Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation


  • K. Furutsu


In a previous paper, the characteristic functional of the wave function has been found to be the solution of a basic equation which has the same form as the original wave equation and is given in terms of the characteristic functional of the refractive index. In this paper, the theory is applied to the case of scalar wave propagation in a medium of randomly distributed particles, and the transport equation is derived by use of an analytical procedure. The renormalization of the medium and of the one-particle scattering matrix is explicitly introduced, and is found to play an essential role for energy conservation. The Fourier transform of the resulting single Green's function (in the infinite medium) has a set of poles of infinite number. The transport equation is expressed by a series of residue values at the poles of the original and complex conjugate Green's functions, and has a wider range of applicability than that of the conventional transport equation. As a method of solving the transport equation, a set of eigenfunctions is introduced, and the solution is obtained in terms of the eigenfunction series. The diffusion function associated with each eigenfunction is also obtained.