Observations at millimetre wavelengths provide a valuable tool to study the small scale dynamics in the solar chromosphere. We evaluate the physical conditions of the atmosphere in the presence of a propagating shock wave and link that to the observable signatures in mm-wavelength radiation, providing valuable insights into the underlying physics of mm-wavelength observations. A realistic numerical simulation from the 3D radiative Magnetohydrodynamic (MHD) code Bifrost is used to interpret changes in the atmosphere caused by shock wave propagation. High-cadence (1 s) time series of brightness temperature (T$_text{b}$) maps are calculated with the Advanced Radiative Transfer (ART) code at the wavelengths $1.309$ mm and $1.204$ mm, which represents opposite sides of spectral band~$6$ of the Atacama Large Millimeter/submillimeter Array (ALMA). An example of shock wave propagation is presented. The brightness temperatures show a strong shock wave signature with large variation in formation height between $sim0.7$ to $1.4$ Mm. The results demonstrate that millimetre brightness temperatures efficiently track upwardly propagating shock waves in the middle chromosphere. In addition, we show that the gradient of the brightness temperature between wavelengths within ALMA band $6$ can potentially be utilised as a diagnostics tool in understanding the small-scale dynamics at the sampled layers.
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