Galaxy cluster merger shocks are the main agent for the thermalization of the intracluster medium and the energization of cosmic ray particles in it. Shock propagation changes the state of the tenuous intracluster plasma, and the corresponding signal variations are measurable with the current generation of X-ray and Sunyaev-Zeldovich (SZ) effect instruments. Additionally, non-thermal electrons (re-)energized by the shocks sometimes give rise to extended and luminous synchrotron sources known as radio relics, which are prominent indicators of shocks propagating roughly in the plane of the sky. In this short review, we discuss how the joint modeling of the non-thermal and thermal signal variations across radio relic shock fronts is helping to advance our knowledge of the gas thermodynamical properties and magnetic field strengths in the cluster outskirts. We describe the first use of the SZ effect to measure the Mach numbers of relic shocks, for both the nearest (Coma) and the farthest (El Gordo) clusters with known radio relics.
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