Showing papers on "Group velocity published in 1993"
TL;DR: In this article, the dispersion of Yanai waves in a reduced gravity equatorial model is studied and the usefulness of the wavelet transform is demonstrated by measuring the group velocity directly over a finite range of wavenumbers by examining the time evolution of the transform.
Abstract: Wavelet analysis is a relatively new technique that is an important addition to standard signal analysis methods. Unlike Fourier analysis that yields an average amplitude and phase for each harmonic in a dataset, the wavelet transform produces an instantaneous estimate or local value for the amplitude and phase of each harmonic. This allows detailed study of nonstationary spatial or time-dependent signal characteristics. The wavelet transform is discussed, examples are given, and some methods for preprocessing data for wavelet analysis are compared. By studying the dispersion of Yanai waves in a reduced gravity equatorial model, the usefulness of the transform is demonstrated. The group velocity is measured directly over a finite range of wavenumbers by examining the time evolution of the transform. The results agree well with linear theory at higher wavenumber but the measured group velocity is reduced at lower wavenumbers, possibly due to interaction with the basin boundaries.
529 citations
TL;DR: In this article, the authors investigated the propagation, growth and absorption of electromagnetic ion cyclotron waves in the Pc 1 frequency range using the HOTRAY ray tracing program for a realistic distribution of thermal plasma (H+, He+ and O+) that is assumed to be in diffusive equilibrium inside the plasmasphere and collisionless in the low-density region outside the Plasmapause.
Abstract: The propagation, growth and absorption of electromagnetic ion cyclotron waves in the Pc 1 frequency range is investigated using the HOTRAY ray tracing program for a realistic distribution of thermal plasma (H+, He+ and O+) that is assumed to be in diffusive equilibrium inside the plasmasphere and collisionless in the low-density region outside the plasmapause. Free energy for L- mode wave growth is provided by a bi- Maxwellian distribution of energetic H+ and O+ with a temperature and density modelled on satellite observations. Solutions to the hot plasma dispersion relation show that inside the plasmasphere the spatial growth rates are small whereas they increase outside the plasmapause with increasing L shell. Ray tracing shows that inside the plasmasphere guided L- mode waves only grow during one crossing of the magnetic equator and only achieve small path-integrated wave gain (≤ 2 e- foldings). At the plasmapause the density gradient enables guided mode waves to grow during several equatorial crossings and the net path-integrated gain is much larger (≃ 8.7 e-foldings). For the largest observed ring current densities of 4 × 106 m−3 at L = 4 the gain is above the critical level (10 e-foldings) for amplification to observable levels. Just outside the plasmapause the waves only grow during the first equatorial crossing and the gain is smaller. In the absence of nonconvective instabilities the path- integrated amplification of the guided mode tends to increase with L shell and reaches the critical level for observable waves only in the outer magnetosphere (L ≥ 7). Unguided L- mode waves have very small wave gain. For L ≥ 7 the plasma beta becomes large (β⊥ > 1) and should lead to the onset of nonconvective instabilities. However, we suggest that inhomogeneities in the medium and quasi-linear scattering will prevent absolute instabilities from occurring and that in reality the waves are propagating with very low group velocities. We suggest that the waves observed by Anderson et al. (1990, 1992a, b) beyond L = 7 near local noon are influenced by the enhanced wave gain due to these very low group velocity waves.
234 citations
TL;DR: In this article, the authors proposed to add artificial selective damping terms to the finite difference scheme to purge the short waves so as to improve the quality of the numerical solution, and demonstrated the effectiveness of such damping coefficients by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.
Abstract: The feasibility of performing direct numerical simulations of acoustic wave propagation problems has recently been demonstrated by a number of investigators. It is easy to show that the computed acoustic wave solutions are good approximations of those of the exact solutions of the linearized Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves are contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration. Numerical results of direct simulations of these waves are reported. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.
171 citations
TL;DR: In this paper, a morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel, and a weakly nonlinear analysis is presented which results in a GinzburgLandau equation.
Abstract: A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a GinzburgLandau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.
142 citations
TL;DR: Two theorems on the group velocity are presented, one of which is more widely applicable to propagation in waveguides or through multilayer dielectrics, and the other to dispersion in a medium with two resonance lines, where the group speed is abnormal within the absorption line and outside the gain line.
Abstract: Two theorems on the group velocity are presented in this paper. First a simple proof is given that for any dispersive dielectric, there must be a frequency at which the group velocity of an electromagnetic pulse becomes abnormal, i.e., greater than the vacuum speed of light, infinite, or negative. Second, at the frequency at which the attenuation (or gain) is a maximum, the group velocity must be abnormal (or normal). This second theorem is more widely applicable, e.g., to propagation in waveguides or through multilayer dielectrics. To illustrate these theorems we discuss dispersion in a medium with two resonance lines, one absorption and the other gain. We find that the group velocity is abnormal within the absorption line and in a transparent region outside the gain line.
142 citations
TL;DR: It can be concluded that the cross correlation technique behaves superiorly especially for a low velocity dispersion, and the standard deviation of the velocity estimate decreases for an increasing sample volume length and package length, while the performance of the conventional Doppler technique is rather independent of the length of the sample volume.
Abstract: In pulsed Doppler systems the received RF (radio frequency) signal is multiplied by a quadrature reference signal and subsequently averaged over a short depth range to obtain a sample of the complex Doppler signal. The mean frequency of the sampled Doppler signal, obtained with the autocorrelation function, reflects the mean velocity of the scatterers moving through the sample volume. An alternative is to evaluate the two-dimensional cross correlation function of a short segment of the RF signals over subsequent lines, giving the mean velocity of the scatterers. Both methods of velocity estimation were applied to computer-generated RF signals with varying RF bandwidth, signal-to-noise ratio, and mean and width of the imposed velocity distribution. The length of the RF signal segment and the number of lines for velocity estimation (package length) affects the accuracy of the velocity estimate. It can be concluded that the cross correlation technique behaves superiorly especially for a low velocity dispersion. Furthermore, the standard deviation of the velocity estimate decreases for an increasing sample volume length and package length, while the performance of the conventional Doppler technique is rather independent of the length of the sample volume. The difference between both techniques decreases for a greater package length or for signals simulating a wide velocity distribution.
112 citations
TL;DR: If the amplitude of the vibrations is increased, the properties of low-amplitude vibrations in dry unconsolidated granular materials are investigated and nonlinearity near the point at which the authors first see hysteretic behavior is observed.
Abstract: We have investigated the properties of low-amplitude vibrations in dry unconsolidated granular materials. The velocity of sound can vary by a factor of 5 depending upon whether one measures either the arrival time of the rising edge of the pulse or the quantity analogous to the group velocity. If we increase the amplitude of the vibrations, we observe nonlinearity near the point at which we first see hysteretic behavior. This is interpreted as due to the presence of force chains. We also find frequency shifts in all of the features that characterize the transmission spectrum.
111 citations
TL;DR: These measurements include a delineation of the unstable region around zero dispersion that is predicted by a recent theoretical treatment of Kerr-lens mode locking.
Abstract: We report the operating characteristics of a Kerr-lens mode-locked Ti:sapphire laser as the intracavity dispersion is varied continuously from negative to positive values. These measurements include a delineation of the unstable region around zero dispersion that is predicted by a recent theoretical treatment of Kerr-lens mode locking. A detailed comparison with the theory is made.
109 citations
TL;DR: In this paper, the authors investigated the performance of an active control of bending wave propagating along a beam using an adaptive digital filter, where the coefficients of the filter were adjusted to minimize the mean square output from a single downstream error sensor (another accelerometer).
68 citations
TL;DR: It is shown that intracavity group-velocity dispersion compensation with the use of prisms composed of conventional optical materials can be accomplished while simultaneously eliminating the round-trip cavity cubic phase.
Abstract: We show that intracavity group-velocity dispersion compensation with the use of prisms composed of conventional optical materials can be accomplished while simultaneously eliminating the round-trip cavity cubic phase. The ability to compensate perfectly both second- and third-order dispersion exists for pulses whose central wavelengths lie within a range that depends on the prism and laser rod materials as well as on the prism angles. In the case of Ti:sapphire and Cr:LiSrAlF6 lasers, Brewster prisms composed of readily available materials can be used to compensate for both group-velocity dispersion and cubic phase over much of the respective tuning ranges.
66 citations
TL;DR: In this article, the authors combined the Kramers-Kronig dispersion relation for n and the classic Sommerfeld-Brillouin argument to show that either n( infinity ) < 1, or the imaginary part of n is negative at least for some ranges of frequency, in which case the vacuum between the mirrors fails to respond to a light probe.
Abstract: Because it is scattered by the zero-point oscillations of the quantized fields, light of frequency omega travelling normally to two parallel mirrors experiences the vacuum between them as a dispersive medium with refractive index n( omega ). An earlier low-frequency result that n(0)<1 is combined with the Kramers-Kronig dispersion relation for n and with the classic Sommerfeld-Brillouin argument to show (under certain physically reasonable assumptions) that either n( infinity )<1, in which case the signal velocity c/n( infinity ) exceeds c; or that the imaginary part of n is negative at least for some ranges of frequency, in which case the vacuum between the mirrors fails to respond to a light probe like a normal passive medium. Further, the optical theorem suggests that n exhibits no dispersion to order e4, i.e. that n( infinity )=n(0) up to corrections of order e6 at most.
TL;DR: In this article, the authors apply the Huygens method and wave theory to plane-wave propagation through a 2D acoustic medium with Gaussian or exponential autocorrelation function of the slowness fluctuations and derive a correction factor which approximately transforms plane wave 2D into spherical-wave 3D velocity shifts; the latter correspond to 3D two-point ray tracing.
Abstract: SUMMARY Seismic waves in a random medium (with standard deviation E and correlation distance a of the relative slowness fluctuations) prefer fast paths, and therefore the apparent velocity of wave propagation is larger than the velocity which corresponds to the volume average of slowness. This velocity shift can be determined by ray perturbation theory (Snieder & Sambridge 1992), by the Huygens method (Podvin & Lecomte 1991) and by wave theory (Muller, Roth & Korn 1992). We apply all three methods to plane-wave propagation through a 2-D acoustic medium with Gaussian or exponential autocorrelation function of the slowness fluctuations. Ray perturbation theory gives numerical and analytical results, but has path-length (L) limitations. The Huygens method, which also gives the ray-theoretical velocity shift, can be used for Lla ratios of seismological interest. Wave theory shows that the velocity shift also depends on the wavelength A and that for Ala less than about 0.1 the velocity shift agrees with the result of the Huygens method. For Ala = 1 the wave-theoretical (i.e. true) shift is lower than the Huygens-method shift by a factor of 0.25 to 0.5. Simple formulae for the E dependence of the Huygens-method shift at long path lengths (Lla 2 80) are given, and a correction factor is derived which approximately transforms plane-wave 2-D into spherical-wave 3-D velocity shifts; the latter correspond to 3-D two-point ray tracing. For short-period seismic waves, propagating to teleseismic distances, mantle heterogeneity with E = 1 per cent and a = 100 km produces a velocity shift of about 0.2 per cent. Shifts of this order can explain the difference in earth models, derived from free oscillations on the one hand and from short-period body waves on the other. A velocity shift (or velocity dispersion) due t o anelasticity would be
TL;DR: In this paper, the influence of the nonlinear refractive index and the group-velocity dispersion of optical fibers on optical-amplifier noise was studied and a new method was used to calculate the spectrum of the amplified spontaneous emission.
Abstract: The influence of the nonlinear refractive index and the group-velocity dispersion of optical fibers on optical-amplifier noise is studied. A new method is used to calculate the spectrum of the amplified spontaneous emission. The result shows that the positive dispersion is favorable for suppressing the enhancement of the amplifier noise. >
TL;DR: In this paper, a femtosecond soliton through a potential barrier-like spectral inhomogeneity of group velocity dispersion (GVD), including the forbidden band of positive GVD, is predicted.
Abstract: A soliton spectral tunnelling effect (SSTE) is predicted. This is characterised in the spectral domain by the passage of a femtosecond soliton through a potential barrier-like spectral inhomogeneity of group velocity dispersion (GVD), including the forbidden band of positive GVD.
TL;DR: In this paper, the effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied.
Abstract: Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters α and β; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.
TL;DR: It is demonstrated that at one-loop order this method is identical to simple, earlier techniques for computing the wall velocity based on computing the friction from particles reflecting off or transmitting through the wall in the ideal gas limit.
Abstract: The limiting bubble wall velocity during a first-order electroweak phase transition is of interest in scenarios for electroweak baryogenesis. Khlebnikov has recently proposed an interesting method for computing this velocity based on the fluctuation-dissipation theorem. It is demonstrated that at one-loop order this method is identical to simple, earlier techniques for computing the wall velocity based on computing the friction from particles reflecting off or transmitting through the wall in the ideal gas limit.
TL;DR: In this paper, a single-band effective mass equation was used to analyze the waveguiding properties of the ballistic electron waveguides with and without spatially varying effective mass, and it was shown that heterostructure wells, homostructure voltage-induced wells, and heterostructured barriers can act as waveguiders for ballistic electrons.
Abstract: Semiconductor slab electron waveguides with and without spatially varying effective mass are analyzed using the single-band effective-mass equation. Starting with ballistic electron incidence on a potential energy/effective mass interface, expressions for the phase shift, the lateral shift, and the time delay upon total internal reflection are found. It is shown that heterostructure wells, homostructure voltage-induced wells, and heterostructure barriers can act as waveguides for ballistic electrons, and that the waveguiding is described by a single dispersion relation. The guided mode wave functions, dispersion curves, cutoffs, group velocity. effective mass, density of states, and ballistic guided current density are determined. >
TL;DR: In this paper, a 10-element hot-film sensor was mounted on a rotatable insert in a swept flate plate with imposed favorable pressure gradient to obtain direction and magnitude of the phase velocity and the group velocity of the traveling instability waves, thus filling a gap in the knowledge of crossflow instability characteristics.
Abstract: Wave propagation phenomena in three-dimensi onal boundary-layer flows with crossflow instability were investigated experimentally. A 10-element hot-film sensor was flush-mounted on a rotatable insert in a swept flate plate with imposed favorable pressure gradient. By means of cross-spectra l analysis it was possible to obtain direction and magnitude of the phase velocity and the group velocity of the traveling instability waves, thus filling a gap in the knowledge of crossflow instability characteristics. The waves were found to propagate approximately normal to the potential streamline direction, according to linear theory. Phase velocity and the resulting wavelengths also agree satisfactorily, whereas the measured direction and magnitude of the group velocity shows significant differences. Nomenclature c = chord length, 0.5 m cgr = group velocity cph = phase velocity E = voltage F = dimensionless frequency, / • / = dimensional frequency Ga = power spectrum of Xa(t) Gab(f) = cross spectrum of Xa(t) and Xb(t) k - complex wave number vector, k = (kr, &/), IA>l=27r/X
TL;DR: Longitudinal coherence in two-beam interferometers means that the two partial wave packets arrive in the plane of interference simultaneously, but this simultaneity can be lost due to a difference in the geometrical path lengths or the optical path length.
Abstract: Longitudinal coherence in two-beam interferometers means that the two partial wave packets arrive in the plane of interference simultaneously. In charge-particle interferometers, this simultaneity can be lost due to a difference in the geometrical path lengths, a difference in the optical path length, or a difference in the group velocities for the two wave packets on parts of or all of the beam paths. Several of those influences can combine to yield a net relative spatial delay between the wave packets in the interference plane, thus causing a reduction of the interference fringe contrast
TL;DR: In this paper, the analytical expressions used to compute the partial derivatives of phase and group velocity of Rayleigh waves with respect to the P- and S-wave velocity and the density are derived and related computer code is developed.
Abstract: The analytical expressions used to compute the partial derivatives of phase and group velocity of Rayleigh waves with respect to the P- and S-wave velocity and the density are derived and the related computer code is developed The results of the analytical computations were satisfactorily tested against numerically determined values Several examples of partial derivatives for a given structural model are presented
TL;DR: In this article, the authors used laser-Doppler velocimetry to make velocity and turbulence measurements across the layer depth and found that the maximum velocity in the layer occurred very near the plate, contrary to the assumption that the velocity outside the boundary layer is everywhere equal to the preimpingement jet velocity.
TL;DR: In this paper, the authors measured mean velocity distributions in a wall jet embedded in a uniform stream for a variety of initial velocity ratios and Reynolds numbers and determined that the bulk of the flow is self-similar, provided the maximum velocity in the jet is twice as large as the freestream velocity.
Abstract: Mean velocity distributions in a plane, turbulent, and fully developed wall jet embedded in a uniform stream were measured for a variety of initial velocity ratios and Reynolds numbers. It was determined that the bulk of the flow is self-similar, provided the maximum velocity in the jet is twice as large as the freestream velocity. The normalized velocity profile depends on two velocity scales and on two length scales that, in turn, depend on the momentum flux at the nozzle, the viscosity, and the initial velocity ratio between the jet and the freestream defined by R≡(U j -U ∞)/(U j +U ∞). The width of the nozzle that was commonly used to reduce these data has no part in the similarity considerations
TL;DR: In this paper, the effect of a mean fluid velocity gradient on the motion of a small solid particle suspended in a turbulent gas is analyzed using Fourier transform techniques; the presence of such a gradient is shown to elevate the particle velocity variance above the level predicted without such gradients; however, the velocity variance in the direction normal to the flow is only indirectly affected by the existence of fluid velocity gradients.
TL;DR: A stationary solitary solution in the accelerated frame is obtained from the model equations and simulations thereof for the laser, plasma and atoms.
Abstract: Physical conditions are obtained to make the propagation velocity of a laser pulse and thus the phase velocity of the excited wake be at any desired value, including that equal to or greater than the speed of light. The provision of an active-plasma laser medium with an appropriately shaped pulse allows not only replenishment of laser energy loss to the wakefield but also acceleration of the group velocity of photons. A stationary solitary solution in the accelerated frame is obtained from the model equations and simulations thereof for the laser, plasma and atoms. This approach has applications in photonics and telecommunications as well as wakefield accelerators.
TL;DR: In this paper, the 2D wavelet functions Arc and Morlet2D are used to decompose a wave field in space b and scale a. As both wavelets are Hardy functions, the transform result is complex, and the phase, φ, is defined over all b.
TL;DR: In this paper, a method for measuring group velocity dispersion using supercontinuum pulses generated in an optical fiber was proposed, which uses graphical observations of group delay against wavelength characteristics and needs no interpolation formula.
Abstract: A novel method for measuring group velocity dispersion has been proposed which uses supercontinuum pulses generated in an optical fibre. Group delays over a continuous 1200–1395 nm wavelength range have been measured simultaneously for two optical fibre samples. The method features instantaneous graphical observations of group delay against wavelength characteristics and needs no interpolation formula.
TL;DR: In this paper, the motion of a wave packet in a dissipative, homogeneous medium is examined by means of a saddle-point method, and the integral representation of the packet is analyzed by using simple "ray-tracing equations" to follow the packet centers in coordinate and Fourier spaces.
Abstract: In order to clarify the role of the imaginary term Im(d(omega)/dk), the motion of a wave packet in a dissipative, homogeneous medium is examined. The integral representation of the packet is analyzed by means of a saddle-point method. It is shown that in a moving frame attached to its maximum the packet looks self-similar. A Gaussian packet keeps its Gaussian identity, as is typical for the case of a nondissipative medium. The central wave number of the packet slowly changes because of a different damping among the Fourier components. Simple 'ray-tracing equations' are derived to follow the packet centers in coordinate and Fourier spaces. The analysis is illustrated with a comparison to geometric optics, and with two applications: the case of a medium with some resonant damping (or growth) and the propagation of whistler waves in a collisional plasma.
TL;DR: A first-order solution to the transport equation for the total power propagating in each degenerate mode group of a multimode fiber is used to describe pulse propagation in multimode fibers.
Abstract: A first-order solution to the transport equation for the total power propagating in each degenerate mode group of a multimode fiber is used to describe pulse propagation in multimode fibers. The basic transport equation is supplemented by models for Rayleigh-scattering-induced losses, mode coupling, and the modal group velocity function. Each of these models can be specified for a particular fiber from a standard set of near-field measurements and then incorporated into pulse propagation simulations, which closely matched the measured pulse response of test fibers.
TL;DR: In this article, the effects of neglecting the binormal components of velocity which cool each sensor are also assessed, when the probe is treated as two uncoupled X-arrays.
Abstract: The necessary assumption that the instantaneous flow field seen by hot-wire probes with two or more sensors is uniform, i.e. that all sensors are cooled by identically the same instantaneous velocity field, is often quite erroneous in highly sheared turbulent flow, such as near the wall in a turbulent boundary layer. Intense local shear layers occur, resulting in large instantaneous velocity gradients across the sensing volume of the probe. The effects of these neglected velocity gradients on the ability of a four-sensor probe, consisting of a pair of orthogonal X-arrays, to measure the three velocity and the streamwise vorticity components is assessed. This is done by determining the synthetic response of the probe to the turbulent boundary layer database of Balint et al. (1991), in which all the velocity gradients are known. The effects of neglecting the binormal components of velocity which cool each sensor are also assessed, when the probe is treated as two uncoupled X-arrays. A small improvement to the probe's coupled X-array response is found when an estimate of the mean wall-normal velocity gradient is incorporated in the response equations.
TL;DR: In this article, the sum-frequency generation obtained by phase-matched optical pulses of different profiles and unequal pulse widths and frequencies is analyzed and expressions that indicate the exact dependence of field and intensity upon phase mismatch, crystal thickness, and group velocity mismatches between the fundamental pulses and between fundamental and the sumfrequency pulses.
Abstract: Sum-frequency generation obtained by phase-matched optical pulses of different profiles and unequal pulse widths and frequencies is analyzed. Expressions are obtained that indicate the exact dependence of field and intensity upon phase mismatch, crystal thickness, and group velocity mismatches between the fundamental pulses and between fundamental and the sum-frequency pulses. The expressions reduce to known results in the case of replica fundamental pulses, in which case autocorrelation is used. A rigorous model is derived and, in the case where the group velocity mismatch between incident and generated pulses for a specific profile can be neglected, analytical and numerical treatments are given. >