No 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.)

No abstract

The object of this paper is to determine all the solutions of the wave equationwhich are of the simple formwhere F denotes an arbitrary function. It will be shown that, in addition to the obvious cases of plane or spherical progressive waves, such solutions exist only when the wave frontsare certain algebraic surfaces of the fourth order, the cyclides of Dupin. These include, as degenerate cases, the sphere, the plane, the cylinder, the cone, and the torus.