Rayleigh expansion

Consider the expansion about the origin of the coordinate system of a simple Gaussian function. We introduce a modified form of the Rayleigh expansion[42]... 

The Rayleigh expansion of a plane wave in a basis of spherical waves [10],... 

The evaluation of the Fourier transform employs Rayleigh expansion of the exponential over spherical harmonics exp(d[Pg.64]

Approximate solutions to Eq. 11-12 have been obtained in two forms. The first, given by Lord Rayleigh [13], is that of a series approximation. The derivation is not repeated here, but for the case of a nearly spherical meniscus, that is, r h, expansion around a deviation function led to the equation... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

Now the bubble collapse is discussed using the Rayleigh-Plesset equation. After the bubble expansion, a bubble collapses. During the bubble collapse, important terms in the Rayleigh-Plesset equation are the two terms in the left hand side of (1.13). Then, the bubble wall acceleration is expressed as follows. 

In Fig. 1.4a, an example of the radius-time curve for a stably pulsating bubble calculated by the modified Keller equation is shown for one acoustic cycle [43]. After the bubble expansion during the rarefaction phase of ultrasound, a bubble strongly collapses, which is the inertial or Rayleigh collapse. After the collapse, there is a bouncing radial motion of a bubble. In Fig. 1.4b, the calculated flux of OH... 

When the size parameter x is sufficiently small, that is, when the particle is small compared with the wavelength of light, only the leading term in the normal mode expansion for the spherical harmonic functions is needed. In this case Eq. (76) reduces to Rayleigh s result, Eq. (47), for the ratio of the scattered irradiance to the incident irradiance. 

In the first case (Figure 8a), the side walls are adiabatic, and the reactor height (2 cm) is low enough to make natural convection unimportant. The fluid-particle trajectories are not perturbed, except for the gas expansion at the beginning of the reactor that is caused by the thermal expansion of the cold gas upon approaching the hot susceptor. On the basis of the mean temperature, the effective Rayleigh number, Rat, is 596, which is less than the Rayleigh number of 1844 necessary for the existence of a two-dimensional, stable, steady-state solution with flow in the transverse direction that was computed for equivalent Boussinesq conditions (188). 

Interesting properties of the Rayleigh relationships can be obtained from the expansion of Equation (13.1) in Fourier series. For E = Eq sin(ut), one obtains ... 

The Rayleigh-Schrodinger perturbation expansion for the exact wave function to the first order is given by... 

Because of the spin-orbit selection rules, only triplet zeroth-order states contribute to the first-order perturbation correction of a singlet wave function. In Rayleigh-Schro dinger perturbation theory, the expansion coefficient a of a triplet zeroth-order state (3spin-orbit matrix element with the electronic ground state (in the numerator) and its energy difference with respect to the latter (in the denominator). 

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