  Gravitational waves (GWs) can provide crucial information about the central
engines of core-collapse supernovae (CCSNe). In order to unveil the nature of
GW emission in CCSNe, we apply perturbative analyses with the same underlying
equations as simulations to diagnose oscillations of the proto-neutron star
(PNS) during $\sim$1 s postbounce. In the pseudo-Newtonian case, we find that
radial profiles of GW emission match well between the perturbative analysis
with $l=2$ and simulations inside the PNS at \emph{any} frequency and time.
This confirms that the GW emission of CCSNe arises from the global PNS
oscillations in the perturbative regime. Based on this, we solve for the
discrete eigenmodes with a free PNS surface and tentatively identify a set of
$g$ modes and the $f$ mode contributing to the peak GW emission. We also offer
a possible explanation for the power gap in the GW spectrum found in
simulations that lies at the frequency with vanishing cumulative emission of
the PNS. Our results enhance the predictive power of perturbative analyses in
the GW signals of CCSNe.
