Front quadratic

Thus the quadratic sum of all the Zemike coefficients gives the rms value of the entire wave-front error by ... 

In the approximate treatment which follows we consider the two parts of the wave separately and see that the leading front can be considered as a cubic Fisher wave and the recovery front by a quadratic form. 

The second front in the pulse, moving with velocity c2, sees the conversion of B to C. Thus 0 must fall from 0+. to zero. There may also be a final decay in the reactant concentration from a+ to zero. However, a+ is already small, so we will make the approximation that a = a+ k2 throughout the whole of this front. Assuming constant a means that again the system is reduced to one governing equation. Substituting a = k2 into eqn (11.52) gives a quadratic wave equation... 

Figure 5.4 Migration distances of the mobile phase on a TLC plate. Over a sequence of regular time intervals, the quadratic progression of the eluant front can be seen. Curve A was obtained in a chamber unsaturated with eluant vapour. Curve B was obtained by saturating the chamber with eluant vapour.

The two solutions of this quadratic equation are precisely the values of p i.e. they are the intersections of the p coordinate with the front and rear parts of the lamp at any value of 6 and ... 

Density plays an important role in the behavior of energetic materials. The pressure in explosions and the impulse produced by the same compound when used as a propellant are related. The shockwave pressure behind the detonation front is proportional to the density squared [3] times the specific impulse [4). The specific impulse itself depends on the volume of gas produced and the heat of combustion per gram of propellant which leads to a further complex dependence on density [5). Thus, the overall dependency of the detonation pressure on the density is greater than quadratic. Two examples of dense energetic materials are the widely used /9-HMX and RDX [6,7], shown in Fig. 1. 

Figure 17 Phase plane for quadratic autocatalysis front described by Eqs. [83]. The front profile corresponds to a trajectory emanating from the origin (saddle point) along the outset and approadiing the singularity at (1, 0) along the (degenerate) eigenvector (Reprinted from Ref. 43 with permission of the American...

Hence for quadratic autocatalysis, constant velocity, constant wave-form propagating fronts are allowed with any velocity c greater than some minimum velocity ... 

The front velocity for cubic autocatalysis may take on any value above some minimum velocity just as for the quadratic case. In dimensionless terms, we have... 

There are a number of important differences between quadratic and cubic fronts however, the most striking is found in their behavior in two- and three-dimensional configurations. We shall focus on the two-dimensional case in which the diffusivities of A and B may take on significantly different values. (Similar behavior is found in the three-dimensional case.) now use a two-dimensional Laplacian defined by -t- dVdV with ... 

The behavior found for the quadratic system in the one-dimensional case is directly applicable to the two-dimensional configuration planar fronts are exhibited with velocities given by Eq. [88] for the case of equal diffusivities of A and B. When the diffusivities significantly differ, planar fronts are still observed however, now the velocity scales with the diffusion coefficient D = Dr according to Eq. [89]. > The one-dimensional solution for the cubic system is also valid for the two-dimensional configuration however, the cubic front may exhibit lateral instabilities that are not observed in the quadratic system. will now consider the stability of cubic autocatalysis fronts. 

Both Hamiltonians (5.11) and (5.21) involve the maximum velocities ajx and aX, respectively. The front velocities tend to infinity in the fast reaction limit when the diffusion approximation is considered. This means that H p) depends quadratically on ap for ap small. The two models (5.11) and (5.20) differ fundamentally with respect to propagating fronts discreteness in time leads to a finite propagation rate, while the continuous-in-time model leads to an infinite velocity of propagation in the limit of fast reaction, r -> oo. The front velocity for model (5.21) is... 

Fig. 6.31 Sequence of grating pairs and prism pairs for the compensation of quadratic and cubic phase dispersion. LL and MM are two phase-fronts. The solid line represents a reference path and the dashed line illustrates the paths for the wave of wavelength X, which is diffracted by an angle p at the first grating and refracted by an angle a against the reference path in a prism [689]...

Optimization is performed solving a Multiobjective Optimization Problem (MOP) using as optimization algorithm a customized Sequential Quadratic Programming (SQP) method (Fletcher 1987). The solution of the MOP provides the Pareto front of the problem which, in our application, consists of 155 optimal solutions. However, as inputs of the equipment reliability and cost models fluctuate according to distribution laws reflecting uncertainty on parameters, objective functions will fluctuate also in repeated runs. 

The 1/2 factor in front of accounts for the linear dependence of the operator on the solute charge (i.e., the quadratic dependence on ). In a more physical description, the same factor is introduced when one considers that half of the interaction energy has been spent in polarizing the solvent and it has not been included in G. 

We note that such a transition from a circular to a quadratic envelope of the crystals has also been reproduced by computer simulations [13,14]. There, this transition is due to the reduction of the growth front nucleation probability. Wliile the disk-like pattern consists of multiple crystals, the square-shaped pattern represents a single crystal. We thus assume that, for the given film thickness, we observed a transition from a poly crystalline structure to a single crystal within the temperature interval from 45 to 50°C. 

This demonstrates a quadratic dependence of burnout on the diameter of the initial liquid droplet. For burning droplets, the droplet flame front acts as a nearby source of heat for droplet vaporization, otherwise the mechanism is the same as pure vaporization. For a singlecomponent, uniform droplet burning under quasi-steady gas-phase conditions... 

In OPLC, the developing system is essentially an S-chamber that resembles an HPLC column. The distance traveled by the solvent front is proportional to time development time does not increase with distance according to the quadratic law (Geiss, 1987) as in capillary-flow TLC. Aqueous solvents that do not wet the layer can be used with RP layers. 

The simplest attack [7] anticipates the fact that cubic autocatalysis generates a steady wave-front of quadratic form ... 

Figure 1.22 Sketch of the top view of the suspended channel with the exception of the front end of the flow near the interface where the flow is not estabhshed, the velocity profile is close to the Poiseuille quadratic profile.

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