Dispersive wave

Nonlinear dispersion becomes relevant at sufficient pulse powers. In some fibre stmctures tire interiDlay between tire nonlinear dispersion and tire group velocity dispersion can be used to produce non-dispersive waves called solitons. Solitons, altliough beyond tire scope of tliis treatment, may revolutionize tire communication systems of tire future. A full treatment of soliton tlieory can be found in [4, 261. 

The rise times of the elastic wave may be quite narrow in elastic single crystals, but in polycrystalline solids the times can be significant due to heterogeneities in physical and chemical composition and residual stresses. In materials such as fused quartz, negative curvature of the stress-volume relation can lead to dispersive waves with slowly rising profiles. 

Figure 1 The frame on the left shows the development of a dispersive wave and the frame in the middle that of a nonlinear wave. When these effects are balanced as in the frame on the right a soliton is formed.

Special conical lenses have been developed for Lamb wave imaging of layered structures because the waves are dispersive the frequency can be varied to tune the Lamb wave angle of the specimen to the angle for which the lens is designed (Atalar and Kbymen 1989). Such lenses could similarly be used for Sezawa wave imaging and measurement, and also for other dispersive waves. It may be that for layered structures measurements of V(f) at fixed defocus will become as prevalent as V(z) (Atalar et al. 1995). 

Another popular approach to wave-particle duality, which originated with Schrodinger, was to view the quantum particle as a wave structure or wave packet. This model goes a long way towards the rationalization of particle-like and wave-like properties in a single construct. However, the simplified textbook discussion, which is unsuitable for the definition of quantum wave packets, relies on the superposition of many waves with a continuous spread of wavelengths, defines a dispersive wave packet, and therefore fails in modelling an electron as a stable particle. 

Such a function will have a large pulse near t = to and it disperses with time. In the two-component system the pair of dispersive waves have different velocities gJiki cu2fc2) and the profile of the wave packet moves with a velocity uj — cu2)/(k — fc2), which is different from the phase velocity (oq + cu2)/(fci + fe2) of the rapidly oscillating part. Velocity of the wave packet is known as the group velocity. If the components are not too different = lo/k and vg = (cox — cu2)/(fci — fc2) = dto/dk. In terms of wavelength... 

In three dimensions a spatial wave group moves around an harmonic ellipsoid and remains compact, in contrast to the dispersive wave packets of classical optics. The distinction is ascribed to the fact that the quantum wave packet is built up from discrete harmonic components, rather than a continuum of waves. The wave mechanics of a hydrogen electron is conjectured to produce wave packets of the same kind. At small quantum numbers the wave spreads around the nucleus and becomes more particle-like, at high quantum numbers, as it approaches the ionization limit where the electron is ejected from the atom. 

Kinra, V. K., "Dispersive Wave Propagation in Random Particulate Composites, Recent Advances in Composites in the U.S. and Japan," ASTM STP 864. 1985, pp. 309-325. 

The average of tiiese two pressiwe drops resulting from die interpolation based on G and x represents die two-phase pressiwe drop for the d ansition region data point. This model predicts 87% of die data widiin 20%. It should be noted diat diese predictions include not only die annular flow region, but also die mist and disperse-wave flow data, whereas die preliminary model of Garimella et al. 

This annular/disperse-wave/mist flow pressure drop model development above uses a physical representation where die liquid forms an annular film aroimd a gas core however, as noted above, the resulting correlation is also... 

A comparison of the measured pressure drops and those calculated using the intermittent and annular/disperse-wave/mist flow models is shown in Figure 11 for each tube considered. In the overlap zone (Figure 7), the flow exhibits both the adjoining mechanisms (intermittent and annular/disperse-wave/mist flow). Therefore, for calculating the pressure drops in the overlap zones in Figure 11, the four-point interpolation scheme described above in connection with the transition between laminar and turbulent data was applied to the pressure drops calculated using the intermittent and annular/disperse-wave/mist flow models. This combined model for the... 

Wise FW, Walmsley lA, Tang CL (1988) Simultaneous formation of solitons and dispersive waves in a femtosecond ring dye-laser. Opt Lett 13 129-131... 

In contrast, under nonlinear conditions peak deformations occur and the retention times become functions of concentration. This leads under thermodynamically controlled ideal conditions to the formation of disperse waves and shock fronts (Figure 2.6e and f). 

The physical concepts to be retained are dispersive waves are formed when isotherms are unfavorable each concentration propagates with a velocity given by the De Vault equation. Compressive waves are formed for favorable isotherms and the physical limit is a shock which propagates with a velocity... 

He, Cunfu et al. 2004. The modal acoustic emission source location technique in pipeline using the wavelet transform of dispersive waves. Journal of Beijing University of Technology 30(9) 96-101. 

Nayfeh, A. H. Hassan, S.D. The method of multiple scales and nonlinear dispersive waves. J. Fluid Mech. 48,463 75 (1971). 

Microrotorcrafts could be used for a variety of missions. Their portability and hovering capabilities make them well suited for military surveillance missions (also categorized as hover-and-stare missions), as well as for other data gathering activities. Furthermore, clusters of microrotorcrafts could be used to provide insight into weather phenomena, pollution dispersal, wave patterns, etc. Finally, because microrotorcrafts are unmanned, they could also be used in hazardous environments (whether due to biological, chemical, or other causes) or for extra-planetary or low-gravity missions, such as exploration of Mars. 

If, on the other hand, the speed of rotation of the sector is increased so that the duration of a period of light or darkness becomes much smaller than the relaxation time of the active centers, everything happens as if the reactor were illuminated all the time by half the intensity I and the rate will now be proportional to (7/2) and is therefore 2 times larger than at low sector speed. If the rate of reaction is plotted versus the logarithm of the frequency of rotation, a dispersion wave will be obtained (Fig. 3.6.1) and the inflection point of the wave corresponds to the mean lifetime of the active centers. 

Kawahara, T., The derivative-expansion method and nonhnear dispersive waves, J. Phys. Soc. Japan, 35 (5), 1537-1544, 1973. 

Theoretical treatments of shock waves in vapour-droplet flows are rare in the literature. Partly dispersed shock waves are discussed by Marble [3] (who made some incorrect deductions concerning the magnitude of the relaxation times), Konorski [4] and Bakhtar and Yousif [5]. Fully dispersed waves dominated by just one relaxation process were treated by Petr [6], but few details of his analysis appear in the paper. No experimental measurement of shock wave structure is available, although the work by Barschdorff [1], and Schnerr [7] shows interesting shock formation patterns and has stimulated the present work. 

From a qualitative point of view, we have described nothing other than the well-known types of stationary wave to be found in all relaxing gas flows. An important result of our work, however, is that fully dispersed waves in vapour droplet flows can naturally be subdivided into three further categories depending on the upstream vapour phase velocity. We thus define... 

In Type I fully dispersed waves, the gradients of the flow properties are comparatively mild and an excellent approximation is to assume that the droplet temperatures and velocity slip instantaneously relax to their equilibrium values. Thus, in equations (9) and (10) we let 0 and Xd - 0, the ratios Au/x and ATf/Xp remaining finite such that (in steady flow) riua ATf ciTg ... 

What are metallic bonds and how do they compare or relate to with the ionic and the covalent bond Metallic bond is an unsaturated covalent bond, i.e., the valence electrons are very weakly bound to the atoms from the metallic bond. In other words, the waves associated with these electrons are much diffused (spread) within the bond space and not concentrated between the bound atoms as in a covalent bond. The dispersed waves of the metallic bond account for its increased electronic conductibility. 

We restrict the attention to periodic solids, molecular crystals. The excitations are characterized by wave vectors q, that lie in the first Brillouin zone of the lattice considered. These excitations are not necessarily pure translational phonons, librons or vibrons, in general they will be mixed. Much experimental information has been collected about such excitations, by infrared and Raman spectroscopy and, in particular, by inelastic neutron scattering. Due to the optical selection rules infrared and Raman spectra can only probe the = 0 excitations. By neutron scattering one can excite states of any given q and thus measure the complete dispersion (wave vector dependence) of the phonon and vibron frequencies. 

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