Propagation diagram of the inner 30% of a red giant model with M = 1.25 M_⊙,  Z = 0.020. Characteristic frequencies $\widehat{S}/(2\pi )$ (orange) and $\widehat{N}/(2\pi )$ (purple) as a function of fractional radius. The dashed blue line and shaded area show the value of ν_max and the range of dipole mode frequencies used in the analysis. In the g-mode cavity of a mode, its frequency lies below and in the p-mode cavity, it lies above the two characteristic frequencies. Where $\widehat{N}/(2\pi )<\nu <\widehat{S}/(2\pi )$, the mode evanesces. The chemical discontinuity caused by the first dredge-up (i.e., the spike in $\widehat{N}/(2\pi )$) is located just below the evanescent zone of the maximum sampled frequency in this model.