We have carried out a numerical analysis of the spatial structure of Alfvén waves in a finite pressure inhomogeneous plasma in a dipole model of the magnetosphere. We have considered three magnetosphere models differing in maximum plasma pressure and pressure gradient. The problem of wave eigenfrequencies was addressed. We have established that the poloidal frequency can be either greater or less than the toroidal frequency, depending on plasma pressure and its gradient. The problem of radial wave vector component eigenvalues was considered. We have found points of Alfvén wave reflection in various magnetosphere models. The wave propagation region in the cold plasma model is shown to be significantly narrower than that in models with finite plasma pressure. We have investigated the structure of the main Alfvén wave harmonic when its polarization changes in three magnetosphere models. A numerical study into the effect of plasma pressure on the structure of behavior of all Alfvén wave electric and magnetic field components has been carried out. We have established that for certain parameters of the magnetosphere model the magnetic field can have three nodes, whereas in the cold plasma model there is only one. Moreover, the longitudinal magnetic field component changes sign twice along the magnetic field line.