Long-wave approximation

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

SRPA itself does not give a recipe to determine Qk f). But choice of these operators can be inspired by physical and computational arguments. The operators should be simple and universal in the sense that they can be applied equally well to all modes and excitation channels. The main idea is that the initial operators should result in exploration of different spatial regions of the system, the surface and interior. This suggests that the leading scaling operator should have the form of the applied external field in the long-wave approximation, for example. 

The nonlinear wave equation thus obtained is the famous sine-Gordon equation, which is well known from soliton theory (see, for example, Dodd et al. [1982] and Rajaraman [1982]). The long wave approximation used to replace the discrete rotor angle < by continuous variable 4>(x, t)... 

When the wavelength of light is much greater than the radius of the electron on the metal site (the long-wave approximation), the transition moment operator is given by the multipole expansion 11... 

The quadmpoie operator transforms as WY V and makes the 1 s —> 3d transition electric quadmpoie allowed when the dY v2 orbital (in molecular coordinates) bisects the k and E vectors (which define the laboratory coordinates). Quadmpoie intensity is usually very low however, at —9000 eV the wavelength of light is — 1.4 A and in this case the long-wave approximation no longer holds and higher terms in the multipole expansion in Equation 1.2 become important. 

At small k(long-wave approximation) in a onemode pole approximation this dependence has the form 4>(k) = [4tt/(1 — 1/e)] (l + Ak2) where A is the correlation length. This form corresponds to a gradient expansion of the Landau free energy functional... 

We use a thin-film model where the film thickness is assumed to be much smaller than the characteristic wavelength of the undulations in the lateral plane. Under such a long-wave approximation the Navier-Stokes equations lead to the following boundary-layer equations [31-33] ... 

Equation 3.2 for the sinuous and varicose growth rates are shown in Figs. 3.3 and 3.4 for gas Weber numbers We = p Uo a/a of 0.5 and 5.0, respectively. Each figure also shows the results for the long wave (Equation 3.4), that tanh( ) ka and short wave (equation 3.6) approximation. Long wave approximation is similar to that of Hagerty and Shea [4]. For a Weg = 5.0, the dimensionless growth rate curves are very similar, except at low values of the dimensionless wave number ka. 

Such decoupling in the liquid may be strictly justified only in the long-wave approximation.In this sense, such a procedure is justified for the macroscopic description. However, one should remember that this is the correct method in a number of cases also for short wavelengths. For example, this is the case for phonons in solids. In other cases, such as the electron gas in metals (plasmons), acoustic phonons in quantum liquids and so on, this decoupling may be considered as the self-consistent field method or the random phase approximation (the analog of the superposition approximation in the classical theory of liquids). 

The space of electronic operators Op is naturally decomposed into subspaces, conforming to irreducible representations of a symmetry group of the Brillouin zone center and having the corresponding tensors i/(r,). To calculate the quantities (142), the parameters of an electron-deformation interaction (eq. 19) are necessary besides only coupling constants with odd optical vibrations are necessary, if acoustic vibrations are considered in the long-wave approximation. Parameters 5/(l 7g(u)) are linear combinations of 5 (f ) (eq. 112) with coefficients defining the expansion of the... 

The full B-M model problem, [(4.3a,b,c), (4.5a,b,c,d), (4.6), (4.7)], is a very complicated problem, but for a thin film flow we can apply the long wave approximation. In this case, we derive (as in next Section 4.2) a simplified Benard-Marangoni Boundary Layer (B-Mbl) model problem for the high Reynolds numbers. 

According to long waves approximation, we assume that the characteristic value of the horizontal (in the x and y directions) wave-length X d. In this case, instead of dimensionless variables (t , x 1, x 2, x 3), it is judicious to introduce the following news dimensionless variables ... 

For every prescribed set of free physical parameters the eigenvalues cj a) could be computed for arbitrary high a > 0 values. But we must keep in mind that assumptions of the long wave approximation introduce some limitations on a. For the problem under consideration there exist indeed two lengths h and hi. By virtue of inequality hi h short waves in h scale could be considered as long wave in /ii scale. Some arbitrary upper boundary a < 10 in the following numerical experiments is used. Let us consider the case of constant surface excess concentration F. 

For optically allowed transitions, the electronic matrix element ( ) in (6.10) is usually a slowly varying function of the nuclear coordinates over the range where Xs Xo significantly differs from zero. Therefore, it may be removed from the inner element of the nuclear integration by using the q-centroid-type approximation. To obtain (6.10), we have assumed that the electric dipole transition is allowed for the decay from state t )j to the ground state and set = 1 in Equation 6.7. This is an excellent approximation for atomic transitions, since called long-wave approximation. 

Womersley, J. R. Oscillatory motion of a viscous liquid in a thin-walled elastic tube. I. The linear approximation for long waves. Phil. Mag. Ser. 7 46 199-221, 1955. 

Another evidently radiation-induced band occurs in the orange part of spectrum. Under long waved UV and visible excitations the band peaking at 600 nm is detected with half-width of 95 nm (Fig. 5.66a). Excitation spectrum of this emission contains for maxima peaking at 345,360 and 410 nm (Fig. 5.66b). The band is evidently not symmetrical with shoulder at 625 nm, but such form remains in all time-resolved spectra with different delays and gates and does not resolved to several emission bands. This band can be detected with extremely narrow gate width, which is a strong evidence that its decay time is very short, approximately 10-12 ns, which is on the border of our experimental system alrility. At 40 K the band becomes extremely intensive, while its spectrum and decay time remain practically the same. 

It follows from the experimental data that the distribution function of SC is bimodal. The fraction with shorter wavelengths in the optical absorption spectrum is characterized by less pronounced spectral difference between SC in this fraction than that for long-wave SC. The decomposition of the optical absorption spectrum of SC into two fractions and a change in its shape (calculated in this approximation) during oxidation are shown in Figure 7.23b and c, respectively. A satisfactory agreement between the experimental and calculated dependences is observed. The difference in the properties of SCs is certainly associated with their spatial structure however, it remains unclear what structural features of quartz glass are responsible for the stabilization of two types of sites. 

The visible spectrum extends approximately from 4,000 to 8,000 A. The human eye is not sensitive to light of wave lengths much shorter or longer than these. The near infrared region extends from 8,000 up to perhaps 200,000 A, and the far infared region extends still further toward the long waves from meters to thousands of meters, which are familiar to anyone who turns a radio dial. On the shorter side of the visible spectrum, we have ultraviolet light... 

If the frequency v and the wave length X are related bv Ea. (1.7V the values of , in Eq. (1.5) will satisfy the equations (1.3). If the velocity were constant, we should have v = v/. the frequency being inversely proportional to the wave length. From Eq. (1.7) we can see that this is the case for long waves, or low frequencies, where we can approximate the sine by the angle. In that limit we have... 

In a quantum-chemical description of the two subsystems Q and S of the preceding section the total wave function—in the long-range approximation—can be written as... 

Unsubstituted 5(6)-nitrobenzimidazole has two absorption bands in the field of 230-235 and 300-309 nm [1186, 1190-1192], Its long-wave band, as compared with the initial benzimidazole, is bathochromically moved by approximately 25 nm and does not show vibrational structure. The short-wave band, on the contrary, undergoes a hypsochromic shift (—10 nm) upon mononitration [709, 1193], Band displacement of this kind and the disappearance of vibrational observed for the long-wave band are responsible for essential differences in the localization and, hence, in the nature of, at least, one of the transitions. For example, from CNDO/S calculations it follows that the long-wave transition of benzimidazole is multicon-figurational (Table 3.70) and results in the total molecular excitement [1193],... 

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