Random Equations

A random equation such as

describes linear waves and does not constitute a linear problem because the mean solutions do not satisfy the mean equation. This is because

In other words, the wave function and the refractive index are not statistically independent. If we try to evaluate (n2(r)T(r)), we must multiply Equation 2.57 by n2(r) and average afterward; this will yield the form n2(ri)n2(r)Ф(г), and

so on.

Keller [30] has obtained an equation for a function generating the entire set of moments. This equation helps with new approximation procedures, but does not solve the problem. The fact that even the lowest order moment of the wave function (Ф(г)) depends upon the infinite set of moments of the refractive index seems to make the problem hopelessly difficult. However, it happens that in certain limiting cases, one may obtain solutions that do not depend upon the entire set of moments of the refractive index.

The perturbation method, described in References 29, 30, gives Bouret’s equation, which depends only on the mean value and the covariance of the refractive index. It is only valid for wavelengths that are long compared to the range of index correlations. Conversely, for the random Taylor expansion (see References 30—32, 45), we need only the probability distribution of the refractive index and some of its derivatives at one fixed point. It is valid for wavelengths that are very short compared to the range of index correlations.

Another case of great interest is when n2(r) a Gaussian random function is. In this case, as was shown in Reference 45, it is then possible to get an exact solution of Equation 2.57 through functional integration, which gives all the moments of the wave functions in terms of mean value and the covariance of n2(r) (see References 45, 46, Chapter 3). Unfortunately, this method cannot be generalized to other equations such as the electromagnetic vector wave equation (2.33). Finally, it must be noted that no rigorous mathematical treatment of Equation 2.57 has been presented until now. This is mainly because we are not able to solve linear partial differential equations with non-constant (e.g., variable) coefficients [45].

All the aspects mentioned above can be usefully used for the description of both scalar and vector stochastic equations for different kinds of random media— quasi-homogeneous isotropic and anisotropic and irregular turbulent. We briefly present below some of these methods for the scalar and electromagnetic stochastic wave equations, based on different approximations. For a detailed description of the problem, we direct the reader to References 45, 46.

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