F. Generalizations: II. Energy injection and refreshed shocks
The simple adiabatic model assumes that the energy of the GRB is constant. However, the energy could change if additional slower material is ejected behind the initial matter. This would be expected generically in the internal shock model. In this model the burst is produced by a series of collisions between shells moving at different velocities. One naturally expect here also slower moving matter that does not catch up initially with the faster moving one. However, as the initially faster moving matter is slowed down by the circum-burst matter this slower matter eventually catch up and produces refreshed shocks [207, 335, 366].
There are two implications for the refreshed shocks. First the additional energy injection will influence the dynamics of the blast wave [335, 366]. This effect can be modelled by modifying E in Eq. 74, but the effect of additional mass carrying the slower energy must be included in some cases. This would change the decay slope from the canonical one and produce a slower decay in the light curve. In the following section Section VIIG I describe a scheme for calculating the light curve resulting from a variable blast wave energy. If the additional matter is emitted sporadically then the shell collision could produce initial temporal variability in the early afterglow signal. Fox et al. , for example, suggest that refreshed shocks are the origin of the variability in the early afterglow of GRB 021004.
A second effect is the production of a reverse shock propagating into the slower material when it catches up with the faster one . This is of course in addition to the forward shock that propagates into the outer shell. This reverse shock could be episodal or long lasting depending on the profile of the additional matter. Kumar and Piran  consider two shells with energies E1 and E2 in the outer and the inner shells respectively. The outer shell is moving with a bulk Lorentz factor 0c ~ 5(t / day)3/8 at the (observed) time, t, of the collision. As the inner shell catches up with the outer one when both shells have comparable Lorentz factors the reverse shocks is always mildly relativistic. The calculation of the shock is slightly different than the calculation of a shell propagating into a cold material (another shell or the ISM) discussed earlier. Here the outer shell has already collided with the ISM. Hence it is hot with internal energy exceeding the rest mass energy. The reverse shock produces emission at a characteristic frequency that is typically much lower than the peak of the emission from the outer shell by a factor of ~ 70c2(E2 / E1)1.1, and the observed flux at this frequency from the reverse shock is larger compared to the flux from the outer shell by a factor of ~ 8(0c E2 / E1)5/3. This emission is typically in the radio or the FIR range.
Kumar and Piran  suggest that due to angular spreading the refreshed shocks produce an enhancement with a typical time scale t ~ t. Granot et al.  stress that the fact that energy necessarily increases in refreshed shocks, the overall light curve must have a step-wise shape (above the continues power-law decline) with a break at the corresponding shocks. This behavior was seen in GRB 030329. However there the transitions are fast with t < t. Granot et al.  point out that if the refreshed shocks take place after the jet break (as is likely the case in GRB 030329) then if the later shells remain cold and do not spread sideways we would have t ~ tjet < t. This explains nicely the fast transitions seen in this burst.