Radiation fields and hyperbolic scattering theory

Abstract

Let u(x, t), where x ∈ 3, t ∈ , be a C∞ solution of the wave equation on {|x| > a} × for some a > 0, and suppose that u = 0 for |x|> a, t < 0. One then also has u = 0 for |x| > a + t, t > 0, so that u(.,t) can be thought of as a wave expanding into a previously undisturbed medium. One can now ask for a description of the asymptotic behaviour of u as |x| → ∞. It turns out that there is a v0(θ, τ) ∈ C∞ (S2 × ) such thatin the topology of C∞ (S2 × ). This limit may be called the radiation field of the expanding wave u. (See (2) and the earlier papers quoted there.)