Waves in magnetized polytropes

We consider the linear oscillations of a plane-parallel semi-infinite electrically conducting atmosphere with a constant temperature gradient, subjected to an imposed uniform gravitational acceleration and uniform magnetic field. The oscillations are treated in the ideal (dissipationless) limit and the uniform gravitational acceleration and magnetic field are taken to be co-aligned with the prevailing temperature gradient. It is demonstrated that atmospheric motions with prescribed horizontal variations of the form exp(ikx), with k real, possess both a discrete set of complex eigenfrequencies omega(n), n=0, 1, 2,..., and a continuous spectrum. These two behaviours derive from a particular fourth-order ordinary differential equation that arises in the solution of the initial value problem via an integral transform and describes the coupled fast- and slow-magnetoatmospheric waves. We devote considerable efforts to document how the discrete spectrum varies in response to incremental changes in the horizontal wavenumber k and we compare and contra-st this behaviour with that found by Lamb for the same atmosphere, but with the magnetic field being absent. Implications for the helioseismology of sunspots are discussed.