  We study generic waves without rotational symmetry in (2+1) - dimensional
noncommutative scalar field theory. In the representation chosen, the radial
coordinate is naturally rendered discrete. Nonlocality along this coordinate,
induced by noncommutativity, accounts for the angular dependence of the fields.
The exact form of standing and propagating waves on such a discrete space is
found in terms of finite series. A precise correspondence is established
between the degree of nonlocality and the angular momentum of a field
configuration. At small distance no classical singularities appear, even at the
location of the sources. At large radius one recovers the usual commutative
behaviour.
