The Oscillations of a Viscous Liquid Drop*

By virtue of Green's Theorem, it is shown that for the diffraction of anarbitrary two-dimensional incident pulse by a wedge of angle n, the ratio of the resultantvelocity potential to the corresponding value of the incident pulse at the corner ofthe wedge at any instant is equal to 2x/ (2x — n); and that for the diffraction of a three-dimensional pulse by a cone of solid angle u>,the ratio at the vertex of the cone is equaltO 4ir/ (47T — co).Two-dimensional space. The statement concerning diffraction of a pulse by awedge is evidently true in the special case of an incident plane Heaviside pulse whichwas solved by Keller and Blank [1]. It therefore also follows for all incident pulses whichare superpositions of plane Heaviside pulses, or limits of such superpositions. Sincethis includes all incident pulses it yields the preceding statement. However, these con-siderations depend upon knowing the exact solution in a special case which the follow-ing proof does not require.**Let t — 0 be the instant at which the incident pulse hits the corner of the wedge,which is located at the origin (xL = 0, x2 = 0). Let h(xi , x2) and k(xx , x2) denote, re-spectively, and at an instant t = — t0 < 0 if the corner is absent. If G representsthe domain in the xx — x2 plane outside which both h and k vanish, then the origin mustlie outside G. When the wedge is present, the region G lies outside the wedge if theincident disturbance (pM has not hit either side of the wedge at t = — t0 < 0. Then theresultant disturbance at any instant tx > — t„ fulfills the wave equation and the sameinitial conditions as that of i.e., in the region exterior to the wedge•j(1) *Received Feb. 10, 1959; revised manuscript received May 27, 1959.**This paragraph is based on a private communication from Prof. J. B. Keller, Institute of Mathe-matical Sciences, New York University.