Eigenfrequencies and optimal driving frequencies of 1D non-uniform magnetic flux tubes

The eigenfrequencies and the optimal driving frequencies for flux tubes embedded in uniform but wave-carrying surroundings are calculated, based on matching conditions formulated in terms of the normal acoustic impedances at the Bur tube boundary. The requirement of the equality of the normal acoustic impedance of the transmitted wave field with the normal acoustic impedance of the outgoing wave field selects the eigenmodes, while the equality of the ingoing and the transmitted normal acoustic impedance selects the optimal driving frequencies (Keppens 1996). Even if the flux tube is uniform, the eigenfrequencies can be complex due to leakage of wave energy into the surroundings. The case of uniform flux tubes has been considered previously (e.g. Cally 1986), and serves as a testcase of our formalism. We extend Cally's results by taking a radial stratification of the flux tube into account. The non-uniformity of the flux tube can introduce another cause for energy loss, namely resonant absorption internal to the flux tube. When resonant absorption occurs. we must incorporate the appropriate jump conditions over the dissipative layer(s). This can be done using a simple numerical scheme as introduced by Stenuit et al. (1995).