during the May 2000 time-reversal experiment using the gp curves.

Upper Crustal Velocity Structure in Slovenia from Rayleigh Wave Dispersion

The inversion of surface-wave dispersion curves can provide information on the average elastic properties of the upper crustal layers that are usually poorly sampled by body waves. The broad band digital records of earthquakes which recently occurred in Slovenia and neighbouring regions are used to extract the group velocity of the fundamental mode of Rayleigh waves, using frequency-time analysis (FTAN). The obtained dispersion curves permit a good resolution for the velocity and the thickness of the upper crust. The thickness of the uppermost sedimentary layer varies between 4 and 6 km and its shear-wave velocity is less than 3 km/s. The lower sedimentary layer is 7 to 9 km thick and its shear-wave velocity ranges from about 3.05 km/s in eastern Slovenia, to about 3.25 km/s in western Slovenia. The shear-wave velocity in the crystalline layer is around 3.5–3.7 km/s in the eastern part, while in the western part it reaches a rather high value of about 3.85 km/s.

Asymptotics of the Critical Regimes of Internal Gravity Wave Generation

The problem of constructing an asymptotic representation of the solution of the internal gravity wave field excited by a source moving at a velocity close to the maximum group velocity of the individual wave mode is considered. For the critical regimes of individual mode generation the asymptotic representation of the solution obtained is expressed in terms of a zero-order Macdonald function. The results of numerical calculations based on the exact and asymptotic formulas are given.

Group velocity, energy, and polarization mode dispersion

The group-velocity concept of a complex of modes is investigated, and the findings are related to power and energy. A special case of interest is polarization mode dispersion. This approach permits a generalization of polarization mode dispersion to the case of polarization mixing of multiple modes. Further, it is shown that first- and second-order polarization mode dispersions are related to an energy matrix.

Changes of the acoustic modal phase velocity, group velocity, and interference distance in an eddy

In this paper, by using the modal dispersion equation with the phase-integral approximation, and considering an eddy (or water mass) as a sound channel disturbance, the effects of the undisturbed channel, cold-core eddy, and warm-core eddy on the acoustic propagation characteristics are discussed. According to the solutions of the dispersion equation, the relation is obtained between the modal phase velocity (Cn), group velocity (Un), and interference distance (Ln), and the eddy intensity. When the plane wave (with an incident angle α) travels toward the center of a warm-core eddy (disturbed intensity BM), ‘‘double channel phenomenon’’ will take place in case of sin2<th>α<BM<2(1−cos<th>α), and then the modal phase velocity and interference distance will have anomalous changes. The interference distance of normal mode has a minimum. The larger the number of modes and the higher the source frequency, then the more obvious the ‘‘hollow case’’ for interference distance. However, for any cold-core eddy, the modal phase velocity or the interference distance will be monotonically changed.

Effects of eddy currents on retarded modes of antiferromagnets

We investigate retarded modes of noninsulating antiferromagnets with a uniaxial anisotropy. The damping caused by the eddy currents and their effects on retarded modes are emphasized. We find that the influences of the eddy currents upon the dispersion properties of the surface and bulk modes are very obvious when the wave numbers are not large; and the damping reaches its maximum at the maximum of the group velocity of the surface mode. The dispersion relations for the surface and bulk modes and the attenuation total reflection are presented numerically.

S -wave Velocity Models in the Scotia Sea Region, Antarctica, from Nonlinear Inversion of Rayleigh Waves Dispersion

—More than 60 events recorded by four recently deployed seismic broadband stations around Scotia Sea, Antarctica, have been collected and processed to obtain a general overview of the crust and upper mantle seismic velocities.¶Group velocity of the fundamental mode of Rayleigh waves in the period between 10 s to 30–40 s is used to obtain the S-wave velocity versus depth along ten different paths crossing the Scotia Sea region. Data recorded by two IRIS (Incorporated Research Institutions for Seismology) stations (PMSA, EFI) and the two stations of the OGS-IAA (Osservatorio Geofisico Sperimentale—Instituto Antarctico Argentino) network (ESPZ, USHU) are used.¶The Frequency-Time Analysis (FTAN) technique is applied to the data set to measure the dispersion properties. A nonlinear inversion procedure, "Hedgehog," is performed to retrieve the S-wave velocity models consistent with the dispersion data.¶The average Moho depth variation on a section North to South is consistent with the topography, geological observations and Scotia Sea tectonic models.¶North Scotia Ridge and South Scotia Ridge models are characterised by similar S-wave velocities ranging between 2.0 km/s at the surface to 3.2 km/s to depths of 8 km/s. In the lower crust the S-wave velocity increases slowly to reach a value of 3.8 km/s. The average Moho depth is estimated between 17 km to 20 km and 16 km to 19 km, respectively, for the North Scotia Ridge and South Scotia Ridge, while the Scotia Sea, bounded by the two ridges, has a faster and thinner crust, with an average Moho depth between 9 km and 12 km.¶On other paths crossing from east to west the southern part of the Scotia plate and the Antarctic plate south of South Scotia Ridge, we observe an average Moho depth between 14 km and 18 km and a very fast upper crust, compared to that of the ridge. The S-wave velocity ranges between 3.0 and 3.6 km/s in the thin (9–13 km) and fast crust of the Drake Passage channel. In contrast the models for the tip of the Antarctic Peninsula consist of two layers with a large velocity gradient (2.3–3.0 km/s) in the upper crust (6-km thick) and a small velocity gradient (3.0–4.0) in the lower crust (14-km thick).

The energy velocity of attenuating guided ultrasonic waves in plates

The understanding of the propagation characteristics of guided waves is fundamental to their exploitation in nondestructive evaluation (NDE) techniques. Long-range testing methods often rely on the detection of the arrival of a wave packet which is understood to travel at the group velocity of the wave mode, and which is easily calculated. However, although in practice the group velocity is a perfectly adequate measure of the energy velocity of the wave packet, it is strictly correct only when the wave mode is lossless. The paper will examine the nature of the energy propagation vectors within a plate and the energy velocity of Lamb waves. Both elastic plates with lossless modes and plates with attenuation will be considered, the latter leading to differences between the group velocity and the energy velocity. Experimental results will be presented as validation, using a plate with material damping under conditions where the difference between group velocity and energy velocity is significant.

Vector solitons in the dynamics of anharmonic monatomic lattices

It is shown that three types of solitary acoustic waves can develop in anharmonic crystal lattices corresponding to the three branches of acoustic phonons. A system of three nonlinear Schrodinger equations is derived to describe this situation. For greatly different group velocities, the interaction between solitons reduces collisions between them. When the group velocities of the different acoustic modes in a lattice are close to one another, bound states of the corresponding types of solitary waves occur. Bound states of this sort are vector solitons, whose polarization varies along the pulse. If the transverse acoustic modes are degenerate in velocity, the situation is extremely similar to the propagation of pulses in optical fibers.

Electromagnetic mode density for finite quasi-periodic structures

The transmission properties of quasi-periodic Cantor-like and Fibonacci photonic bandgap structures that have the same optical paths are studied and compared with one-dimensional, finite, N-period stacks. The electromagnetic mode densities and group velocities are also discussed. We show that the density of modes at the edge of the bandgap is greater for Cantor-like multilayers, so at the band edge a sharp decrease in the group velocity is found.

Numerical analysis on inversed bottom geoacoustic models for predicting a shallow-water acoustic field

The bottom reflection coefficient as a function of the grazing angle is calculated for a liquid bottom with depth-dependent sound velocity, density, and attenuation. The bottom geoacoustic parameters of a simplified model, which describes only the depth dependence of sound velocity and treats the density and attenuation as the depth independent, are derived from the bottom reflection coefficients by the least-squares fit. The inversion accuracy in reproducing an acoustic field in the water column is tested by the performance of matched field processing, the sound propagation losses, modal group velocities, and modal eigenvalues for numerical examples. In general, the test results are good for single frequencies, but degrade for broader frequency bands. The possible distortion of the depth dependence of sound velocity from the inversion under the simplified model is discussed.

EurID: a regionalized 3-D seismological model of Europe

This paper presents a 3-D regionalized model of the European crust and mantle velocity structure down to depths of approximately 300 km. The model is entirely based upon the existing geological and geophysical literature and it incorporates topographic and bathymetric features. The resulting model, which we call EurID (European I-Data set), consists of approximately 6400 1-D vertical structures having an average separation of about 100 km. EurID is compared quantitatively with EUR-S91, a recent S-wave velocity tomographic model obtained from body and surface wave waveform inversion [Zielhuis, A., Nolet, G., 1994. Shear-wave velocity variations in the upper mantle beneath central Europe, Geophys. J. Int., 117: 695–715]. It is found that the S-wave velocities in EurID and in EUR-S91 (Zielhuis and Nolet, 1994) do not differ by more than about 4% over most of the area. To validate EurID, surface wave dispersion analysis is performed along some selected paths. The fundamental mode group–velocity curves obtained using 1-D average models derived from EurID are found in good agreement with those determined from observations. Broadband waveform synthetics are also calculated along the same paths and compared with the observed time series. Overall, we believe that EurID is representative of our current knowledge of the crust/upper mantle structure beneath Europe and it can be used to model realistically ground motion for periods above 10 s within structurally homogeneous regions. The first digital version of this structure database is available for public use.

Hydromagnetic wave propagation and coupling in a magnetotail waveguide

For some time the magnetotail has been considered as a possible region where hydromagnetic waves can propagate as waveguide modes. Recently, attention has turned to the magnetospheric flanks as waveguides, and much useful insight has been gained into propagation of fast waveguide modes there, and the structure of the field line resonances they can drive. We return to the magnetotail and investigate hydromagnetic wave propagation and coupling in a magnetotail waveguide. This problem is significantly different from the flank waveguide as the ambient magnetic field is directed along the waveguide rather than across. Field line resonances of the flank type are not possible in the lobe waveguide. We describe a numerical simulation of a model waveguide in which the Alfvén speed decreases across the waveguide to the central plasma sheet. The waveguide is stimulated by a short compressional perturbation located in the far tail. The cross‐tail spatial structure is chosen to give relatively weak coupling between fast and Alfvén modes so that phase and group velocities of uncoupled fast modes can be used to interpret the results. We find that the perturbation propagates dispersively down the waveguide in the form of fast waveguide modes. Fourier components with small parallel wavenumber contain most of the energy, and propagate relatively slowly toward the “Earth.” These act as moving sources which launch Alfvén waves continuously earthward. The wave dispersion relations are such that the waveguide modes couple with Alfvén waves only in a limited region of the transverse Alfvén speed gradient. The Alfvén waves travel at the local Alfvén speed along each field line, so that as they travel the wave on a given field line becomes increasingly out of phase with waves on adjacent field lines. The phase mixing in our model is novel in that it includes the effects of transverse gradients in both Alfvén frequency and parallel wavenumber which tend to cancel each other out. Nevertheless, the phase‐mixing process leads to increasingly fine transverse structure as the waves progress down the waveguide. The results are likely to be applicable in regions such as the plasma sheet boundary layer and the plasma mantle.

Numerical Studies of Two-Dimensional Selective Withdrawal in a Stratified Fluid.

Numerical investigations are made of establishment of two-dimensional selective withdrawal through a line sink of a linearly stratified Boussinesq fluid in a finite-depth duct. The flow is mainly characterized by a Froude number. When the Froude number is small, the internal wave modes of horizontal wavenumber and frequency 0 which are also called columnar disturbances propagate horizontally upstream against the induced uniform velocity and the selective withdrawal is established. We introduce the new method to specify the propagation of internal wave mode from numerically obtained velocity distribution. Using this method, we investigate its behavior especially focusing our attention on its vertical structure, amplitude and horizontal group velocity. The numerical results show that the amplitude becomes smaller as the Froude number or the mode number increases, and the group velocity of higher modes is inversely proportional to the mode number.

Ultrasonic Guided Wave NDT for Hidden Corrosion Detection

An experimental study of hidden corrosion detection by using ultrasonic guided waves is presented combined with a BEM numerical simulation. Both corrosion simulation specimens by machine cutting and real corrosion specimens by electrochemical processing were used in the investigation, with a range of corrosion depths from 0.02 to 0.4 mm (1.5 to 20% of the original plate thickness). Various wave modes were subsequently generated on these specimens to examine the implications of thinning on mode cutoff, group velocity, and transmission and reflection amplitudes. The transmission and reflection of guided waves upon entering the corrosion zone were simulated by a hybrid BEM calculation that combines a normal mode expansion technique of Lamb waves for far fields with the boundary element representation for the scattered near fields. A quantitative technique for hidden corrosion depth with guided waves is developed based on a frequency compensation concept. The estimated depth of the real corrosion by this method shows good agreement with that by an optical microscope.

Temperature evolution of the upper ocean in the Greenland Sea January to March 1989

Tomographic data obtained during early 1989 in the Greenland Sea have been analyzed at 4–8 hour resolution to give the range‐averaged vertical temperature evolution in the upper 500m for a 106km path. The tomographic inversions used both ray travel time data and normal mode group velocity data in order to maximize near‐surface resolution. Two major events are apparent in the results. The first is the warming of a cold (−1.9°C) 100m thick surface layer, and the second, 10 days later, is the cooling of a relatively warm (−0.9°C) subsurface layer between 300m and 500m depth. This warm subsurface layer is a critical source of salinity and buoyancy for deep convection. The surface layer warming is consistent with a mixed layer deepening over a portion of the path, bringing up water from below. Special Sensor Microwave Imager (SSM/I) ice data indicate that the local ice field disappears 3–4 days after the surface warming. The cooling of the warm 300m to 500m layer is also consistent with a vertical process. There is no ice cover at this time, and so surface heat fluxes are large. A northerly wind event occurs at the onset of the cooling of the 300–500 m layer, suggesting that wind‐induced mixing may have played a role in initiating the process. There is evidence of southward flow advecting warm water into the area both before and after the two events studied in detail here.

Ray/mode duality and turning point filters

In ocean acoustic tomography, the travel time of ray arrivals are used to infer the sound-speed field. In typically dispersive channels, the ray arrivals are followed by energy, which is more easily described by vertical modes. Using a vertical array, the analyst can separate the modal energy in order to construct a dispersion diagram; a measurement of mode group velocity versus frequency for each mode number. The difference between the measured and computed group velocities would then be used in a stochastic inverse for the sound-speed field. In the WKBJ approximation, the expression for mode group velocity can be written as a function of a single parameter, mode axial inclination angle instead of the two parameters, mode number, and frequency. This fact suggests that information from modes with the same inclination angle (or equivalently the same turning point depths), ought to be combined. A turning point filter can be built that does exactly this. Perhaps it is not surprising that nature constructs rays from modes with the same turning point depths also. There is some advantage in this method over standard linear beamforming. [Work supported by the Strategic Environmental Research and Development Program through ARPA.]

Interpretation of frequency-dependent transmission-loss interference patterns

Experimental observations of broad-band acoustic propagation in a known geological region of the Atlantic Generating Station (AGS) site (1994) has prompted a new approach to understanding frequency-dependent behavior in shallow-water regions. A modal-based theory is presented to explain quantitatively the interference patterns of transmission loss versus frequency observed in the experimental data. It is shown that the higher modes are responsible for observed interference patterns and that these can be related to modal group velocities using an ideal waveguide model. This may provide new insights applicable to existing inverse techniques.

Group Velocity Measurement of Surface Waves by the Wavelet Transform.

A new time-series analysis called is applied to measure the group velocity of surface waves as compared with that obtained by the conventional Fourier transform. We use vertical-component Rayleigh waves for both synthetic seismograms and GDSN long-period data of oceanic paths. The results of this study are summarized as follows: for synthetic seismograms, moving-window analysis using the Fourier transform can measure the group velocity of the fundamental mode correctly, while the group velocity of the first-higher mode is systematically larger than the correct value. In contrast, the wavelet transform measures the group velocity of both modes precisely although the resolution in frequency may not be sufficiently high. For GDSN data propagating along the Pacific Ocean, both methods provide stable results for the group velocity of the fundamental mode in the period range of 20 to 100 s. Using the Fourier transform, we obtain the group velocities of the first-higher mode between 20 and 40 s although these values seem unreliable. In contrast, the wavelet transform can measure both modes precisely in the period range of 20 to 100 s for non-shallow events and even for shallow events with relatively small noise in the data. Another advantage of the wavelet analysis is that we can specify resolving power in group velocity measurement rigorously.