Let us now nd the free oscillations that satisfy

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2 4 2 k0 k2 e2 k0 w k dij 1 e2 k0 m k ki kj hEj k i 0:

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There are two kinds of oscillations: a) Transverse oscillations. Here hei and k are perpendicular. The dispersion equation is

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2 4 k0 k2 e2 k0 w k 0:

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b) Longitudinal oscillations. Here hei and k are parallel. The dispersion equation is

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2 1 e2 k0 w k k2 m k 0:

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Let us also nd the renormalized wave number K for transverse waves. We take for this purpose the correlation function exp R= . After some straightforward manipulations we nd that for k ( 1 ( l 2 1 2 w K 2 1 2iK 2 O 4 K : 3 3k0 3:153

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The dispersion equation for transverse oscillations is solved for the renormalized wave number ! 1 1 2 2 2 K k0 1 e2 k0 w k 1=2 % k0 1 e2 e2 k0 2 1 2ik0 : 6 3 3:154

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FUNDAMENTALS OF WAVE PROPAGATION IN RANDOM MEDIA

We compare this result to the corresponding formula for the scalar wave equation obtained in References [16 18] or deduced from Keller s result [14] with a covariance function of exp R= ! 1 2 3:155 K k0 1 e2 k0 2 1 2ik0 : 2 First of all, the imaginary part of K in (3.155) has been reduced by factor $(1/3) with respect to that in (3.154), the damping length of the mean wave have thus increased by 50%. Secondly, due to the additional negative term $ 1k0 e2 in (3.154) 6 2 compared with (3.155), the real part of K is less than the real part of k0 if 2k0 2 < 1. As we assumed that k ( 1 ( l , this is satis ed. We conclude that the effective phase velocity of transverse waves increases at long wavelengths, instead of decreasing as is the case for the scalar wave equation. This needs some explanation. There are two wave modes actually in this medium: the transverse mode, whose phase velocity is approximately o=k0, and the longitudinal wave mode, whose phase velocity is much longer (in nite in the nonrandom case). Due to the term ki kj Ej of (3.143), the wave modes are coupled and part of the mean transverse wave has traveled part of its way as a longitudinal wave. The traveling time being thus decreased, the phase velocity is increased. Without this coupling it would be impossible to explain the increase of the phase velocity. As the additional term 1k0 e2 6 does not depend on , it is possible that it corresponds rather to a diffraction effect by the scattering blobs (whose sizes are small compared to the wavelength), than to a volume scattering effect. 3.8. PROPAGATION IN STATISTICALLY INHOMOGENEOUS MEDIA In this section we assume that the mean refractive index is constant through space, but that its random part is not strictly stationary with respect to space translations. The correlation functions G x; x0 are functions of (x x0 ) and also of (x x0 )/2. We shall assume that this additional space dependence has a scale of variations h that is large compared to the wavelength. As there is no homogeneous turbulence in nature, this is a very common situation. The FT m k of such a slowly varying random function does not satisfy the wave vector conservation condition hm k1 m k2 . . . m kp i 0; if k1 k2 kp 6 0 3:156

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and does not give rise to any singular terms in the perturbation series. All arguments based upon the extraction of the leading singular terms seem to disappear suddenly. We show however that if the condition e2 K 4 3 h ) 1 is satis ed, in addition to the usual condition K ( 1, nothing is changed, because we have pseudo singular terms. We assume that the additional space variation of the correlation functions h i X r1 ; r2 ; . . . ; rn is given by a factor exp

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