An analytical and numerical study is made of the influence of plasma finite pressure on the structure of Alfv¿n waves with large numbers of the azimuthal wave number, m ≫ 1, as well as of the conditions under which the waves can be poloidally polarized. The study is based on using the axisymmetric model of the magnetosphere assuming the background plasma inhomogeneity along field lines and across magnetic shells, the curvature of field lines, and an equilibrium electric field. The poloidality condition of the mode can imply that in the region where many azimuthal wavelengths fit in the region where the propagation of the wave is possible (the transparent region). The broader is the transparent region, the less stringent conditions are imposed on azimuthal wave numbers of the poloidally polarized Alfv¿n wave. In this paper it is shown that with nonzero plasma pressure, the transparent region is much broader when compared with cold plasma case. For instance, if finite pressure for the second standing harmonic along a field line of the wave (N = 2) is accounted for, the width of the transparent regions several thousand kilometers, and poloidal Alfv¿n waves can exist when m ≫ 10. This is consistent with experimental data on radially polarized Pc4 pulsations in the magnetosphere which have m from 50 to 150, whereas in the case of zero pressure the transparent region does not exceed a few tens of kilometers; therefore, in ¿ = 0 case, poloidally polarized waves must have too large values of m in excess of 1000, which is not observed. A further important result of this study implies that near the ring current maximum there must exist a cavity, and high-m oscillations enclosed within this cavity must be standing oscillations across magnetic shells.