Abstract
The theory of Alfv\'en resonance effects on the wave modes of a tokamak is extended beyond the incompressible magnetohydrodynamic model to include finite-($\frac{\ensuremath{\omega}}{{\ensuremath{\Omega}}_{i}}$) effects and compressibility. The discrete spectrum of compressional Alfv\'en waves consists of a sequence of frequencies with finite damping decrements resulting from the Alfv\'en resonance. The finite-frequency effects can cause the damping to almost vanish. This permits Alfv\'en resonance heating via high-$Q$ eigenmodes in large tokamaks.
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