  Hydromagnetic waves, especially those of frequencies in the range of a few
milli-Hz to a few Hz observed in the Earth's magnetosphere, are categorized as
Ultra Low Frequency (ULF) waves or pulsations. They have been extensively
studied due to their importance in the interaction with radiation belt
particles and in probing the structures of the magnetosphere. We developed an
approach in examining the toroidal standing Aflv\'{e}n waves in a background
magnetic field by recasting the wave equation into a Klein-Gordon (KG) form
along individual field lines. The eigenvalue solutions to the system are
characteristic of a propagation type when the corresponding eigen-frequency is
greater than a cut-off frequency and an evanescent type otherwise. We apply the
approach to a compressed dipole magnetic field model of the inner
magnetosphere, and obtain the spatial profiles of relevant parameters and the
spatial wave forms of harmonic oscillations. We further extend the approach to
poloidal mode standing Alfv\'{e}n waves along field lines. In particular, we
present a quantitative comparison with a recent spacecraft observation of a
poloidal standing Alfv\'{e}n wave in the Earth's magnetosphere. Our analysis
based on KG equation yields consistent results which agree with the spacecraft
measurements of the wave period and the amplitude ratio between the magnetic
field and electric field perturbations. We also present computational results
of eigenvalue solutions to the compressional poloidal mode waves in the
compressed dipole magnetic field geometry.
