To Oscillation period

When looking at the snapshots in Figure A3.13.6 we see that the position of maximal probability oscillates back and forth along the stretching coordinate between the walls at = -20 and +25 pm, with an approximate period of 12 fs, which corresponds to the classical oscillation period r = 1 / v of a pendulum with... 

In addition to tire period-doubling route to chaos tliere are otlier routes tliat are chemically important mixed-mode oscillations (MMOs), intennittency and quasi-periodicity. Their signature is easily recognized in chemical experiments, so tliat tliey were seen early in the history of chemical chaos. 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

As the Reynolds number rises above about 40, the wake begins to display periodic instabiUties, and the standing eddies themselves begin to oscillate laterally and to shed some rotating fluid every half cycle. These still laminar vortices are convected downstream as a vortex street. The frequency at which they are shed is normally expressed as a dimensionless Strouhal number which, for Reynolds numbers in excess of 300, is roughly constant ... 

Place a high-frequency capacitor (ceramic or film) across the primary winding of the transformer, rectifier, or the element to be snubbed. Determine the capacitor value that produces an oscillation period which is three times the original period (Co). 

The above results show that the structure of the system with the molecules self-assembled into the internal films is determined by their correlation functions. In contrast to simple fluids, the four-point correlation functions are as important as the two-point correlation functions for the description of the structure in this case. The oil or water domain size is related to the period of oscillations A of the two-point functions. The connectivity of the oil and water domains, related to the sign of K, is determined by the way four moleeules at distanees eomparable to their sizes are eorrelated. For > 0 surfactant molecules are correlated in such a way that preferred orientations... 

In general, parametric excitation (or action) may be defined as follows if a parameter of an oscillatory system is made to vary periodically with frequency 2/, / being the free frequency of the system, the latter begins to oscillate with its own frequency. 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

Assume that we have a pendulum (Fig. 6-14) provided with a piece of soft iron P placed coaxially with a coil C carrying an alternating current that is, the axis of the coil coincides with the longitudinal axis OP of the pendulum at rest. If the coil is excited, one finds that the pendulum in due course begins to oscillate, and th oscillations finally reach a stationary amplitude. It is important to note that between the period of oscillation of the pendulum and the period of the alternating current there exists no rational ratio, so that the question of the subharmonic effect is ruled out. 

Interesting results have also been obtained with light-induced oscillations of silicon in contact with ammonium fluoride solutions. The quantum efficiency was found to oscillate complementarity with the PMC signal. The calculated surface recombination rate also oscillated comple-mentarily with the charge transfer rate.27,28 The explanation was a periodically oscillating silicon oxide surface layer. Because of a periodically changing space charge layer, the situation turned out to be nevertheless relatively complicated. 

If T is the oscillation time of pendulum or the time the pendulum needs to swing from the angle to (half period ), then integration of Equation (3.38) gives... 

Potential differences between the nitrobenzene and aqueous phases at the interfaces in the presence [Fig. 2(B)] and absence of surfactant (C) were measured simultaneously. KCl salt bridges were inserted into the octanol phase to monitor potential. Oscillation measurement data across the nitrobenzene membrane are given in Fig. 2(A) for comparison. The oscillation mode in Fig. 2(C) is virtually the same as that in (A) with respect to oscillatory period and amplitude but quite different with that in (B). Although the potential across the nitrobenzene membrane (A) was not recorded simultaneously with that between nitrobenzene-water phases (B) and (C) but successively, it was noted that the algebraic sum of (B) and (C) should be essentially the same as (A). This is an indication that potential oscillation across the nitrobenzene membrane is likely generated at the interface between the nitrobenzene phase and aqueous phase initially containing no surfactant. 

In Chapter 3 the steady-state hydrodynamic aspects of two-phase flow were discussed and reference was made to their potential for instabilities. The instability of a system may be either static or dynamic. A flow is subject to a static instability if, when the flow conditions change by a small step from the original steady-state ones, another steady state is not possible in the vicinity of the original state. The cause of the phenomenon lies in the steady-state laws hence, the threshold of the instability can be predicted only by using steady-state laws. A static instability can lead either to a different steady-state condition or to a periodic behavior (Boure et al., 1973). A flow is subject to a dynamic instability when the inertia and other feedback effects have an essential part in the process. The system behaves like a servomechanism, and knowledge of the steady-state laws is not sufficient even for the threshold prediction. The steady-state may be a solution of the equations of the system, but is not the only solution. The above-mentioned fluctuations in a steady flow may be sufficient to start the instability. Three conditions are required for a system to possess a potential for oscillating instabilities ... 

The transit time from the hole to the reflector and back again corresponds to the period of oscillation (v). Thus the microwave frequency can be tuned (over a small range) by adjusting the physical distance between the anode and... 

At the maximum time shown in Figure 9B, the outlet CO level had only risen to about 60% of the rich steady-state outlet level, which can be seen on the left side of Figure 9A. Approximately 25 s were required for the outlet CO to reach the new steady-state level after the lean-to-rich step. This time is much shorter than that mentioned above for catalysts in S02 free simulated exhaust, but is still long with respect to the periods of the exhaust composition oscillations observed during actual automotive operation. 

The same vibration leads to a periodical shift of the surface-state energy levels via the exchange coupling of the surface spin to the bulk magnetization. Bovensiepen and coworkers observed this oscillation of the binding energy, which with the aid of DFT calculations they translated to the interlayer spacing in picometer scale [20,23],... 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

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