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 $:=∂Σ≈S2 of a spacelike
hypersurface Σ, is equivalent to the statement that the Cauchy
development D(Σ) 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(Σ). The metric on D(Σ)
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(Σ) 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 Σ 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.Comment: 15 pages, Plain Tex, no figure