  We investigate fixed energy scattering from conical potentials having an
irregular cross-section. The incident wave can be any arbitrary non-trivial
Herglotz wave. We show that a large number of such local conical scatterers
scatter all incident waves, meaning that the far-field will always be non-zero.
In essence there are no incident waves for which these potentials would seem
transparent at any given energy. We show more specifically that there is a
large collection of star-shaped cones whose local geometries always produce a
scattered wave. In fact, except for a countable set, all cones from a family of
deformations between a circular and a star-shaped cone will always scatter any
non-trivial incident Herglotz wave. Our methods are based on the use of
spherical harmonics and a deformation argument. We also investigate the related
problem for sources. In particular if the support of the source is locally a
thin cone, with an arbitrary cross-section, then it will produce a non-zero
far-field.
