  Recent developments in string theory have brought forth a considerable
interest in time-dependent hair on extended objects. This novel new hair is
typically characterized by a wave profile along the horizon and angular
momentum quantum numbers $l,m$ in the transverse space. In this work, we
present an extensive treatment of such oscillating black objects, focusing on
their geometric properties. We first give a theorem of purely geometric nature,
stating that such wavy hair cannot be detected by any scalar invariant built
out of the curvature and/or matter fields. However, we show that the tidal
forces detected by an infalling observer diverge at the `horizon' of a black
string superposed with a vibration in any mode with $l \ge 1$. The same
argument applied to longitudinal ($l=0$) waves detects only finite tidal
forces. We also provide an example with a manifestly smooth metric, proving
that at least a certain class of these longitudinal waves have regular
horizons.
