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

Abstract
It was shown in an earlier paper that, if u ( x 1 , x 2 , x 3 , t ) = u ( x, t ) satisfies the wave equation u tt = ∆ u in the exterior of some fixed sphere r = │ x │ = a and vanishes for t ≤ r , then ru ( rξ , t ) ~ f ( ξ , t — r ) as r → ∞, provided that ξ is a fixed unit vector and t — r remains bounded. It was also shown that the 'radiation field' ’ f ( ξ , s ) determines u ( x , t ) uniquely in r ≥ a . In the present paper it is assumed that the Laplace transform of u with respect to t exists. This is found to imply that the Laplace transform of f with respect to s also exists, and is an analytic function of ξ that is regular for all complex unit vectors ξ . From this it can be inferred that, if f itself vanishes for all 8 , and for all ξ in any open subset of the (real) unit sphere, then f ≡ 0, and hence u ≡ 0 in r ≥ a . Furthermore, an integral representation of the Laplace transform of u in terms of the Laplace transform of f is obtained, which generalizes Weyl’s integral representation of diverging spherical waves in terms of plane waves with complex propagation vectors.

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