In Section 2.21 we reviewed the paper of Vainio and Schlickeiser (1999a) on the bulk transport of CR particles caused by the quasi-linear interactions with transverse, parallel-propagating plasma waves. Vainio and Schlickeiser (1999a) note that in cosmic shock waves particles can gain energy through first and second order Fermi mechanisms by multiple shock crossings and stochastic downstream acceleration, respectively. When first-order Fermi acceleration dominates, the spectral index of the shock accelerated particles is and is thus determined by the scattering-center compression ratio of the shock, where u1 and u2 are the upstream and downstream flow speeds of the plasma in the shock-frame, and V and V2 are the respective relative bulk speeds of particles owed to finite phase speed of the waves (see Section 2.21). Vainio and Schlickeiser (1999a) examine the effects of the non-zero wave speeds at the first-order Fermi acceleration of CR. In the upstream region of the shock they assume that all waves are propagating against the flow (backward waves, w < 0) if they are self-generated by the accelerated particles through the streaming instability. Then they assumed that backward waves are generated at all frequencies -1 < f' < Op . All upstream waves that have - w < u will then be converted to the shock and become downstream waves. In the downstream region, owing to the interaction of the upstream waves with the shock, waves propagating in both directions will be present. For Alfven waves Vainio and Schlickeiser (1999b,c) showed that the dominating downstream wave components are the backward ones. However, since it was assumed that the waves are generated in the upstream region it can not have downstream backward waves propagating faster than the shock relative to the downstream plasma. This implies that there are no backward waves at frequencies r = (( + 2)/( -1)
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