  In coalescing neutron star (NS) binaries, tidal force can resonantly excite
low-frequency (< 500 Hz) oscillation modes in the NS, transferring energy
between the orbit and the NS. This resonant tide can induce phase shift in the
gravitational waveforms, and potentially provide a new window of studying NS
interior using gravitational waves. Previous works have considered tidal
excitations of pure g-modes (due to stable stratification of the star) and pure
inertial modes (due to Coriolis force), with the rotational effect treated in
an approximate manner. However, for realistic NSs, the buoyancy and rotational
effects can be comparable, giving rise to mixed inertial-gravity modes. We
develop a non-perturbative numerical spectral code to compute the frequencies
and tidal coupling coefficients of these modes. We then calculate the phase
shift in the gravitational waveform due to each resonance during binary
inspiral. We adopt polytropic NS models with a parameterized stratification. We
derive relevant scaling relations and survey how the phase shift depends on
various properties of the NS. We find that for canonical NSs (with mass M =
1.4M_sun and radius R = 10 km) and modest rotation rates (< 300 Hz), the
gravitational wave phase shift due to a resonance is generally less than 0.01
radian. But the phase shift is a strong function of R and M, and can reach a
radian or more for low-mass NSs with larger radii (R > 15 km). Significant
phase shift can also be produced when the combination of stratification and
rotation gives rise to a very low frequency (< 20 Hz in the inertial frame)
modified g-mode. We also find that some inertial modes can be strongly affected
by stratification, and that the m = 1 r-mode, previously identified to have a
small but finite inertial-frame frequency based on the Cowling approximation,
in fact has essentially zero frequency, and therefore cannot be excited.
