Damping time ratio as a function of density ratio (left) and as a function of the average density of the tube (right). Red asterisks represent the results from simulations. Solid black lines correspond to the results from the eigenvalue problem. Horizontal dashed lines represent the case of tubes with uniform density. The red dotted lines represent the approximations given by Eqs. (24) and (25). The blue dotted-dashed lines in the left and right panels represent the fits given by Eqs. (26) and (27), respectively.