  The dynamics of collapsing and expanding cylindrically symmetric
gravitational and matter fields with lightlike wave-fronts is studied in
General Relativity, using the Barrabes-Israel method. As an application of the
general formulae developed, the collapse of a matter field that satisfies the
condition R_{AB}g^{AB} = 0, (A, B = z, phi), in an otherwise flat spacetime
background is studied. In particular, it is found that the gravitational
collapse of a purely gravitational wave or a null dust fluid cannot be realized
in a flat spacetime background. The studies are further specified to the
collapse of purely gravitational waves and the general conditions for such
collapse are found. It is shown that after the waves arrive at the axis, in
general, part of them is reflected to spacelike infinity along the future light
cone, and part of it is focused to form spacetime singularities on the symmetry
axis. The cases where the collapse does not result in the formation of
spacetime singularities are also identified.
