It was shown in an earlier paper that, if u(x1, x2, x3, t) = u(x, t) satisfies the wave equation utt = ∆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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F. G. Friedlander (1964) studied this question.