Theoretical models of different possible MHD equilibria and wave propagation are required to better explain the ever increasing number of hi-resolution solar observations thanks to modern ground- and space-based instruments (SDO, Hinode, Solar Orbiter, DST, SST, DKIST). In this work (Skirvin et. al, MNRAS 2021), a numerical approach has been used to obtain the dispersion diagrams and eigenfunctions of any arbitrarily-symmetric inhomogeneous equilibrium. The proposed technique implements the shooting method to match necessary boundary conditions on continuity of displacement and total pressure of the slab or cylindrical waveguide. This approach was tested against well-known analytical solutions for MHD waves in uniform waveguides. This work is then extended further by investigating the dispersion diagrams and eigenfunctions when the equilibrium plasma density and background flow are radially inhomogeneous modelled as a Gaussian and sinc(x) profile. The resulting eigenfunctions of perturbed total pressure and displacement in an inhomogeneous plasma are compared with the uniform model and the physical differences in spatial structure are discussed which have implications for observers. This developed numerical technique can be used to analyse the MHD wave generation by small scale photospheric vortices under equilibria which are routinely observed in the solar atmosphere. The Solar Orbiter and DKIST era will provide observational data at much greater spatial resolution such that this new tool can be used in parallel to gain a better understanding of the waves which are observed.