  Consider the energy-critical focusing wave equation in space dimension $N\geq
3$. The equation has a nonzero radial stationary solution $W$, which is unique
up to scaling and sign change. It is conjectured (soliton resolution) that any
radial, bounded in the energy norm solution of the equation behaves
asymptotically as a sum of modulated $W$s, decoupled by the scaling, and a
radiation term. A nonradiative solution of the equation is by definition a
solution whose energy in the exterior $\{|x|>|t|\}$ of the wave cone vanishes
asymptotically as $t\to +\infty$ and $t\to -\infty$. In a previous work
(Cambridge Journal of Mathematics 2013, arXiv:1204.0031), we have proved that
the only radial nonradiative solutions of the equation in three space
dimensions are, up to scaling, $0$ and $\pm W$. This was crucial in the proof
of soliton resolution in 3 space dimension.
  In this paper, we prove that the initial data of a radial nonradiative
solution in odd space dimension have a prescribed asymptotic behaviour as $r\to
\infty$. We will use this property for the proof of soliton resolution, for
radial data, in all odd space dimensions. The proof uses the characterization
of nonradiative solutions of the linear wave equation in odd space dimensions
obtained by Lawrie, Liu, Schlag and the second author (Advances in Mathematics,
2015, arXiv:1409.3643) .
  We also study the propagation of the support of nonzero radial solutions with
compactly supported initial data, and prove that these solutions cannot be
nonradiative.
