  In this paper, we follow up on the discovery of a new type of solution in the
Einstein-Maxwell system coupled minimally to a self-interacting complex scalar
field. For sufficiently large gravitational coupling and sufficiently small
electromagnetic coupling we demonstrate that boson stars as well as black holes
can carry scalar hair that shows a distinct new feature~: a number of spatial
oscillations in the scalar field away from the core or horizon, respectively.
These spatial oscillations appear also in the curvature invariants and hence
should be a detectable feature of the space-time. As a first hint that this is
true, we show that the effective potential for null geodesics in this
space-time possesses a local minimum indicating that in the spatial region
where oscillations occur a new stable photon sphere should be possible. We also
study the interior of the black holes with scalar hair and show that the
curvature singularity appears at a finite value of the radius and that black
holes with wavy scalar hair have this singularity very close to the center.
