Statistical Theory of Wave Propagation in a Random Medium and the Irradiance Distribution Function

Abstract

The complete statistical description of the waves in a random medium can be obtained from the characteristic functional or the cumulant functional of the wave function. The basic equations of these functionals are found first for the gaussian medium and then for the nongaussian medium. As an application of those equations, the approximate equations of the arbitrary-order coherence functions are derived, which are found to be equivalent to those recently obtained by several authors using different methods. The operational method is introduced to solve the equation of the νth-order moment of the irradiance, and an exact solution is obtained for the particular correlation function of the medium, assuming the gaussian form of the incident wave. The irradiance probability-density function is obtained without approximation by the use of the νth-order moment of irradiance for this particular medium, and is found to be exactly the Rice–Nakagami distribution with respect to the log irradiance. This distribution approaches the log-normal distribution in the outside domain of the wave beam, and is also checked in several points. Finally, the operator representation of the physical variables is introduced, and it is shown that the various equations, e.g., the wave equation and the energy-conservation equation, in the statistical system of the wave plus the random medium can be represented by the same equations as the corresponding equations in the deterministic medium. The discussion is also extended on the basis of DeWolf’s result for the irradiance-distribution function.