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In this paper we develop the hydromagnetic wave equations for\udtoroidal Alfvén waves in a background axi-symmetric magnetic\udfield. In the case where spatial variations are directed along\udthe ambient magnetic field direction, the equations can be cast\udin a Klein-Gordon form in which the adiabatic-geometric amplitude\udfactor of the perturbations varies as √ρL5sin5θ along a magnetic field line (where θ is\udcolatitude and L the L-shell number) and the cut-off\udfrequency, associated with the Klein-Gordon form, displays an\udastonishing variation with distance along a field line (see\udEqs. 35 and 37 of the text), in\udthe case of a dipole magnetic field. We compute the eigenvalues\udand eigenfunctions for the Earth's dipole field which are relevant\udto geomagnetic pulsations

Klein-Gordon equations for toroidal hydromagnetic waves in an axi-symmetric field