  Context. Radio relics in galaxy clusters are known to be good laboratories
for verification of the applicability of the diffusive shock acceleration (DSA)
model in its canonical version. The need for such verification stems from the
inconsistencies in the shock properties resulting from radio observations
compared to X-ray observations. Aims. In this article we aim to explore how the
presence of a second shock in the vicinity of a relic modifies the spectrum of
accelerated electrons and decipher which of the involved parameters can have a
significant impact on their shape. Methods. We analytically studied DSA of
cosmic rays in two stationary shocks aiming to investigate the change of the
distribution function. The latter eventually leads to spectrum slope deviations
visible in different observations and simulations that do not appear to be
explained by the case wherein cosmic rays interact with a single shock wave.
Results. We obtain a complex distribution function $f(x,p)$ depending on many
parameters (distance between two shocks, compression ratios, spatial diffusion
coefficients, injection value, etc.). This function reveals modifications that
occur because of the coupled acceleration in both shocks. Apparently,
deviations in the particle spectrum from the pure power law depend on a few
parameters 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 as
an example, we demonstrate a potential cause of spectral disturbances in radio
relics. In general terms, our findings appear to correlate with results from
the literature when the distance between the shocks is of the order of the
width of a radio relic and $\kappa_{1}/\kappa_{2} \propto 3$.
