Effective wave number

Under the conditions of scattering the phase velocity of capillary waves will deviate from the value for the medium without scatterers. Therefore, it is natural to introduce an effective wave number K for the former case and to choose the following trial functions for the coefficients C ... 

Eqs. (9)-(12) allow us to obtain the following expression for the complex effective wave number after rather cumbersome transformations (Noskov 1998)... 

Relation (13) is the main result. It allows us to determine the effective wave number at the propagation of surface waves along the interface with two-dimensional scatterers on the basis of the scattering characteristics on a single particle. As it could be expected, the difference from the three-dimensional case only consists in the form of coefficients f(0) and f(it) and also in the form of the expressions for the scattered amplitudes in themselves. 

The coupled-phase c-p model is based on analysis of a two-phase fluid by solving four differential equations that govern the motion of the two-phase mixture. For the effective wave number k, the solution gives... 

From the effective wave number, one obtains the effective wave velocity V = Re( oo/k) and attenuation a = Im(k). 

The effective wave number k is related to the frequency o) by the dispersion relation, D being the total physical thickness of the structure ... 

As a result, the long-wavelength equation for the effective wave number takes the form... 

Thus, the procedure is to first find the singular solution of the Lorentz-Lorenz law together with the dispersion relation for the effective wave number Kg, and then to determine the arbitrary constant from the scalar equations (2.190) or (2.191). After the conditional average of the scattered field coefficients has been evaluated, the coherent reflected field can be computed by taking the configurational average of (2.171), i.e.,... 

EFMEDT90 computes the effective wave number of a half-space with randomly distributed particles... 

The symbols in the second column represent the electronic state in particular the first number is the total quantum number of the excited electron. We shall see later that in one case at least the symbol is probably incorrect. The third column gives the wave-number of the lowest oscillational-rotational level, the fourth the effective quantum number, the fifth and sixth the oscillational wave-number and the average intemuclear distance for the lowest oscillational-rotational level. The data for H2+ were obtained by extrapolation, except rQ, which is Burrau s theoretical value (Section Via). 

The effective atomic numbers in the radial wave functions cannot be evaluated by minimizing the energy integral, because of neglect of inner shells. In all the calculations reported the effective atomic numbers were given the value 1. 

The parameter Z is an effective atomic number whose value is determined by the minimization of in equation (9.2). Since the hydrogen-like wave functions 01 and 02 are normalized, we have... 

For these complexes, the extent of back donation increases as the number of CO ligands increases, which causes the stretching frequencies to be found at lower wave numbers. A similar trend is seen for the following complexes (all of which obey the 18-electron rule), showing the effect of the charge on the metal ion ... 

Indeed, we have discussed the matrix elements involved in these formulas (see Eqs. (36), (52), and (56)) as well as the physical meaning of the Fourier coefficients pk p] t). However, the mathematical expressions are often rather involved and it is convenient, especially in specific applications, to introduce a diagram technique in order to represent the various terms of these general formulas.28 We first notice that in Eqs. (41) and (42), the momenta p,- essentially appear as parameters indeed, according to Eq. (52) only the wave-vectors are explicitly modified by the interactions. This is the reason why we shall only represent these wave numbers graphically it should, however, be kept in mind that the momenta are effectively affected by the interactions through the differential operators d/dp ... 

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