  In the present paper the determination of the {\it pp}-wave metric form the
geometry of certain spacelike two-surfaces is considered. It has been shown
that the vanishing of the Dougan--Mason quasi-local mass $m_{\$}$, associated
with the smooth boundary $\$:=\partial\Sigma\approx S^2$ of a spacelike
hypersurface $\Sigma$, is equivalent to the statement that the Cauchy
development $D(\Sigma)$ is of a {\it pp}-wave type geometry with pure
radiation, provided the ingoing null normals are not diverging on $\$ $ and the
dominant energy condition holds on $D(\Sigma)$. The metric on $D(\Sigma)$
itself, however, has not been determined. Here, assuming that the matter is a
zero-rest-mass-field, it is shown that both the matter field and the {\it
pp}-wave metric of $D(\Sigma)$ are completely determined by the value of the
zero-rest-mass-field on $\$ $ and the two dimensional Sen--geometry of $\$ $
provided a convexity condition, slightly stronger than above, holds. Thus the
{\it pp}-waves can be characterized not only by the usual Cauchy data on a {\it
three} dimensional $\Sigma$ but by data on its {\it two} dimensional boundary
$\$ $ too. In addition, it is shown that the Ludvigsen--Vickers quasi-local
angular momentum of axially symmetric {\it pp}-wave geometries has the familiar
properties known for pure (matter) radiation.
