  An initial data characterization for vacuum $pp$-wave spacetimes in dimension
four is constructed. This is a vacuum initial data set plus some extra
conditions guaranteeing that the data development is a subset of a vacuum
$pp$-wave. Some of the extra conditions only depend on the same quantities used
to construct the vacuum initial data, namely the first and the second
fundamental forms while others are related to a conformal Killing initial data
characterization (CKID).
