Rayleigh phase function

If the atmosphere is perfectly clear (neglecting the presence of solid and liquid particulates), the scattering is due only to air molecules and Rayleigh theory can be applied. The corresponding attenuation coefficient is given by the equation (4.46), and the phase function is given by ... 

If the size parameter is small (x < 0.05), then the phase function can be approximated by the Rayleigh phase function, which reads as... 

Treating a molecule as a classical harmonic oscillator, one finds that the phase function for molecular (Rayleigh) scattering becomes ... 

It is clear that the nematic phase exhibits a featureless Rayleigh wing and that several distinct solid phases can be formed depending on cooling rate [79]. This includes an apparently glassy phase. The vertical tick marks indicate the calculated frequencies of vibrational modes as obtained from density functional methods. 

This is the fundamental distillation equation, often referred to as the Rayleigh law when in its integrated form (Rayleigh, 1896). As far as Dt is considered to be a function of F, this equation applies to the change of any species concentration in the course of phase separation. Liquid-vapor or solid-solid fractionations are liable to the same formulation. 

As shown by TalkneP there is a direct connection between the Rayleigh quotient method and the reactive flux method. Two conditions must be met. The first is that phase space regions of products must be absorbing. In different terms, the trial function must decay to zero in the products region. The second condition is that the reduced barrier height pyl" 1. As already mentioned above, differences between the two methods will be of the order e P. ... 

When kx < k the last term in the numerator and the denominator are real, and the reflectance function is real, so that there is no variation of phase with incident angle. When kx = k all terms except the first term in the numerator and the denominator vanish, and R(kx) = 1. When k square brackets depends only on kx and the properties of the solid. Provided that there is no dissipation in the solid, it vanishes when kx is equal to the Rayleigh wavenumber kR = co/vR, as can be seen by substituting X = (ks/kR)2, Y = (k /ks)2, and comparing with (6.55). 

The exponential term can be thought of as a Green function, with the time dependence always implicit. Thus an excitation at x causes a response at x whose phase is delayed by the distance between them multiplied by the real part of kp (this corresponds approximately to 2 /Ar), and whose amplitude is decreased exponentially by the distance between them multiplied by the imaginary part of kp (this corresponds to the decay associated with the propagation of the leaky Rayleigh wave). The magnitude x — x is used because... 

A well known result states that the values of the discrete Fourier transform of a stationary random process are normally distributed complex variables when the length of the Fourier transform is large enough (compared to the decay rate of the noise correlation function) [Brillinger, 1981], This asymptotic normal behavior leads to a Rayleigh distributed magnitude and a uniformly distributed phase (see [McAulay and Malpass, 1980, Ephraim andMalah, 1984] and [Papoulis, 1991]). 

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