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A uniqueness theorem is presented for reflector mappings in the complex plane. These arise in geometrical optics, in the synthesis of a reflector surface to produce a ray beam with specified angular distribution of energy when illuminated by a non-isotropic point-source. The mappings may be represented as solutions of a nonlinear boundary-value problem involving the Monge-Ampère equation. When the equation is elliptic there are at most two solutions, provided that either the incident or far-field ray cone satisfies a convexity condition, which is always the case if one cone is circular. The result has practical application to the design of single and offset dual reflectors used for radar and communications purposes.

The reflexion of outward travelling cylindrical gravitational waves by a thin cylindrical shell of matter is discussed. The shell is regarded as a surface distribution, and it is shown that if it is perfectly rigid, total reflexion occurs. Quasi-periodic incident waves are found to lead to a completely periodic standing wave state between the reflector and a central wave-source, the field outside the system remaining static. In the treatment of standing waves the components of the stress-energy tensor in the shell are determined and the construction for a model source is given.

A device making use of the linearity of one of the field equations for Rosen’s cylindrically symmetric metric is employed to demonstrate a correspondence between some outgoing toroidal gravitational waves and imploding-exploding cylindrical waves. An exact form of axially symmetric metric for a smooth toroidal pulse in the region ｜ z ｜ < t (in coordinates which approximate to cylindrical polars when the field is weak) is constructed, and extensions to the region ｜ z ｜ < t are considered. The solution remains valid throughout the reflexion of the wave as it meets itself on the symmetry axis, and displays this process clearly. As a particular example, a solution of Weber & Wheeler for a non-singular cylindrical pulse is converted into one for a time-symmetric toroidal pulse, and the distant field at the moment of time-symmetry is briefly examined.

Known cylindrical wave solutions of the free-space gravitational field equations are superimposed on a static field and extended to the interior of a generating cylinder of matter. It is shown that the waves propagate energy. The interior and exterior field of a static cylinder of approximately constant density is obtained.

A construction is given for an exact solution of Einstein’s empty space-time field equations representing an axially sjonmetric pulse wave with the properties that (i) the wave front is a sphere, outside of which all space-time is flat; (ii) a finite, expanding line singularity forms the axial diameter of this sphere. Because of (i), the solution would need to be generalized before a physical source could be introduced. The presence of radiation is confirmed by the fact that at a great distance from the centre of the wave the Riemann tensor is asymptotically of Petrov’s canonical type II (with vanishing scalar invariants).

Further properties of a vacuum metric obtained in an earlier paper, representing a spherical-fronted gravitational wave (which, however, possesses cylindrical symmetry) are derived. This metric is constructed by introducing ‘bending terms’ into Rosen’s metric for cylindrical waves. The way the transition occurs from properties characteristic of cylindrical waves to those of spherical waves is examined. The asymptotic behaviour of the optical parameters and the null tetrad components of the Weyl (Riemann) tensor along an outgoing null geodesic is determined. An interpretation of the axial singularity is given by comparison with an analogous feature in classical electromagnetic wave propagation.

The geometry of an extended region of space-time is not fully determined by the vanishing of the Riemain curvature tensor. This suggests the possible existence of a non-trivial gravitational field where space-time is flat. Two examples of such fields are considered with reference to their sources. It is quite familiar that certain Riemannian spaces, such as the plane, cylinder and cone, are locally isometric. They differ only in their topology. Little attention, however, has yet been paid to an analogous situation concerning space-times in general relativity. In this paper it is shown that Einstein's gravitational field equations admit exact solutions whose empty regions are locally flat (in that the Riemann tensor vanishes everywhere), but for which there exists no universal Minkowskian co-ordinate system. Two examples are given, and an investigation of geodesics shows that each corresponds to a non-trivial gravitational field. The first example relates to a cylindrically symmetric field, for which a model source with uniform positive density is constructed. This field is important to the problem of cylindrical gravitational waves. In the second example a toroidal source is assumed, although here it appears that no physically sensible interior solution is possible. Nevertheless, the example is interesting as one in which, for topological reasons, the metric of a solution for the exterior of a finite source does not determine the field completely.