Response of a premixed turbulent flame stabilized behind a bluff body in a cylindrical duct to weak, one-dimensional, uniform or non-uniform ( the so-called "axial convective waves''), harmonic perturbations of the oncoming flow velocity is studied by extending the well-known kinematic model of such flame oscillations in order to allow for flame speed increase with distance from the bluff body for a long flame. Such an increase is an important peculiarity of practical premixed turbulent flames, which develop as convected by the mean flow from a flameholder. Comparison of results computed in the cases of a constant or increasing flame speed shows that the flame speed development notably affects both the gain and phase of the flame transfer function, the effects being more pronounced for the axial convective waves as far as the gain of the transfer function is concerned.