  We propose a novel approach for studying $\nu_\mu \to \nu_\tau$ oscillations
with extragalactic neutrinos. Active Galactic Nuclei and Gamma Ray Bursts are
believed to be sources of ultrahigh energy muon neutrinos. With distances of
100 Mpc or more, they provide an unusually long baseline for possible detection
of $\nu_\mu \to \nu_\tau$ with mixing parameters $\Delta m^2$ down to
$10^{-17}$eV$^2$, many orders of magnitude below the current accelerator
experiments. By solving the coupled transport equations, we show that
high-energy $\nu_\tau$'s, as they propagate through the earth, cascade down in
energy, producing the enhancement of the incoming $\nu_\tau$ flux in the low
energy region, in contrast to the high-energy $\nu_\mu$'s, which get absorbed.
For an AGN quasar model we find the $\nu_\tau$ flux to be a factor of 2 to 2.5
larger than the incoming flux in the energy range between $10^2$ GeV and $10^4$
GeV, while for a GRB fireball model, the enhancement is 10%-27% in the same
energy range and for zero nadir angle. This enhancement decreases with larger
nadir angle, thus providing a novel way to search for $\nu_\tau$ appearance by
measuring the angular dependence of the muons. To illustrate how the cascade
effect and the $\nu_\tau$ final flux depend on the steepness of the incoming
$\nu_\tau$, we show the energy and angular distributions for several generic
cases of the incoming tau neutrino flux, $F_\nu^0 \sim E^{-n}$ for n=1,2 and
3.6. We show that for the incoming flux that is not too steep, the signal for
the appearance of high-energy $\nu_\tau$ is the enhanced production of lower
energy $\mu$ and their distinctive angular dependence, due to the contribution
from the $\tau$ decay into $\mu$ just below the detector.
