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~ k') [2 + (~) '] +47rn Tt l (~ - Mn) ~ 0(6.1.55)
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(6.1.56) Solving (6.1.55) gives
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1.3 Dispersion Relations at HhrlJer Frequencies
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K r . The final result is 2 K 2 =k 2 + 3fk y [1+i~k3a3 y 1 - fy 3 1 - fy
41rn o roodrr2[g(r)_I]}] Jo (6.1.60) where f = n o 41ra 3/3 is the fractional volume of scatterers. The result in (6.1.60) is the same as the result obtained through the simple model in 10 of Volume II. Using the Percus-Yevick pair-distribution function from 8 of Volume II for g(r), the result for the effective propagation constant is
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In Figs. 6.1.3 and 6.1.4, we show, respectively, the normalilled phase velocity V p = k/ K r and the effective loss tangent LT = 2K;jK 1 as a function of fractional volume of scatterers f using the result of (6.1.61) for ka = 0.1. We note that the phase velocity decreases with f. The loss tangent first increases with f, then saturates and decreases as f further increases.
1.3 Dispersion Relations at Higher Frequencies
For larger values of ka, (6.1.46a) and (6.1.46b) can be solved numerically. In this section we shall show the numerical results of the effective propagation constant K for ka ranging from 0.05 to 2.5 and f ranging from a to 0.4. For these values of ka and f, the determinant of the coefficient matrix was computed numerically by retaining a maximum of eight simultaneous homogeneous complex equations. The Wigner 3j symbols are generated by the computer code, as discussed in 10 of Volume II, and checked against tabulated value" [Rotenberg et al. 1959]. The elements of Mp(k, Klb) for p = 0,1, .. , ,8 were computed by numerically evaluating the integral in (6.1.39), for r between band 4b. For f between 0 and 0.4, the value of g(r)-1 is practically zero for r larger than 4b.
0.70 '--_-'-_--L_ _L - _ - ' - _ - - ' - _ - - - ' L - _ - ' - _ - - ' - _ o om 0.10 0.11I 0.20 o.a 0.30 0.3lI o.~
Figure 6.1.3 Normalized phase velocity k/Kr in the Rayleigh limit as a function of fractional volume of scatterers f for ka = 0.1, and s = 3.24 0'
0.14 0.12
Figure 6.1.4 Effective loss tangent 2KdK r as a function of fractional volume of scatterers
f for ka = 0.1 and s = 3.24 0'
1.3 Dispersion RelcltionB nt Higher FrequencieB
For a given value of ka and f, the roots of the determinant were searched for in the complex I< plane (I<r + iI<i) using Muller's method. There are two good initial guesses:
(i) We first note that for media with sparse concentration of particles, the
exciting field will be approximately the same as the incident field, so that X~M) and X~N) both are approximately equal to 1. Replacing them by 1 in the generalized Ewald-Oseen extinction theorem of (6.1.49) gives the result of an effective propagation constant I F) under Foldy's approximation
I F) =
k - rr'tn o "'(2n + 1) (T(M) k 2 Ln
+ T(N ) n
(ii) The second initial guess is provided by the low-frequency solution in
(6.1.60). These two guesses could be used systematically to obtain quick convergence of roots at increasingly higher values of ka. In the following, we shall illustrate results for (a) the normalized phase velocity v p defined as k/ I<r and (b) the effective loss tangent (LT) defined as 2I<dI<r. The Percus-Yevick pair-distribution function is used unless otherwise specified. In Figs. 6.1.5 and 6.1.6, we plot, respectively, v p = k / I<,. and LT = Es . We note that the phase velocity first decreases with increasing frequency and then oscillates as frequency increases further. The oscillation is a characteristic of resonant scattering at higher frequencies. The loss tangent first increases rapidly with frequency and then saturates at high frequencies. In Figs. 6.1.7 and 6.1.8, we plot v p and LT versus ka for two different f values and for complex Es representing absorptive scatterers. On comparing the results of Figs. 6.1.6 and 6.1.8, we note that at low frequencies, absorption dominates over scattering as the loss tangent in Fig. 6.1.8 is much larger than that in Fig. 6.1.6. At higher frequencies the loss tangents are comparable in magnitude in the two cases. In Figs. 6.1.9 and 6.1.10 we show, respectively, vp and LT versus f for two different ka values. We note that the phase velocity decreases as f increases since the phase velocity for the scatterer is lower. As a function of j, LT first rises to a peak and then decreases as f further increases. In Figs. 6.1.11 and 6.1.12 we compare the results based on the PercusYevick (PY) pair function and the hole-correction (HC) pair function for ka = 0.5. For the HC pair function we have gHc(r) = 0 for r :S band gHc(r) = 1 for r > b. We note that the HC result for v p agrees very well