  We investigate the primordial power spectra for general kinetic inflation
models that support a period of kinetic dominance in the case of curved
universes. We present derivations of the Mukhanov-Sasaki equations with a
nonstandard scalar kinetic Lagrangian which manifests itself through the
inflationary sound speed $c_s^2$. We extend the analytical approximations
exploited in Contaldi et al [1] and Thavanesan et al [2] to general kinetic
Lagrangians and show the effect of $k$-inflation on the primordial power
spectra for models with curvature. In particular, the interplay between sound
speed and curvature results in a natural low wavenumber cutoff for the power
spectra in the case of closed universes. Using the analytical approximation, we
further show that a change in the inflationary sound speed between different
epochs of inflation results in non-decaying oscillations in the resultant power
spectra for the comoving curvature perturbation.
