Dependence on the axial wavenumber (k) of the real part of the mode frequency (ℜΩ) pertaining to the classic BVP. The density contrast is fixed at ρ_i/ρ_e = 5, whereas several steepness parameters (μ) are examined in different panels as labeled. The magenta dashed lines represent ℜΩ = kv_Ai and ℜΩ = kv_Ae, the latter separating discrete leaky modes (DLMs, to its left) from trapped modes (to its right). The trapped modes are identical to the proper eigenmodes in Fig. 2; the mode labeling convention therein applies here. Mode labels with j ≥ 2 also make sense for the DLMs, as illustrated in panel a.