Context: As coronal loops are spatially at least partially resolved in the longitudinal direction, attempts have been made to use the longitudinal profiles of the oscillation amplitudes as a seismological tool. Aims: We aim to derive simple formulae to assess which oscillation modes and which quantities of the oscillation (displacement or compression) are most prone to modifications induced by stratification of the equilibrium density along the loop. We furthermore clarify and quantify that the potential of such a method could be enhanced if observational profiles of the compression in the oscillations could be determined. Methods: By means of a linear expansion in the longitudinal stratification along with the “thin tube” approximation, the modifications to the eigenfunctions are calculated analytically. The results are validated by direct numerical computations. Results: Higher axial overtones are found to be more affected by equilibrium stratification and hence would provide a much better tool if observed. For the k-1th overtone the compression is found to be around (k + 2)^2/k2 times more sensitive to longitudinal density variation than the displacement. While the linear formulae do give a good indication of the strength of the effects of longitudinal density stratification, the numerical computations indicate that the corrections to the approximate analytical results are significant and cannot be neglected under the expected coronal conditions.