  Gravitational waves (GW) generated during a core-collapse supernova open a
window into the heart of the explosion. At core bounce, progenitors with rapid
core rotation rates exhibit a characteristic GW signal which can be used to
constrain the properties of the core of the progenitor star. We investigate the
dynamics of rapidly rotating core collapse, focusing on hydrodynamic waves
generated by the core bounce and the GW spectrum they produce. The centrifugal
distortion of the rapidly rotating proto-neutron star (PNS) leads to the
generation of axisymmetric quadrupolar oscillations within the PNS and
surrounding envelope. Using linear perturbation theory, we estimate the
frequencies, amplitudes, damping times, and GW spectra of the oscillations. Our
analysis provides a qualitative explanation for several features of the GW
spectrum and shows reasonable agreement with nonlinear hydrodynamic
simulations, although a few discrepancies due to non-linear/rotational effects
are evident. The dominant early postbounce GW signal is produced by the
fundamental quadrupolar oscillation mode of the PNS, at a frequency $0.70 \,
{\rm kHz} \lesssim f \lesssim 0.80\,{\rm kHz}$, whose energy is largely trapped
within the PNS and leaks out on a $\sim\!10$ ms timescale. Quasi-radial
oscillations are not trapped within the PNS and quickly propagate outwards
until they steepen into shocks. Both the PNS structure and Coriolis/centrifugal
forces have a strong impact on the GW spectrum, and a detection of the GW
signal can therefore be used to constrain progenitor properties.
