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Alfven-wave transmission through a parallel shock in case of an anisotropic gas pressure ($P_{\parallel }\neq P_{\perp } $) is considered from the point of view of test particle acceleration in such shock waves. Treating the shock's gas compression ratio and the firehose factor, $ \phi =1-4\pi (P_{\parallel }-P_{\perp })/B^{2} $, on both sides of the shock as given parameters, we calculate the wave-transmission coefficients for circularly-polarized Alfven waves of finite amplitude. We also give an equation for the shock's gas compression ratio including the effects of pressure anisotropy and waves. We study wave transmission and test-particle acceleration in a model, where the downstream plasma is isotropic and the upstream plasma remains stable against the firehose instability ($ \phi > 0 $). The results can be arranged in a following manner with respect to the upstream parallel plasma beta, $\beta _{\parallel }\equiv 8\pi P_{\parallel }/B_{0}^{2} $ (with $ B_{0} $ being the ordered magnetic field): (i) for shocks with $ \beta _{\parallel }\ll 2 $ the results for particle acceleration are only quantitatively different for anisotropic upstream plasma; (ii) for $ \beta _{\parallel }\sim 2 $, large anisotropies alter the wave-transmission and particle acceleration picture completely relative to the isotropic case yielding much harder energy spectra with spectral indices $ 1< \Gamma < 2 $ for all shocks with $ 1< r< 4 $; (iii) for very hot and firehose-stable plasmas, the qualitative changes in particle acceleration due to pressure anisotropies are limited to the weakest shocks and, thus, should be analyzed in a non-linear manner to confirm the results.

The effect of anisotropic gas pressure on Alfvén-wave transmission and test-particle acceleration at parallel shock waves