Rayleigh Distribution

The field radiated into the coupling medium by such a distribution of sources may be obtained by means of the well-known Rayleigh integral. The field at the considered point r is computed by a simple integral over the whole radiating surface of the contributions of each elementary source acting as a hemispherical point source. 

Distribution of radiation for (a) Rayleigh scattering and (b) large-particle scattering. 

There are other distributions that can be used in a variety of reliability models. The Poisson, the extreme value, gamma, binomial, and Rayleigh distributions are sometimes used in specialized models. 

Thermally driven convective instabilities in fluid flow, and, more specifically, Rayleigh-B6nard instabilities are favorite working examples in the area of low-dimensional dynamics of distributed systems (see (14 and references therein). By appropriately choosing the cell dimensions (aspect ratio) we can either drive the system to temporal chaos while keeping it spatially coherent, or, alternatively, produce complex spatial patterns. 

Figure 1.3 Field distributions along the Ag-tip surface and corresponding Ag-tip geometry. z = 0 corresponds to the Au-substrate. r/R is the normalized radius from the pointdirectly beneath the tip (R is the Rayleigh length R = /2n). Reprinted with permission from S. Klein, Electrochemistry, 71, 114 (2003). Copyright 2003, The Electrochemical Society of Japan.

Fluctuations near Tc behave as classical fields in sense of Rayleigh-Jeans 3-momentum distribution is n(p) T/E(p) in the vicinity of... 

The velocity, density and temperature of a streaming gas can be determined by measuring the magnitude, frequency and spectral distribution of Rayleigh-scattered light from two simultaneously pulsed ruby lasers with parallel beams and slightly different frequencies 246)... 

In this section, the parameters influencing the mean diameter dR resulting from the Rayleigh instability are examined. Hereafter, the second fragmentation regime is not considered because a narrow size distribution is already obtained after the first one. The parameters that influence dR are the applied stress a, the viscosity ratio p, the rheological behavior, and the way the shear is applied. 

Size distribution plays a major role in the microbubble stability, behavior in vivo, and the microbubble acoustic response. The Rayleigh-Plesset equation which describes the microbubble response to pressure waves suggests that ultrasound scattering is proportional to the sixth power of the microbubble diameter [46]. It is not possible, however, to inject large bubbles (e.g., 0.1 or 1 mm in diameter) in the bloodstream, because they would be immediately lodged in the vasculature as emboli, severely limiting the blood flow. Fortunately, microbubbles with the size of several micrometers are still quite echogenic in the ultrasound... 

Measurements of extinction by small particles are easier to interpret and to compare with theory if the particles are segregated somehow into a population with sufficiently small sizes. The reason for this will become clear, we hope, from inspection of Fig. 12.12, where normalized cross sections using Mie theory and bulk optical constants of MgO, Si02, and SiC are shown as functions of radius the normahzation factor is the cross section in the Rayleigh limit. It is the maximum infrared cross section, the position of which can shift appreciably with radius, that is shown. The most important conclusion to be drawn from these curves is that the mass attenuation coefficient (cross section per unit particle mass) is independent of size below a radius that depends on the material (between about 0.5 and 1.0 fim for the materials considered here). This provides a strong incentive for deahng only with small particles provided that the total particle mass is accurately measured, comparison between theory and experiment can be made without worrying about size distributions or arbitrary normalization. 

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