  We carried out a fully non-adiabatic analysis for nonradial oscillations of
pre-white dwarfs evolved from the post-Asymptotic Giant Branch (AGB) with
hydrogen-dominated envelopes. It is shown that nuclear reactions in the
hydrogen burning-shell excite low-degree g-modes in the period range of about
40-200 s for the pre-white dwarf models with Teff=40,000 K - 300,000 K. It is
also shown that the amount of hydrogen {has} a significant influence on the
instability domain of such pre-white dwarfs in the Hertzsprung-Russel (H-R)
diagram. Thus, the thickness of hydrogen-dominated envelopes may be well
constrained by observing the presence of the g-mode oscillations.
