We analyze the evolution of shock waves in high-resolution 3D radiative MHD simulations of the quiet Sun and their synthetic emission characteristics. The simulations model the dynamics of a 12.8x12.8x15.2 Mm quiet-Sun region (including a 5.2 Mm layer of the upper convection zone and a 10 Mm atmosphere from the photosphere to corona) with an initially uniform vertical magnetic field of 10 G, naturally driven by convective flows. We synthesize the Mg II and C II spectral lines observed by the IRIS satellite and EUV emission observed by the SDO/AIA telescope. Synthetic observations are obtained using the RH1.5D radiative transfer code and temperature response functions at both the numerical and instrumental resolutions. We found that the Doppler velocity jumps of the C II 1334.5 A IRIS line and a relative enhancement of the emission in the 335 A SDO/AIA channel are the best proxies for the enthalpy deposited by shock waves into the corona (with Kendalls $tau$ correlation coefficients of 0.59 and 0.38, respectively). The synthetic emission of the lines and extreme ultraviolet passbands are correlated with each other during the shock wave propagation. All studied shocks are mostly hydrodynamic (i.e., the magnetic energy carried by horizontal fields is $leq{}$2.6% of the enthalpy for all events) and have Mach numbers > 1.0-1.2 in the low corona. The study reveals the possibility of diagnosing energy transport by shock waves into the solar corona, as well as their other properties, by using IRIS and SDO/AIA sensing observations.
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