Dependence of the cut-off period for longitudinal waves in an exponentially divergent magnetic flux tube on the vertical coordinate z. The colours correspond to different values of the magnetic field scale height, L = 0.2H (red), 0.5H (purple), H (black), 2H (green), 5H (blue). The dashed lines mark the acoustic cut-off period (lower) and Brunt-Väisälä period (upper). The panels show the cut-off periods for the high a), of the order of unity b), low c), and very low d) plasma β in the solar corona (see the corresponding values of the Alfvén speed and sound speed in the text).