Kink motions with (a) S​_x and (b) A_x patterns in a two-circular-tube configuration. Plotted are the time sequences of the x speed sampled at the left tube center (v_x(−d/2, 0; t), the solid curves) together with the best-fit damping envelopes (dashed). The time-dependent computations pertain to a fixed combination, [ρ_i/ρ_e = 5, ℓ/ℛ = 0.4, d/ℛ = 2.5, kb = π/30], with ℓ and ℛ being the nonuniform layer width and mean tube radius implemented by Ruderman & Roberts (2002), respectively. A fitting procedure with an exponentially damped cosine was performed over the duration exactly encompassing the extrema indicated by the red asterisks, even though the best-fit envelopes are plotted for the entire time interval. Kink motions are excited with the perturbation described by Eq. (20) with appropriate signs.