Neutrino oscillations and the atmospheric neutrino fluxes

We have examined the effects of a possible ${\nu }_e-{\nu }_{\mu }$ oscillation on the atmospheric neutrino fluxes. We have included the influence of the Earth's medium on the oscillations and also considered averaged fluxes in order to account for the inherent uncertainties in (i) the neutrino path lengths and (ii) zenith angles of neutrino arrival. The restrictions provided by our simple criteria for the detectability or for the neglect of oscillation effects in the fluxes are compared with the limits set by a recent reactor experiment. This comparison reveals that the oscillation effects are ignorable in the downward fluxes, but can be sizable in the upward fluxes provided the oscillation length is in the range 600-24 000 km. Incorporating these oscillation effects, we have also obtained lower and upper limits on the ratios of (i) upward and downward fluxes of ($\nu +\overline{\nu }$) of a given flavor, and (ii) fluxes of ($\nu +\overline{\nu }$) of $e$ and $\mu$ flavors traveling in the upward direction.