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 ∼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 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.