  Gravitational waves (GWs) provide unobscured insight into the birthplace of
neutron stars (NSs) and black holes in core-collapse supernovae (CCSNe). The
nuclear equation of state (EOS) describing these dense environments is yet
uncertain, and variations in its prescription affect the proto-neutron star
(PNS) and the post-bounce dynamics in CCSNe simulations, subsequently impacting
the GW emission. We perform axisymmetric simulations of CCSNe with Skyrme-type
EOSs to study how the GW signal and PNS convection zone are impacted by two
experimentally accessible EOS parameters, (1) the effective mass of nucleons,
$m^\star$, which is crucial in setting the thermal dependence of the EOS, and
(2) the isoscalar incompressibility modulus, $K_{\rm{sat}}$. While
$K_{\rm{sat}}$ shows little impact, the peak frequency of the GWs has a strong
effective mass dependence due to faster contraction of the PNS for higher
values of $m^\star$ owing to a decreased thermal pressure. These more compact
PNSs also exhibit more neutrino heating which drives earlier explosions and
correlates with the GW amplitude via accretion plumes striking the PNS,
exciting the oscillations. We investigate the spatial origin of the GWs and
show the agreement between a frequency-radial distribution of the GW emission
and a perturbation analysis. We do not rule out overshoot from below via PNS
convection as another moderately strong excitation mechanism in our
simulations. We also study the combined effect of effective mass and rotation.
In all our simulations we find evidence for a power gap near $\sim$1250 Hz, we
investigate its origin and report its EOS dependence.
