We examine the effect of a uniform vertical magnetic field on the modes of an isothermal stratified atmosphere. The present investigation is a continuation of earlier work by Hasan & Christensen-Dalsgaard in which this problem was studied for rigid boundary conditions. In this paper, the earlier results are extended to different sets of boundary conditions. We demonstrate explicitly how these boundary conditions affect the various elementary wave modes present in the atmosphere. In the weak-field limit, an analytic expression for the dispersion relation is derived, which allows the effect of a weak magnetic field on the modes to be studied. We show that, to lowest order in our perturbation expansion, the oscillation spectrum can be analyzed in terms of (a) p- and g-like modes; (b) a magnetic Lamb mode; (c) magnetic or slow modes; and (d) a gravity-Lamb mode. The first three of these were present in the previous analysis for rigid boundaries, whereas the last is a consequence of the vertical gradients of the displacements at the boundaries. We focus our attention on the properties of this mode and show that it is present even in the moderate to strong field case as a magnetogravity-Lamb mode. The recognition and physical interpretation of this mode is a new feature of the present analysis. We also examine the nature of the eigenfrequency curves in the diagnostic (or K-Ω) diagram and find that, similar to the previous analysis, the modes undergo avoided crossings. However, the nature of the solutions in the present case is more complicated, especially when triple-mode interactions occur. Furthermore, the connectivity of the curves in the K-Ω diagram can be strongly influenced by the choice of boundary conditions. Our results, though somewhat idealized, find application in the analysis of waves in sunspots. It is conjectured that conditions for the existence of the magnetogravity-Lamb mode may also be satisfied in the subphotospheric layers of the Sun.
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