On Scattering of Waves by Objects Imbedded in Random Media: Stochastic Linear Partial Differential Equations and Scattering of Waves by Conducting Sphere Imbedded in Random Media

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Abstract
A new inhomogeneous linear partial differential equation satisfied by the mean value of the solution of the corresponding inhomogeneous stochastic linear partial differential equation is derived. This new equation has the interesting phenomenon that the differential operators couple with the inhomogeneous terms to form new inhomogeneous terms of the equation. Physically, this means that the randomness of medium and source are coupled together to form new sources. The above approach is then used to derive the equation characterizing wave motions in random media due to random sources. Finally, the problem of scattering of a plane wave by a perfectly conducting sphere of radius a imbedded in a random medium is considered. By utilizing a “pseudopotential” to incorporate the effect of the boundary condition into the reduced wave equation and by the above result, for both ka large and small it is found that up to and including terms of order ε2 (ε = perturbation parameter) the mean value of the scattered field can be calculated from the same deterministic scattering problem with k replaced by an effective propagation constant kñ. The specialization of the new formulation to the problem of scattering of a plane wave by a perfectly conducting semi-infinite space checks with a previous result of Chen.

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