On the renormalization method in random wave propagation

In this work it is shown that the diagrams that correspond to the first renormalization equation for the coherent field, in a statistically homogeneous and isotropic medium, are the dominant diagrams. To study the cumulative effect of the omitted diagrams, which increase in number with each power of the expansion parameter e, we sum the next largest omitted diagrams. This sum is proportional to the propagation path length times the usual first renormalization solution. The above analysis and a more compact one that utilizes the full Dyson equation both yield a validity criterion that is basically distance independent. The derivation of the validity criterion is primarily based on two observations that are not used by other authors. First we note and account for the fact that the solution of the first renormalization integral equation is always approximated. Secondly, we utilize the simple observation that phase accumulation and rate of decay are unimportant if the amplitude of the wave is already insignificant. We obtain the same criterion as other authors for k0l » 1, where k0 is the free space wave number and l is the correlation length of the dielectric fluctuations; for k0l « 1 the criterion is new. Finally, a relatively simple method of solving the first renormalization integral equation for the correlation function of the field is presented for the case k0l » 1 and some justification for the validity of this equation is given.