Curvature wave front

It can be derived from the transport of intensity equation that this signal has two terms at the edges of the pupil it is proportional to the local wavefront gradient in the direction normal to the pupil edge (i.e. radial in the case of a circular pupil) and elsewhere in the pupil it is proportional to the local wave-front curvature (Roddier, 1988). The signal is more intense when the planes are nearer the telescope focus, but diffraction will limit the spatial resolution more. Thus there is a trade-off between resolution and signal-to-noise (see Ch. 24). 

The deficiency can be made up, if no energy is added from the outside, only from the detonation products, with a resulting drop in their temperature from the isochoric adiabatic flame temperature. This may quench the chemical reaction. The deficiency diminishes with decrease of wave-front curvature. For point initiation , enough energy must be added from the outside to make up for the total deficit which accumulates during the time the wave is reaching a diameter at which the curvature drops below a critical value... 

In the abstract of the paper (Ref 36a, p 1920) it is stated The limiting slope of he detonation velociry-wave front curvature locus for small- velocity deficits is obtained under an assumption concerning the "reaction zone length as related to the charge diameter and the radius of curvature of the wave front. The model is an extension to two dimensions of von Neumann s classical theory of the plane wave detonation... 

In order to minimize this additional broadening, the radius of curvature has to be made as large as possible. If A

transit-time broadening. This imposes the condition R w /X on the radius of curvature. 

The next important feature in fast beam laser spectroscopy concerns the velocity distribution. Due to kinematic velocity compression, the initial thermal distribution of velocities in the ion source is reduced by a factor R = 4cT/eV, where T is the ion source temperature and V the acceleration voltage. Including the voltage spread 6V, the Doppler width in our accelerator is 20-200 MHz, depending on mass and acceleration voltage. The transverse velocity distribution (particle wave front curvature) not affected during acceleration, contributes to the linewidth. 

Going into the fundamental resolution limits, the transit times, the second-order Doppler effect, and the second-order laser wave-front curvatures, limit the obtainable resolution, as was also experimentally investigated in the 8 3 2 " Ps/zTPA in Bil, where a mainly transit time limited linewidth of -400 kHz was obtained. 

Theoretical considerations of charges of limited diameter have taken one of two forms. The former assumes that the effects are best described as a result of the curvature of the wave front in the explosive (this can be demonstrated experimentally), or of reduction of the driving pressure by lateral expansion. Solutions of this type have been given by Eyring and co-workers and by Jones. Alternatively, the variation in velocity of... 

Curvature of Wave Fronts. In many discussions of stable detonation waves plane wave fronts are assumed to exist. Actually stable, plane wave fronts do not exist, at least in condensed expls as shown by Cook et ai (Ref 1)... 

Jones assumed. It has, in fact, been shown experimentally by Herzberg et al (Ref 2, p 119) that the detonation wave fronts are curved and that the curvature increases as the charge diameter is reduced. A simple method by which this,curvature can be illustrated is provided by the use of a high-speed Streak Camera (same as "Smear Camera" described in Vol 2 of Encycl, p C16-L). 

Eyring et al (Refs 1 3) first investigated the effect of curvature of the wave front on the detonation velocity. They obtd a relation betw the ratio of the actual to the ideal wave velocities (D/Dq) on the one hand and the ratio of the reaction zone length to the radius of curvature of the front (X/r) on the other. The reaction zone was defined as the zone betw the Cj (Chapman-Jouguet)-layer. If the wave front is assumed to maintain its... 

Detonation Front and Shock Front. Detonation Zone and Shock Zone. The shape of the detonation wave and density-distance particle velocity-distance relations behind the wave front are of considerable practical theoretical importance. The deton wave emerging from the end of an unconfined cylindrical chge of a condensed expl is in general spherical in shape. The curvature of this front has a marked effect on both rate pressure of deton. It has been found that there is a minimum radius of convex curvature for each expl, below which deton will not propagate. The min radius of curvature is primarily that at which the divergence is so great that the energy released from the chem reaction of the very small vol of expl involved is insufficient to compensate for the rapid increase in area in the deton front. 

The objective of the work of Wood Kirkwood described in Ref 36a is somewhat different, namely, to give an account of the relation between velocity and radius of curvature of the wave front, rather than the charge radius. The work of W K is closely related to the Curved Front Theory of Eyring et al, although the basic model, as well as the objective, is considerably different. 

W.B.. Cybulsky et al, PrRoySoc 197A, 51-72 (1949) (Explosion waves and shock waves). 22a) H. Eyring et al, Chem Revs 4 69 (1949) (See in the text under "Radius of the Curvature of the Detonation Wave Front 23) G.I. Taylor,... 

The near-field correction is calculated conceptually by repositioning the detector positions from a linear arrangement to a spherical arrangement that matches the curvature of the incoming wavefront. The theoretical phase delay from a point source at normal incidence is subtracted from the measured phase. As shown in those Figures, the curvature of the phase (indicating a curved wave front) is removed by the near-field correction, yielding a linear dependence of phase on detector position. The slopes of the near-field-corrected phase versus detector position plots indicate the direction to the source. 

Clavin [5] performed quasi-steady analysis of the direct initiation process. They developed the critical curvature model, which states that the failure mechanism of the detonation is mainly caused by the nonlinear curvature effect of the wave front. Eckett et al. [6] proposed the critical decay-rate model and pointed out that the critical mechanism of a failed detonation initiation process is due to the unsteadiness of the reacting flow. Their theory for spherical detonation initiation has been supported by numerical simulation and experimental data. 

Many definitions of spiral tips are in use. One of them considers the spiral tip as the point of the spiral wave front with maximal curvature [8]. A practical way to determine the tip position in an experiment is to look at spiral wave fronts extracted from two consecutive frames of a digitized recording. The intersection of these lines is also considered a tip position... 

Fig. 9.1. Different representations of a rigidly rotating spiral wave, (a) Involute of a hole, (b) Solution of the kinematical equation with linear velocity-curvature relationship, (c) Archimedean spiral, (d) Superposition of the three wave fronts where the dotted, dashed and solid lines correspond to (a), (b), and (c), respectively. Far from the rotation center the fronts practically coincide.

A more elaborated kinematical description of a freely rotating spiral wave in a uniform medium is based on the assumption that the normal velocity c of a curved wave front is not a constant, but depends on its curvature [34]. The simplest approximation of this relationship is a linear... 

It is important to stress that the above three descriptions of the spiral wave practically coincide far away from the rotation center. Moreover, already at a relatively small distance ta from the rotation center, the Archimedean spiral becomes very close to the curvature affected spiral obtained from Eq. (9.3), as can bee seen in Fig. 9.1(d). In this example ta can be estimated as ss 9.0 A. Recent computations performed with the Oregonator model [40] and experiments with the BZ reaction [43] also confirm that an Archimedean spiral provides a suitable approximation of the wave front except in a relatively small region of radius A near the rotation center. Even the shape of a slightly meandering spiral waves exhibits only small oscillations around an Archimedean shape, and the amplitude of these oscillations vanishes very quickly with r [44]. Therefore, the Archimedean spiral approximation will be used below to specify the shape of the wave front. 

The opposite is true along the concave segments, where the wave front is retarded relative to the planar front. Here, there is a diffusive focusing of B into the region ahead of the front, leading to a local increase in wave velocity. These local increases and decreases in propagation velocity tend to eliminate die local curvature. Thus we may say that the diffusion of the autocatalyst has a stabiliz-ing effect on the planar wave. 

For the fundamental or TEMooq mode, the Hermite polynomials are unity. X is the free-space radiation wavelength, d is the mirror separation, and q + 1 the number of half-wavelengths between the mirrors. The beam waist, Wg, is the distance from the center of the cavity to the 1/e points of the field strength. The beam waist is a maximum when d = R, the confocal arrangement, and falls to zero when d = 0 or d = 2R. Using the radius of curvature of one of our mirrors, 84 cm, and a mirror separation of 70 cm, the beam waist at 10 GHz is 12.6 cm. The k/>2/2R factor accounts for curvature of the wave front arising because of the curved mirrors. The phase front is... 

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