701 



SURFACE WAVES PRODUCED BY UNDERWATER EXPLOSIONS 



COMPARISON OF THE THEORY OF W. G. PENNEY WITH 



EXPERIMENTAL RESULTS FOR A 32-lb. CHARGE 



A. R. Bryant 



Road Research Laboratory, 

 London 



September 1945 



Summary, 



The amolitude of surface waves oroouceo by a 32-lD, charge oetonatso in water at a oepth 

 of 8 feet has been calculated for a coint 56.5 feet from the chtirge by w.G. "enney's theory'^'. 

 Calculations extend uo to the thiro trough. 



The resjlts havt been comoarea with exccrimentrtl oD'^rvaticns for a similar charge'^', 

 Ajreemcnt is excellent u: to the arrival of ths seconfl crest, out the calculateO thiro trough is 

 larger than that observea. Reason^iole agret^ment uc to the third trough is obtained If the shaoe 

 of the initial c.ivity ?.sS'im?d in "innoy's theory is modiflefl to jive a shallower cavity of the 

 same total volume. 



Equations used . 



'enney's theory assumes that the waves are sroouced by the filling in of an Initial 

 cavity, whose shace Is given By 



3 [(h^rr^ TJTT^j 



(1) 



where y Is the initial elevation of the free surface above the original undlsturbeo water level 

 it a ooint oistant r from i vertical axis of symmetry through the cnargp. It can be shown that 

 this Is a reasonable r=orssontat ion of the covity which might be exoecteo to O; oroduceo when a gas 

 bubble at its maximum radius h just touches the top of the 'done" creiteo by the charge. The 

 Shaoe of this cavity is shown in Figure l. 



At any time t later the system of waves sroOjced by this cavity is described Dy the 

 following relation between surface elevation y ana distance r. 



where S^ = h^ + r^ (j) 



7D 



ana °n is the nth Legenore oolynonial. This formula is only suitable for comoutation it small 

 values of the time. For later times it is* more convenient to use ths following looroxiroate 

 integral methca jiven Oy "tnnpy. 



If <x> is the distance cf a small element in the water surface of area 05 from the coint at 

 which the wave height y is measured, and if r be the oistr'.nce of thi? same area from the vertical 

 axis through the charge, then 



y, (r) 



(3) 



m 



