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of covariance function, for example hm 0 m0 0 i lim hm 0 m h i hm 0 m 0 i G0 0  0 h 3:113
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because G is an even function. The mean value in (3.112) is easily related to the characteristic function of m 0 , m0 0 , m00 0 , and can be calculated in terms of G; this gives hC x i Y x e
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 ! 2 k0 e2 x2 x2 00 3 1 G 0 O x : exp 2 12
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Let us compare this Equation to the exact solution as in (3.60) k2 e2 hC x i Y x eik0 x exp4 0 2 2
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Expanding the covariance function in power of x and integrating it we get exactly the same result as in (3.114). If the condition  2 2   k0 e    G00 0  ) 1
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is satis ed, we can use the so-called random variable approximation [31] hC x i Y x eik0 x exp ! 2 k0 e2 x2 : 2 3:116
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An equivalent condition is that the damping length xd 1=k0 e l=e corresponding to this approximation should be much shorter than the range of random correlations jG00 0 j 1=2 . If it is satis ed, the wave cannot escape the region where the random index is properly approximated by a random variable. The random variable approximation is easily applied to any propagation equation because we only need to solve a partial differential equation with constant coef cients, and average afterwards. If we want a higher order approximation we must solve a partial differential equation with linear or quadratic coef cients. The case of linear coef cients can, in principle, be solved by means of a generalized Laplace transformation, but this is rather complicated. In Section 3.8 we shall apply the random variable method to the coupled wave equations at short wavelength. Now we derive the short wave approximation for the scalar wave equation with point source:
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2 C r k0 1 em C r d r :
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We assume m to be a centered Gaussian random variable and hm2 i 1. Solving (3.117) we get C r expfik0 1 em Rg ; 4pR R jrj: 3:118
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Taking the mean value of this wave function we nd 2 exp ik0 R exp 1e2 k0 R2 2 : hC r i 4pR
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The damping due to phase mixing is thus exponential with a damping length Rd ejk0 j 1 > l: The condition Rd ( can be written as ejk0 j ) 1: It is thus a short wavelength condition ) l . The result of (3.119) disagrees with a result derived by Tatarskii [16 18] for jk0 j ) 1. His mean wave function hC r i exp ik0 R 1 2 4pR 1 e2 k0 R 1=2 3:121 3:120
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has not an exponential decrease, but has a damping length of
2 Rd e2 k0 1 ( l:
His result is expressed as a certain integral over the solution of the Bourret s equation (3.84), and this integral is calculated by the method of stationary phase. The expansion of the solution of the Bourret s equation used by Tatarskii is only valid for jk0 j ( 1. If the proper expansion is used, the result becomes identical with the one mentioned previously. This can also be checked on the 1D-model for which the calculation is easier.
3.6. AN EXACT SOLUTION OF THE SCALAR WAVE EQUATION In References [1 4,16 18], it was suggested that functional space integration might be used to solve these random variables equations. It suggested that it would be necessary to generalize the Wiener measure to more general stochastic processes
besides the Brownian motion. Nevertheless, it was shown in [31] that the reduced scalar wave equation can be related to a complex version of the heat equation and solved via a complex Wiener measure. The method indicated here can also be used for the Schrodinger equation with a random potential. Let us show how by following Reference [31] one can obtain the solution of the reduced scalar wave equation with random refractive index