In this paper we study how the normal fast modes of a coronal loop are modified by the addition of density structure along the loop axis. Using isobaric and thermal equilibria with different heating functions, a density profile for a loop is derived, and then its oscillatory modes are studied with techniques similar to those in our previous works. The main result is that the frequency and spatial structure of the trapped modes are very sensitive to density variations inside the loop, especially at the apex, where the density depends strongly on the footpoint density and the ratio of heating to thermal conduction. Moreover, different tested heating profiles do not introduce important differences with respect to the case of uniform heating along the loop.