  The periodic standing wave method studies circular orbits of compact objects
coupled to helically symmetric standing wave gravitational fields. From this
solution an approximation is extracted for the strong field, slowly
inspiralling motion of black holes and binary stars. Previous work on this
model has dealt with nonlinear scalar models, and with linearized general
relativity. Here we present the results of the method for the post-Minkowski
(PM) approximation to general relativity, the first step beyond linearized
gravity. We compute the PM approximation in two ways: first, via the standard
approach of computing linearized gravitational fields and constructing from
them quadratic driving sources for second-order fields, and second, by solving
the second-order equations as an ``exact'' nonlinear system. The results of
these computations have two distinct applications: (i) The computational
infrastructure for the ``exact'' PM solution will be directly applicable to
full general relativity. (ii) The results will allow us to begin supplying
initial data to collaborators running general relativistic evolution codes.
