Dimensional resonances

A further resonance effect depends upon the dimensions of the specimen and so is referred to as dimensional resonance . The explanation is as follows. In a loss-free medium the propagation velocity for electromagnetic waves is given by... 

Fig. 9.32 The effect of dimensional resonance on permeability for two MnZn ferrite components of different sizes (a) larger and (b) smaller. Permeability values are expressed relative to those measured at 1 kHz.

Unlike the case of the Faraday rotation isolator, even an elementary description of the operational principle of the circulator is not easy, involving as it does the dimensional resonance of the microwave field within the ferrite cylinder. In this context the word resonance does not signify gyromagnetic resonance but a standing-wave resonance determined by the dimensions of the... 

The motion along the one-dimensional resonance line called Arnold diffusion is prominent at lower-order resonances when nonlinearity is weak. In fact, the motion with the residence time distribution of the power 3/2 is observed for low-order resonance with weak nonlinearity. On the other hand, overlapped resonances allow the motion across resonances which leads to Brownian motion at a two-dimensional region. Indeed, the distribution with the power 2 is observed at higher-order resonances, and it is more frequently observed with stronger nonlinearity. Hence, one can distinguish clearly the Arnold diffusion from the motion induced by resonance overlaps by the power of the residence time distribution at each resonance condition. 

It follows that the wisely intentional use of any psychedelic drug is as a self-integrating, self-empowering catalyst. In this way the gods (Devas, Archons, spirits, belief complexes, etc.) cannot coerce our worship -- we coerce theirs in the form of enhanced personal power. Obviously it behooves all psychedelic explorers to prudently evaluate the kinds of "allies" they choose to integrate into their psyches in this way. The higher the level of unification the better otherwise it is seductively easy to become entrapped within dimensional resonances of dubious ultimate value. 

Approximate one-dimensional resonance energy obtained from the solution of equation (3). 

In classical descriptions, thermal runaway is attributed to a strong increase of dielectric losses because of heating. So, the energy provided by microwave irradiation increases with temperature. The authors have shown it is possible to achieve thermal runaway with dielectric losses decreasing with temperature as a result of dimensional resonance or focusing effects of an electromagnetic held within the dielectric sample [128, 129]. 

D. Stueega, The future of thermal microwaves Challenges of thermal uniformity, dimensional resonances, chemical and hydrodynamic selec-tivities HDR dissertation (in French), University of Burgundy, July 13, 1994. 

An important phenomenon which can seriously affect the performance of a ferrite device is the dimensional resonance, which occurs when the wavelength of the signal propagating through the ferrite is comparable to the dimensions of the specimen. The propagation velocity of... 

The agreement noted between the ground and excited bend-state reduced dimensionality resonances and the CS ones of Schatz is quite significant. It indicates that the adiabatic bend approximation for this reaction continues to be a realistic description of the dynamics even for total energies up to 1.20 eV. This should stimulate further inquiry about the realism of the adiabatic bend approximation in reactive scattering. 

Air-packaged microresonators are widely used in a variety of low-frequency applications such as accelerometers, gyroscopes, and sensors. In many of these applications, the quality factor of the resonator, defined as the ratio of total energy to energy dissipated per cycle, is a key performance variable but is very difficult to predict. The problem is that the dominant loss mechanism for many air-packaged resonators is viscous drag, and to predict drag, it is necessary to accurately determine the detailed airflow in these typically very complicated three-dimensional resonators. 

The resonant frequency of the quartz crystal is largely influenced by the shape of the cavity resonator which is composed of the piezoelectric quartz material. A quartz crystal resonator does not behave as a onedimensional system however, the basic principles behind three-dimensional resonance calculations are described in a one-dimensional picture. The equation describing free vibration in a one-dimensional system is... 

Bard PY, Bouchon M (1985) The two-dimensional resonance of sediment-filled valleys. Bull Seismol Soc Am 75(2) 519-541... 

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