In this article, cosmic ray propagation is studied in the hydrodynamic approach. We establish a test particle picture with emphasis on the interaction with waves, in particular, the stochastic acceleration. The interactions between the energetic particles, the waves and the background plasma depend on the waves themselves. The test particle picture is nonlinear in the outset. The model can be completely classified by two parameters: the Alfven Mach number of the background flow and the ratio between the wave-action integral and the streaming integral. Several types of physical solutions are presented. Physical solutions are those with non-negative pressures and satisfying certain 'natural' boundary conditions. We study the cases where both far upstream and far downstream states are uniform. In a uni-directional wave system, cosmic ray pressure increases or decreases monotonically according to whether backward waves or forward waves are absent. When both waves are present cosmic rays can be accelerated efficiently in the sense that there is no cosmic ray far upstream but a finite amount far downstream. In most cases cosmic ray pressure increases non-monotonically, and in some cases it may even increase in two stages. We also examine the cases where the far downstream state is uniform while the upstream state is driven by some external source. Many more solutions are allowed, and solutions in constrast to the previous cases are presented. Stochastic acceleration may not be the dominant contributor to the energy content of cosmic rays. However, it plays the role of a catalyst or a trigger and have some subtle effect on the pressure profiles of cosmic rays and waves.
