Standing/stationary waves

Let us consider thermal radiation in a certain cavity at a temperature T. By the term thermal radiation we mean that the radiation field is in thermal equilibrium with its surroundings, the power absorbed by the cavity walls, Fa (v), being equal to the emitted power, Pe v), for all the frequencies v. Under this condition, the superposition of the different electromagnetic waves in the cavity results in standing waves, as required by the stationary radiation field configuration. These standing waves are called cavity modes. 

Detonation Waves, Stabilized- or Standing. See under Detonation Waves, Stationary., Standing-, or Stabilized... 

Detonation Waves, Stationary, Stand -ing-, or Stabilized. Under these terms are known waves which remain stationary relative to laboratory coordinates According to Nicholls et al (Refs 63,... 

Detonation waves, stationary, standing, or stabilized 4 D700... 

Statement 3. If the concentrations are fixed, iVa = const, Nb = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for k k the stationary (quasi-steady-state) solution holds, whereas for k < k a regular (quasi-regular) oscillations in the correlation functions like standing waves... 

The solution of the first kind is stable and arises as the limit, t —> oo, of the non-stationary kinetic equations. Contrary, the solution of the second kind is unstable, i.e., the solution of non-stationary kinetic equations oscillates periodically in time. The joint density of similar particles remains monotonously increasing with coordinate r, unlike that for dissimilar particles. The autowave motion observed could be classified as the non-linear standing waves. Note however, that by nature these waves are not standing waves of concentrations in a real 3d space, but these are more the waves of the joint correlation functions, whose oscillation period does not coincide with that for concentrations. Speaking of the auto-oscillatory regime, we mean first of all the asymptotic solution, as t —> oo. For small t the transient regime holds depending on the initial conditions. 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

This statement could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

Continuing the explanation as was noted above, V x A = 0 outside the solenoid and the situation must be redefined in the following way. An electron on path 1 will interact with the A field oriented in the positive direction. Conversely, an electron on path 2 will interact with the A field oriented in the negative direction. Furthermore, the B field can be defined with respect to a local stationary component / i that is confined to the solenoid and a component By which is either a standing wave or propagates ... 

Note that wavefunction

time independent. The wave functions obtained from a time-independent equation are called standing (or stationary) waves. To obtain such an equation, we assume

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The observation that the wavelength of light is linked to the particle-like momentum of a photon prompted de Broglie to postulate the likelihood of an inverse situation whereby particulate objects may exhibit wave-like properties. Hence, an electron with linear momentum p could under appropriate conditions exhibit a wavelength A = h/p. The demonstration that an electron beam was diffracted by periodic crystals in exactly the same way as X-radiation confirmed de Broglie s postulate and provided an alternative description of the electronic stationary states on an atom. Instead of an accelerated particle the orbiting electron could be described as a standing wave. To avoid self-destruction by wave interference it is necessary to assume an... 

A little-known paper of fundamental importance to modern atomic theory was published by Hantaro Nagaoka in 1904 [10]. Apart from oblique citation, it was soon buried and forgotten. With hindsight it deserved better than that. It contained the seminal ideas underlying the nuclear model of the atom, the standing-wave nature of orbital electrons and radiationless stationary states. It was so far ahead of contemporary thinking that later imitators either failed to appreciate its significance, or pretended to be unaware of it. 

Fig. 57. Numerical, one-parameter bifurcation diagram displaying the global current of stable solutions as a function of the applied potential. In the case of the standing waves (SW), the global current is oscillatory and the open circles denote maximum and minimum of the oscillations. (hom.a. = homogeneous active stationary state hom.p. = homogeneous passive stationary state.) (Reproduced from J. Lee, J. Christoph, P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 115 (2001), 1485 by permission of the American Institute of Physics.)...

This argumentation can be easily extended to two-variable (H)N-NDR systems. Performing a linear stability analysis of a homogeneous stationary state, it is straightforward to show that a homogeneous stationary state can never become unstable in a nontrivial Hopf bifurcation with n = 1. Thus, whenever a Hopf bifurcation occurs, a homogeneous limit cycle is born. Standing waves and pulses are therefore not to be expected under current control. 

H)N-NDR systems DL activator accelerated waves, inhomogeneous (BF unstable) oscillations stationary domains, standing waves, anti-phase oscillations with n = 1 pulses, target patterns, clusters enhanced acceleration of potential fronts, mainly homogenizing effect cluster formation in the oscillatory region... 

S-NDR systems (4DL inhibitor Turing-like structures (n > 1) standing waves, anti-phase oscillations with n = 1 or mixed-mode structures with n > 1 pulses stationary domains (n = 1 or n > 1)... 

Standing wave a stationary wave as on a string of a musical instrument in the wave mechanical model, the electron in the hydrogen atom is considered to be a standing wave. 

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