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The mean energy of the second wave mode at time t is thus *   q + 2eb2 2 1 cos t o1 o2 2 4b2 e2 E2 t hjQ2 t j i o1 o2 2 4b2 e2 3:207 or in terms of the probability density P(b) of the random variable b E2 t
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jQ2 t j2 P b db:
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Let us rst assume that o1 o2 , then 1 E2 t 2
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as t ! 1 and then E2 t ! 1. In the more general case o1 6 o2 , we take the 2 probability density as P b s 1 : 2 s2 p b 3:210
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The asymptotic energy distribution is easily calculated by means of a LT. lim hE2 t i 1 2es 2 2es jo2 o1 j 1 : 2 3:211
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The condition of effective energy transfer is thus jo2 o1 j 2es: 3:212
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It is found that the time required to reach the equilibrium energy distribution is of the order of magnitude of es 1 . The situation is the same as in the long wavelength approximation, that is:
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We have an important energy transfer in a medium with very small random uctuations, but this requires a very long time.
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Also, we note that in the long time behavior of a random medium, energy transfer is always an important process between waves whose frequencies are not very different (see condition (3.203) at long wavelengths, and condition (3.212) at short wavelengths).
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[1] Dyson, F., The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev., vol. 75, 1949, pp. 486 497. [2] Foldy, L. L., The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., vol. 67, 1945, pp. 107 119. [3] Lax, M., Multiple scattering of waves, Rev. Modern Phys., vol. 23, 1951, pp. 287 310. [4] Lax, M., Multiple scattering of waves. II. The effective eld in dense systems, Phys. Rev., vol. 85, 1952, pp. 621 629. [5] Furutsu, K., On the group velocity, wave path and their relations to the Poynting vector of E-M eld in an absorbing medium, J. Phys. Soc. Japan, vol. 7, 1952, pp. 458 478. [6] Salpeter, E. E., and H. A. Bethe, A relative equation for bound-state problems, Phys. Rev., vol. 84, 1951, pp. 1232 1239. [7] Furutsu, K., On the statistical theory of electromagnetic waves in a uctuating medium (I), J. Res. NBS (Radio Prop.), vol. 67D, 1963, pp. 303 323. [8] Matsubara, T., A new approach to quantum-statistical mechanics, Prog. Theoret. Phys., vol. 14, 1955, pp. 351 361. [9] Martin, P. C., and J. Schwinger, Theory of many-particle systems (I), Phys. Rev., vol. 115, 1959, pp. 1342 1349. [10] Schwinger, J., On the Green s functions of quantized elds, I, II, Proc. Natl. Acad. Sci., vol. 37, 1951, 452 455. [11] Twersky, V., Multiple scattering of electromagnetic waves by arbitrary con gurations, J. Math. Phys., vol. 8, 1967, pp. 569 610. [12] Buslaev, V. S., in Birman, M. Sh., ed., Topics in Mathematical Physics, Consultans Bureau, New York, vol. 2, 1968. [13] Bourret, R. C., Fiction theory of dynamical systems with noisy parameters, Can. J. Phys., vol. 43, 1965, pp. 619 627. [14] Keller, J. B., Stochastic equations and wave propagation in random media, Proc. Sympos. Appl. Math., vol. 13, Amer. Math. Soc., Providence, R. I., 1964, pp. 145 147. [15] Chernov, L. A., Wave Propagation in a Random Medium, McGraw-Hill, New York, 1960. [16] Tatarskii, V. I., Wave Propagation in a Turbulent Medium, McGraw-Hill, New York, 1961. [17] Tatarskii, V. I., and M. E. Gertsenshtein, Propagation of wave in a medium with strong uctuations of the refractive index, JEFT, vol. 17, 1967, pp. 548 563. [18] Charnotskii, M. I., J. Gozani, V. I. Tatarskii, et al., in Wolf, E., ed., Progress in Optics, Elsevier, Amsterdam Holland, Vol. 32, 1993. [19] Bassanini, P., Wave propagation in a one-dimensional random medium, Radio Sci., vol. 2, 1967, pp. 429 436. [20] Kieburts, R. B., Application of coherence theory to propagation of a quasi-monochromatic eld in a random medium, IEEE Trans. Antennas and Propag., vol. AP-15, No. 1, pp. 76 80. [21] Brown, W. P. Jr, Propagation in random media-Cumulative effect of weak inhomogeneities, IEEE Trans. Antennas and Propag., vol. AP-15, 1967, pp. 81 89.
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