  We study the propagation properties of slow magneto-acoustic waves in a
multi-thermal coronal loop using a 3D MHD model, for the first time. A bundle
of 33 vertical cylinders, each of 100{\,}km radius, randomly distributed over a
circular region of radius 1{\,}Mm is considered to represent the coronal loop.
The slow waves are driven by perturbing the vertical velocity ($v_z$) at the
base of the loop. We apply forward modelling to the simulation results to
generate synthetic images in the coronal channels of SDO/AIA. Furthermore, we
add appropriate data noise to enable direct comparison with the real
observations. It is found that the synthetic images at the instrument
resolution show non-cospatial features in different temperature channels in
agreement with previous observations. Time-distance maps are constructed from
the synthetic data to study the propagation properties. The results indicate
that the oscillations are only visible in specific channels depending on the
temperature range of plasma existing within the loop. Additionally, the
propagation speed of slow waves is also found to be sensitive to the available
temperature range. Overall, we propose that the cross-field thermal properties
of coronal structures can be inferred using a combination of numerical
simulations and observations of slow magneto-acoustic waves.
