The effects of collisions and radial inhomogeneity of the plasma density on the behaviour of the azimuthally-symmetric Trivelpiece-Gould modes (including the waveguide resonance and the wave resonance absorption at the upper hybrid frequency) are treated in this study in terms of their relation to gas-discharge production in an external magnetic field. Numerical solutions for the wave-field patterns and for the propagation properties of the waves are presented and their changes are discussed with a varying averaged plasma density, for various plasma-density profiles. First modifications due to collisions in homogeneous plasmas are presented for different values of the external magnetic field. Then, for a given magnetic field, the combined effects of collisions and Bessel-type radial profiles of the plasma density are emphasized. After which, for a given profile, the influence is discussed of varying the magnetic field. It became evident that, for low collision frequencies and a cyclotron frequency less than the wave frequency, the plasma-density profile shape has a strong impact on the results and that the wave properties, as the average density is varied for dominantly diffusive charged particle losses, are badly represented by wave properties in uniform plasmas. The behaviour discussed here should be useful in understanding the structure of diffusion-controlled discharges maintained in the fields of propagating Trivelpiece-Gould modes.