The low frequency wave and instability spectrum of a cylindrical, uniform density, zero beta plasma in a Bessel function magnetic field is analysed. The analysis has application to wave heating of reversed field pinches and energy propagation in, and instabilities of, twisted solar coronal loops. A surface Alfven wave which can be Alfven resonance damped is identified, and its connection to the unstable external kink mode is discussed. A distinctive extra mode is identified as a poloidal fast Alfven wave with wave fields concentrated at the position of zero axial equilibrium magnetic field and with a dispersion relation dependent dominantly on the poloidal wavenumber and only weakly on the axial wavenumber. Finally, the spectrum of discrete Alfven waves, or global Alfven eigenmodes, is also identified.