Self-consistent shock structure in the presence of cosmic rays and Alfven waves

We analyse the structure of a parallel shock including the back-reaction from cosmic rays and Alfven waves generated by the streaming instability. The investigation is facilitated by obtaining an integral of the wave growth equation which describes how the wave magnetic pressure is amplified by both adiabatic and instability effects. The results imply that the overall efficiency of the conversion of flow energy to cosmic ray energy may only become high provided that the wave power is saturated at some modest level by nonlinear effects