  R-modes of neutron stars could be a source of gravitational waves for ground
based detectors. If the precise frequency $\sigma$ is known, guided
gravitational wave searches with enhanced detectability are possible. Because
of its physical importance many authors have calculated the r-mode frequency.
For the dominant mode, the associated gravitational wave frequency is 4/3 times
the angular velocity of the star $\Omega$, subject to various corrections of
which relativistic and rotational corrections are the most important. This has
led several authors to investigate the dependence of the r-mode frequency on
factors such as the relativistic compactness parameter ($M/R$) and the angular
velocity of stars with different equations of state. The results found so far,
however, are almost independent of the equation of state. Here we investigate
the effect of rapid rotation and differential rotation on $\sigma$. We evolve
the perturbation equations using the Cowling approximation by applying finite
differencing methods to compute the r-mode frequency for a series of rotating
neutron stars with polytropic equations of state. We find that rotational
effects in the r-mode frequency can be larger than relativistic effects for
rapidly spinning stars with low compactness, reducing the observed frequency.
Differential rotation also acts to decrease $\sigma$, but its effect inscreases
with the compactness. The results presented here are relevant to the design of
gravitational wave and electromagnetic r-mode searches.
