High-frequency approximation

High frequency approximations in the solution of an acoustic wave equation The reader, familiar with the background of seismic exploration methods, should recall that many successful seismic interpretation algorithms are based on the simple principles of geometrical seismics, which resembles the ideas of geometrical optics. The question is how this simple but powerful approach is connected with the... 

We look for the value of X at which p is a maximum, using (as appropriate) the short-wavelength (high-frequency) approximation... 

We use the relationships (4)—(7) to calculate the water spectra. In such a calculation for ice the static permittivity, s(ice) is not involved. For molecules reorienting in the hat well the high-frequency approximation is employed. The complex permittivity of the LIB state is represented, instead of Eq. (4), as... 

We consider collective motion of pairs of water molecules. Let the unit volume of the medium comprise Avlb/2 of such pairs, with Ny being the concentration of molecules suffering elastic vibration. Using the high-frequency approximation, we calculate the complex susceptibility /vib — Xvib + Yvib °f the medium pertinent to harmonic vibration of the HB particles (we omit the complex-conjugation symbol). We assume that for an instant just after a strong collision, the velocities and position coordinates of the particles have Boltzmann distributions. Then the elastic-vibration complex susceptibility /vib and permittivity Asvib in view of TGN are determined by the formulas... 

We do not consider the low-frequency spectra for ice, since the contribution to complex permittivity of rigid reorienting dipoles is calculated from the simplified expression (A29), which is applicable only in the high-frequency approximation. Indeed, the ice permittivity is found for v > 0.1 cm-1 (see Figs. 20a,b and 24a), while for liquid water Eq. (4) is used, applicable also in the relaxation region. 

We recall the difference in notations The symbol A denotes that the corresponding quantity is found in the high-frequency approximation, so that its real part vanishes at v —> oo, while or(oc) differs from zero. For water we have aor(oo) = nlo 1-7 for ice this quantity is greater. 

Let a unit volume of an isotropic medium comprise Vvib/2 of such pairs (nonrigid dipoles). We shall calculate the generated complex susceptibility x by using the high-frequency approximation for which it is assumed that at the instant just after a strong collision the velocities and position coordinates are given by the Boltzmann distribution (marked by the subscript B). Then, in view of Eq. (3.5) in GT1, the complex susceptibility x is proportional to the spectral function L ... 

For practical calculations of the transverse susceptibility x and of the relevant contribution As of transverse vibrations to the total permittivity, given in the high-frequency approximation by... 

In our work the SF has a twofold application. First, the formula (254) is used for description of thefar-IR spectrum in the high-frequency approximation. Such a spectrum accounts for resonance or, more often, quasi-resonance interaction of dipoles with radiation. An absorption peak arises when the radiation frequency co is near the mean thermal frequency determined by the Boltzmann energy distributions. 

In the microwave region and at lower frequencies the high frequency approximation, used in this work for ice, becomes inapplicable. In particular, this approach does not describe a deep loss minimum e"(v), denoted in Fig. 26b by the dashed line. Such a behavior is typical for ice in a wide temperature range (Fig. 36), but has no analogy in the case of water, where the loss minimum of the g"(v) curve is not emphasized. On the contrary, curve e"(v) exhibits a shallow minimum in a much narrower frequency range (Fig. 37a). Moreover, in ice, only transverse vibration constitutes the loss minimum, while as depicted in Fig. 37c, in water all mechanisms, especially mechanisms a and d, contribute to the loss minimum. 

Equation (3.25) shows that at very low frequenciesRxcu,thus R-+0 as 0, There will be an intermediate frequency range (assuming 4= Ty) in which R is independent o fo) but at high frequencies (co 41 At, 1 Ae) will vary as m " Since in practice tt and tjg both usually come within the range 10—0.1 s, for most applications the high frequency approximation of (3.25) is valid, i.e.,... 

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