Turbulence transmission in parallel modified shocks using ray tracing
  We apply a semi-classical approach of handling waves as quasiparticle gas in a slowly varying flow -- analogous to ray tracing -- to calculate the Alfven wave transmission parameters, the resulting cross-helicity of the waves and the scattering-centre compression ratio, for cases where the shock thickness is large enough for the turbulent waves in the plasma to see the transition of the background flow parameters as smooth and slowly varying. For nonrelativistic shocks the wave transmission produces similar effects on the downstream turbulence and the scattering-centre compression ratio as does the transmission through a step shock: the downstream Alfven waves propagate predominantly towards the shock in the local plasma frame and, thus, the scattering-centre compression ratio is larger than the gas compression ratio. For thick relativistic shocks, however, we find qualitative differences with respect to the step-shock case: for low-Alfvenic-Mach-number shocks the downstream waves propagate predominantly away from the shock, and the scattering-centre compression ratio is lower than that of the gas. Thus, when taken into account, the Alfven wave transmission can decrease the efficiency of the first-order Fermi acceleration in a thick relativistic shock. 
Joni Tammi and Rami Vainio