We present a time-dependent cosmic-ray modified shock model for which the calculated H-alpha emissivity profile agrees well with the H-alpha flux increase ahead of the Balmer-dominated shock at knot g in Tychos supernova remnant, observed by Lee et al (2007). The backreaction of the cosmic ray component on the thermal component is treated in the two-fluid approximation, and we include thermal particle injection and energy transfer due to the acoustic instability in the precursor. The transient state of our model that describes the current state of the shock at knot g, occurs during the evolution from a thermal gas dominated shock to a smooth cosmic-ray dominated shock. Assuming a distance of 2.3 kpc to Tychos remnant we obtain values for the cosmic ray diffusion coefficient, the injection parameter, and the time scale for the energy transfer of 10^{24} cm^{2} s^{-1}, 4.2x10^{-3}, and 426 y, respectively. We have also studied the parameter space for fast (300 km s^{-1} - 3000 km s^{-1}), time-asymptotically steady shocks and have identified a branch of solutions, for which the temperature in the cosmic ray precursor typically reaches 2-6x10^{4} K and the bulk acceleration of the flow through the precursor is less than 10 km s^{-1}. These solutions fall into the low cosmic ray acceleration efficiency regime and are relatively insensitive to shock parameters. This low cosmic ray acceleration efficiency branch of solutions may provide a natural explanation for the line broadening of the H-alpha narrow component observed in non-radiative shocks in many supernova remnants.
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