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Chapter 1:
Solving problems of wave propagation in turbulence is a field that occupies the services of a small group of researchers. The methods used in this community and the results obtained are not generally known by researchers in other communities. The main reason is that the field is considered difficult, and if there is not an obvious need to investigate the effects of turbulence, they are neglected. The difficulty arises from the need to solve stochastic differential equations. Advances made by Tatarski and Rytov reduce problems to multiple integrals. These integrals are often difficult to evaluate since fractional exponents of functions appear in integrands. The final step in most cases is to evaluate these integrals numerically and to present the results as parametric curves. Many cases are run to develop some insight into how a quantity of interest varies with parameters. Becoming an expert in this field requires a great deal of time to become familiar with these graphical results so that one has some insight into various effects. As pointed out above, there is a formalism for reducing a problem to quadratures. This process is lengthy, and there are several ways of doing it. Different workers use different methods to get at the same result. This makes it difficult for the novice to understand the literature and to realize that there is some underlying order. This discourages a person with only a casual interest from developing a facility in this field. It was to make the solution of these problems more algorithmic that the methods expounded in this book were developed. In this book I use the Rytov approximation to reduce a very general problem to a triple integral. I develop techniques that allow one to evaluate these integrals analytically. The integrals that one encounters contain products of functions of which one or more is a Bessel function. Workers in the field look for these integrals in integral tables, and if unsuccessful, resort to numerical analysis. Even numerically, some of these integrals are difficult to evaluate. The integrand is often either the product of a function that goes to infinity multiplied by one that goes to zero at one of the integration limits, or the difference of two functions that each lead to a divergent integral. Great care must be exercised in evaluating these integrals.
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