CRs propagate approximately a diffusion length, $x_d = \kappa/u$, on each side of the shock before their return. Since the CRs move at a speed $v$, the average time between a pair of shock crossings is $2(x_d(1) + x_d(2))/v$, where the subscripts refer to the upstream and downstream values for $x_d$. From this the mean acceleration time is (e.g., Lagage & Cesarsky 1983) $$t_{acc} = \frac{3}{u_1-u_2}\int^{p_{max}}_{p_{min}} 
\left [ \frac{\kappa_1}{u_1} + \frac{\kappa_2}{u_2}\right ] \frac{dp}{p}.$$ This expression has been verified by a variety of numerical approaches. In particular it provides a reasonable estimate of the upper cutoff to the CR spectrum one expects if the particles are injected locally at the shock.

The most obvious nonlinear complication in DSA physics comes from a calculation of the pressure associated with the CRs. That is $$P_{cr} = \frac{4}{3} \int^{p_{max}}_{p_{min}} v p^4 f(p) d\ln p.$$ It is clear that for the strong shock limit ($q = 4$) $P_{cr}$ diverges logarithmically with $p_{max}$. Now, that momentum will in practice always be limited by some constraint, such as those we have discussed. Still, it is apparent that the pressure in CRs can potentially compete with the thermalized plasma or even dominate it. This is quite an important 