Profiles of a subsonic equilibrium flow (solid) and a decreasing magnetic field (dashed). The segment of the tube with decelerating flow corresponds to rapid expansion of the flux tube. Smooth profiles of the Alfvén speed with c_A(0) = 1.5 and c_A(0) = 4 are plotted with dotted lines. The lower two panels display the continuous variation of c_A(0) from to 1.5 to 4 and the corresponding real and imaginary frequencies of the eigenmodes. The dashed lines represent an unstable mode. Strongly damped modes are not shown.