2 hC x i exp e2 k0 2 1 X e2 k2 2 n 0 n 0

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1 : 2 ik ik0 e2 k0 n=

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2 From (3.69), one can see that hC k i has the poles kn k0 ie2 k0 in= , which correspond to more and more damped partial waves in r-space. If ek0 ( 1, we can approximate hC k i by the rst partial wave n 0 that 2 gives again (3.66), apart from a factor exp e2 k0 2 6 1.

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3.4. APPROXIMATIONS OF THE PERTURBATION METHOD In this section, we try to justify certain approximation procedures using the formal perturbation series of Section 3.2. We recall, once more, that the random function m r is strictly stationary with respect to space transfer. Our investigations are only concerned with the mean propagator hGi. 3.4.1. Low Order Approximations In the section above, the mean wave function of the 1D-model was shown to be damped through destructive phase mixing; the damping length associated with this phenomenon is very large compared to the wavelength. It is therefore necessary to get approximate solutions for the mean propagator that are valid at long distances compared to the wavelength (radiation problem), and valid at long times compared to the period (initial wave problem). We show below the lower-order perturbation approximations to satisfy this condition, even the very small strength of random uctuations e. The Born approximation for the mean propagator is the lowest order non vanishing approximation of the corresponding perturbation series, that is,

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APPROXIMATIONS OF THE PERTURBATION METHOD

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and explicitly in k-space as hGi c2 e2 z4 c2 G k k0 3 0 d k 2 z2 c2 k02 2 c2 k2 z z2 c2 k2 3:71

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If so, G k k0 g k k0 ; k0 k is the FT of the covariance function. The squared perturbed propagator appearing in the second term on the random homogeneous space of (3.71) is the consequence of the wave vector conservation conditions. This term has two double poles z cK cjkj. It is well known that such double poles will yield two contributions to the inverse Laplace transform hG k; t i, proportional to t exp icKt and t exp icKt . In other words, the rst perturbation term has a singular behavior, that is, it increases without any limit as t ! 1. As this prevents any damping of the mean propagator, we conclude:

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The Born approximation is only a short time (or short distance) approximation.

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Next, it will be useful to have a better knowledge of the time dependence of the Born approximation. Let us, for example, take the covariance function as G r exp ! R ; R jrj: 3:72

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Here , as above, is the range of refraction index correlations. To evaluate the convolution integral 2 c G k k0 3 0 d k z2 c 2 k 0 2 we note that it is the FT of ! R exp izR=c expf i=c z ic= Rg exp 4pR 4pR and changing z into z ic= , yields and e2 c2 z4 z2 c2 k2 2 f z ic= 2 c2 k2 g : 3:74 c2 z ic= 2 c2 k2 3:73

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Besides the double poles z cK, two other poles have appeared in (3.74): z cK ic= . The corresponding contributions to LT and hG k; t i are proportional to expf icKtg expfic t= g. They are thus damped with a damping

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FUNDAMENTALS OF WAVE PROPAGATION IN RANDOM MEDIA

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time td =c. This damping time is the time the wave takes to travel a distance equal to the size of the scattering blobs. We call it, as in quantum eld theory, the interaction time tint . We turn back now to the perturbation series for the mean propagator, and show that, as we take more and more perturbation terms we get more and more divergent singular terms. The fourth order diagrams are (see Section 3.2):

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Because of the wave vector conservation condition, the same vector occurs in the middle and at the terminals of the diagram

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c The factor G 0 k; z z2 c2 k2 occurs thus twice in this diagram, producing singular terms proportional to t2 exp icKt and t2 exp icKt . More generally, any unconnected diagram that is the product of p connected diagrams produces singular terms proportional to tp exp icKt and tp exp icKt . We call them the leading terms of the diagram. Besides the leading terms, there are other singular terms with lower power of t, and also damped terms, with a damping time found to be always of the order of tint =c. If the damping time of the mean propagator is much longer than tint , the asymptotic time dependence of the mean propagator will be governed essentially by the leading terms. Let us show that singular terms arise in the radiation problem too. We are now looking for hG r i, whose FT may be obtained by changing z into ck0 in hG k; z i. Taking again the covariance function exp R= , we obtain

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