Effect of finite dimension of shock wave

In the Section 4.21.3 it was supposed that the plane shock wave front is infinite. In fact the dimension of the shock wave is limited. Berezhko et al. (M1988) show that the limiting of the shock wave's dimension can be roughly taken into account even in the simple one-dimensional approximation used in Section 4.21.3. Let us consider an accelerated particle which after interaction with the shock wave diffuses before the front. When a particle diffuses over the distance x > Rs, where Rs is the dimension of shock wave, it will have very little chance of coming back to the front and being accelerated again (in comparison with the case of an infinite plane shock wave). Therefore we can consider such particles as running away from the acceleration process. It is equivalent to putting at the distance Rs before the shock wave front an absorber of accelerated particles, so the boundary condition described by Eq. 4.21.14 will be changed to

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