Nonlinear Interaction of a Small Cold Beam and a Plasma. Part II

A single-wave model was previously used to construct a properly scaled universal solution for the initial nonlinear interaction between a small cold beam and a plasma. Two experimentally motivated extensions of the single-wave model are now related back to this universal solution. The first extension shows that although the universal solution was originally obtained for boundary conditions where the wave grows temporally, it is, with proper scaling, equally valid for the more realistic boundary conditions in which the wave grows spatially from the point where the beam enters the plasma. The second extension concerns the statistical properties of the single-wave model. The universal solution is generalized so that it depends analytically on the initial phase and amplitude of the single wave, and the ensemble average of this generalized solution is then taken over all possible values of the initial phase and amplitude. This is a straightforward average of the solution over the initial conditions, and it does not rely on the random phase approximation usually invoked in the theory of weak turbulence. The ensemble average corresponds to the typical experimental practice of time averaging the received signal, provided the ergodic theorem applies.