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Context. Radio relics in galaxy clusters are known to be good laboratoriesfor verification of the applicability of the diffusive shock acceleration (DSA)model in its canonical version. The need for such verification stems from theinconsistencies in the shock properties resulting from radio observationscompared to X-ray observations. Aims. In this article we aim to explore how thepresence of a second shock in the vicinity of a relic modifies the spectrum ofaccelerated electrons and decipher which of the involved parameters can have asignificant impact on their shape. Methods. We analytically studied DSA ofcosmic rays in two stationary shocks aiming to investigate the change of thedistribution function. The latter eventually leads to spectrum slope deviationsvisible in different observations and simulations that do not appear to beexplained by the case wherein cosmic rays interact with a single shock wave.Results. We obtain a complex distribution function $f(x,p)$ depending on manyparameters (distance between two shocks, compression ratios, spatial diffusioncoefficients, injection value, etc.). This function reveals modifications thatoccur because of the coupled acceleration in both shocks. Apparently,deviations in the particle spectrum from the pure power law depend on a fewparameters such as $Q_{1}/Q_{2} $, $\kappa_{1}/\kappa_{2} $, $r_{1}/r_{2} $,and $L$. Although we do not verify this idea by taking a particular cluster asan example, we demonstrate a potential cause of spectral disturbances in radiorelics. In general terms, our findings appear to correlate with results fromthe literature when the distance between the shocks is of the order of thewidth of a radio relic and $\kappa_{1}/\kappa_{2} \propto 3$.

Acceleration of cosmic rays by double shock waves in galaxy clusters: application to radio relics