  Beetle and Burko recently introduced a background--independent scalar
curvature invariant for general relativity that carries information only about
the gravitational radiation in generic spacetimes, in cases where such
radiation is incontrovertibly defined. In this paper we adopt a formalism that
only uses spatial data as they are used in numerical relativity and compute the
Beetle--Burko radiation scalar for a number of analytical examples,
specifically linearized Einstein--Rosen cylindrical waves, linearized
quadrupole waves, the Kerr spacetime, Bowen--York initial data, and the Kasner
spacetime. These examples illustrate how the Beetle--Burko radiation scalar can
be used to examine the gravitational wave content of numerically generated
spacetimes, and how it may provide a useful diagnostic for initial data sets.
