  In the past decade, surveys of the stellar component of the Galaxy have
revealed a number of streams from tidally disrupted dwarf galaxies and globular
clusters. Simulations of hierarchical structure formation in LCDM cosmologies
predict that the dark matter halo of a galaxy like the Milky Way contains
hundreds of subhalos with masses of ~10^8 solar masses and greater, and it has
been suggested that the existence of coherent tidal streams is incompatible
with the expected abundance of substructure. We investigate the effects of dark
matter substructure on tidal streams by simulating the disruption of a
self-gravitating satellite on a wide range of orbits in different host models
both with and without substructure. We find that the halo shape and the
specific orbital path more strongly determine the overall degree of disruption
of the satellite than does the presence or absence of substructure, i.e., the
changes in the large-scale properties of the tidal debris due to substructure
are small compared to variations in the debris from different orbits in a
smooth potential. Substructure typically leads to an increase in the degree of
clumpiness of the tidal debris in sky projection, and in some cases a more
compact distribution in line-of-sight velocity. Substructure also leads to
differences in the location of sections of debris compared to the results of
the smooth halo model, which may have important implications for the
interpretation of observed tidal streams. A unique signature of the presence of
substructure in the halo which may be detectable by upcoming surveys is
identified. We conclude, however, that predicted levels of substructure are
consistent with a detection of a coherent tidal stream from a dwarf galaxy.
