Simplification of an analytical solution of the fourth-moment equation

The equation for the fourth moment of a wave propagating in a multiply scattering random medium has been solved by various methods. When the analytical solutions are compared with numerical solutions of the equation it is found that the fundamental solution together with a first-order correction term agree very closely with the numerical results over a wide range of distances and scattering strengths. Unfortunately, the correction term involves multiple integrals and so is difficult to evaluate. This paper shows how some of these integrations can be carried out and the results combined in such a way that an analytical form similar to the fundamental solution is obtained involving only a single integral. This simplified combined solution also agrees very closely with the numerical results.