  In the oscillation spectra of giant stars, nonradial modes may be seen to
undergo avoided crossings, which produce a characteristic "mode bumping" of the
otherwise uniform asymptotic p- and g-mode patterns in their respective echelle
diagrams. Avoided crossings evolve very quickly relative to typical
observational errors, and are therefore extremely useful in determining precise
ages of stars, particularly in subgiants. This phenomenon is caused by coupling
between modes in the p- and g-mode cavities that are near resonance with each
other. Most theoretical analyses of the coupling between these mode cavities
rely on the JWKB approach, which is strictly speaking inapplicable for the
low-order g-modes observed in subgiants, or the low-order p-modes seen in very
evolved red giants. We present both a nonasymptotic prescription for isolating
the two mode cavities, as well as a perturbative (and also nonasymptotic)
description of the coupling between them, which we show to hold good for the
low-order g- and p-modes in these physical situations. Finally, we discuss how
these results may be applied to modelling subgiant stars and determining their
global properties from oscillation frequencies. We also make our code for all
of these computations publicly available.
