Dependence on angular frequency ω of the real (k_R, the solid dots) and imaginary (k_I, open dots) parts of the longitudinal wavenumber. Two density ratios ρ_i/ρ_e = 4 and 100 are examined, and are given by the symbols and curves in red and blue, respectively. The vertical dash-dotted lines correspond to the critical angular frequency ω_c, which separates the leaky (to the left of ω_c) from trapped (right) regimes. Figure 1a presents an overview of the dispersion curves, while Fig. 1b (Figs. 1c and d) examines the portion where ω is close to ω_c (ω approaches zero). The dots are found by numerically solving the full dispersion relation (DR, Eq. (7)) by assuming a real ω but a complex-valued k = k_R + ik_I, while the curves represent the solutions to the approximate DR (see text for details).