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, where s is given vector. This
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PROPAGATION IN STATISTICALLY INHOMOGENEOUS MEDIA
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is not of course the most general case, but it will be suf cient for our purpose. The scale of variation of this additional factor is h 1=jsj . The Fourier transform of h i exp
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conservation condition becomes thus
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s s s X r1 ; r2 ; . . . ; rn is X k1 n; k2 n; . . . ; rn n . The wave vector
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k1 k2 kn s 0: If we apply this to a connected diagram in k-space, such as
3:157
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we nd that k k0 s: 3:158
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Because of the condition h ) l, which can also be written jsj < K, the wave vectors at the terminals of a connected diagram are almost equal. Instead of a squared introduce a factor unperturbed propagator, the terminals of    ! c2 c2 c 1 1 1 1 1 1 0 : z2 c2 K 2 z2 c2 K 02 2 K 2 K 02 K z cK z cK K z cK 0 z cK 0 3:159 Let us nd the corresponding contribution to the inverse LT. It is proportional to 1 e icKt eic Kt e icK t eicK t K 2 K 02 K K0
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then, using the fact that K K 0 is small compared to K, we approximate (3.160) by 1 e icKt 1 eic K K t e icKt 1 e ic K K t 2K 2 K K0 K K0
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3:161
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For c K K 0 ( 1, we can make a Taylor expansion of (3.161) and nd ic te icKt teicKt : 2K 2 3:162
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This expression is not really singular, because it is only valid for cjK K 0 jt ( 1; this condition can also be written t ( h=c. If the damping time td , corresponding to the Bourret s approximation in the stationary case is small compared to h=c,
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then because the expression behaves exactly as a singular term, we call it a 1 pseudo secular term. As td $ e2 cK 4 3 , the condition td ( h=c can be written as e2 K 4 3 h ) 1 3:163
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and because of K ( 1 and e2 < 1, h must be very large compared to the wavelength.
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3.9. PROPAGATION IN HOMOGENEOUS ANISOTROPIC MEDIA Waves in anisotropic media, such as ionospheric plasma in the presence of an ambient magnetic eld (called geomagnetic), obey some partial differential equations that may be much more involved than the scalar wave equations. The wave function may have several components corresponding to a perturbed density, a perturbed velocity, a perturbed magnetic eld, and so on (see 9). Instead of a single dispersion equation we may have several equations corresponding to different wave modes. The wave modes are de ned to be the time harmonic functions of the propagation equations, with the boundary conditions taken into account. If we change the shape of the boundaries, we change also the nature of the wave modes. 3.9.1. Coupling Between Wave Modes Here we shall only consider waves in free space because the eigenfunctions are easily found by means of a FT. Let us rst consider the nonrandom case in order to introduce some de nitions and notations. Nonrandom Case. If the medium has constant parameters the propagation equations have constant coef cients. We assume that they can be written as a system of rst order partial differential equations (this is the most frequent case) as: @Cj r; t bjlm rl Cm r; t ; @t l 1; 2; 3; j; m 1; 2; . . . ; n 3:164
where n being the number of unknowns. Introducing now the FT Cj k; t exp ikr Cj r; t d3 r Equation (3.164) becomes @Cj k; t ibjlm kl Cm k; t @t 3:165
3:166
PROPAGATION IN HOMOGENEOUS ANISOTROPIC MEDIA
or in matrix notation i with Ajm k bjlm kl : 3:168 @C k; t A k C k; t @t 3:167
We assume that A k is diagonalizable, that is, there exists a matrix S k such that S 1 k A k S k D k : 3:169
D k being a diagonal matrix whose elements are the solution of the value equation det o A k 0: This equation has n solutions (distinct or not) o oj k ; j 1; 2; . . . ; n:
We call Equation (3.169) the dispersion equation of the jth mode in an anisotropic medium. For mathematical convenience, we shall take n modes even if some of them are not physically distinct such as the x and y polarization of an electromagnetic wave $ E exp ikz in an isotropic medium. We introduce the wave mode amplitude vector Q k; t S 1 k C k; t ; which satis es the following diagonal equation i @Q k; t D k Q k; t : @t 3:171 3:170
The jth component of Q k; t is called the complex amplitude of the jth wave mode; it satis es a separate propagation equation i @Qj k; t oj k Qj k; t @t 3:172
without summation on j. We may conclude that in a nonrandom anisotropic medium different wave modes are uncoupled. We also de ne the spectral energy density of the jth wave mode: Ej k; t jQj k; t j2 : 3:173