207 



THE BODILY MOTION OF A SPHER!! SUBJECTED TO THE 

 P RESSURE PULSE FROM AN UNDERWATER EXPLOSION 



E. K. Fox 



February 1943 



y: ******* * 



Summary . 



(a) The exact solution is ^iven for the bodily motion of a rijid sphere suDjecteo to'an 



underwater pressure pulse on the assumptions that the- pressure pulst can be treated as of small 

 amplitude and that the bodily motion is small compared with the radius of the sphere. 



(d) The conclusion jiven by G.I. Taylor, that a sphere which is just buoyant does not reversi; 



in notion if given an initial vt-locity, is not necessarily true for the motion produced by j 

 pressure pulse in the water. Quantitatively, however, the revjrse motion in the lattor casc is 

 jf very small veljcity Cjmparej t.' the fjrwjrd n.tijn, 



(c) Results of the analysis illustrate the equalisation of long pulses rojnd the sphere and 

 show that the impulsive effect of a positive pulse, even of short duration, can be subject 

 eventually to appreciable diffraction. 



(d) In the cases of a large explosion (lonj pulse) at a larje distance and of a small explosion 

 (short pulse) at n near distance it is concluded that the major damaje caused by the pressure pulse 

 will in jener.il be reasorably independent of whether the target is fixeo or not. 



Introduction , 



In connection with the use of suspenJed tarjets or gauges in underwater explosion experiments 



it is sometimes desirable to obtain an estimate of the bodily motion due to the pressure pulse and 



for this purpose the solution for the case of a spherical target, which can be simply obtained 

 theoretically. Is of value. 



The diffraction problem of a spherical wave- striking a sphere, not necessarily fixed but of 

 unyielding surface. Is one capable of formal mathematical solution by assuming in addition to the 

 incident wave an infinite series of wave-solut Ijns .;f the form. 



00 



j=o 



f i (ct - ., , 



-^ I Pj (Cosi9) (i) 



where r, 9, are spherical— polar co-ordinates with centre of sphere as origin, t is time, c is wave 

 velocity, j Is integral, and P is Legendre's function of order j. 



For the special case of a plane wave striking a fixed rigia sphere such formal mathematical 

 solution was considered by the writer and A.J. Harris in connection with the diffraction of blast- 

 waves In air. It was then found that, while excessive algebra and numerical calculation would be 

 necessary to evaluate the pressure at various points, the net force acting on the sphere could be 

 simply obtained since It oependeo on only one term (j = l) in the infinite series. 



Since the only knowledge of pressure required to determine the bodily motion of the sphere 

 when not fixed is the total net force on the sphere, tne solution for such motion also depends only 

 on the one term in the complete series solution and can be derived directly In the manner given 

 below without recourse to the complete solution. 



General solution . 



Consider a sphere of radius a subjected to a pressure pulse from a point source distance 

 Xa from the centre of the sphere. The pressure pulse will be assumed of small amplitude at the 

 sphere so that the usual wave equation holds and it will also be assumea that the movement of the 

 sphere is small compared with its raaius. 



The 



