On the radiation field of pulse solutions of the wave equation. Ill

Abstract

It was shown in two earlier papers that, if u ( x, t ) = u ( x 1 , x 2 , x 3 , t ) satisfies the wave equation in the exterior of some fixed sphere r = a and vanishes for t ≤ r , then ru ( rξ , r + τ ) ~ f ( ξ, τ ) as r →∞, provided that ξ is a fixed unit vector and t — r = τ remains bounded. The function f was called the radiation field of the pulse solution u . The problem of determining u when f is given was then considered: this is called the radiation field problem in the present paper. It was proved that (i) the radiation field determines the pulse solution uniquely, and (ii) that if two radiation fields f 1 and f 2 coincide, for all τ ≥ 0, in any open set of the unit sphere, then f 1 = f 2 . In the present paper new proofs of these results are given, by means of a simple transformation of variables and an appeal to Holmgren’s uniqueness theorem. The same method is then applied to self-adjoint linear second order equations of normal hyperbolic type, which can be considered as wave equation in a curved spacetime. Radiation fields are defined by means of a family of characteristics that represent outgoing wave fronts. It is shown that if the metric, expressed in coordinates adapted to these characteristics, satisfies certain conditions at infinity (similar to those that have been used in the theory of gravitational waves by Bondi, van den Berg & Metzner, and Sachs), then radiation fields exist. It is also shown that the radiation field determines the associated pulse solution uniquely, and that radiation fields form a coherent family of functions with unique continuation properties.