The derivation and solution of the modal flux equations for backscattering in a duct in the presence of random space and time fluctuations

In this paper we first derive the equations governing the energy fluxes propagating in each of the modes of a duct. In each mode there is a forward and backward component and the equations are intended to treat ducts in which backscattering plays a major role. The modal fluxes are coupled since there is transfer of energy between the modes that occurs as a result of random time and space sound-speed fluctuations in the medium in the duct. Since the fluctuations are both space and time dependent the governing equations are radiation transport equations. This is not the case if the fluctuations depend only on space. The basic method is to develop a coupled set of equations for the energy spectra in the modes and then to integrate over the frequency to obtain the fluxes. In the second section of this paper the modal flux equations are solved. A numerical result is presented to show how energy is transferred between modes. It is also shown how the reflected energy varies as a function of duct length.