679 



GRAVITY WAVES PRODUCED BY SURFACE AND 



UNDERWATER EXPLOSIO NS 



V. G. Penney 

 March 1945 



Summary . 



The waves protJuced by explosions above, on or under the surface of the water are considered. 

 General arguments -re advanced to show that the greatest waves from an underwater explosion are obtained 

 when the initial depth just' equals the radius of tho buOole produced. The waves at a distance less than 

 10-20 times this radius may be regarded as started by the initial inflo/i of water into a crater. Two 

 thousand tons H.E. exploded at a dipth 250-300 feet will give a s&ries of waves at 2300 feet the largest 

 of which is 18 feet high while the fallowing trough is about equally deep. At U600 feet, the greatest 

 wave and the following trough art , ^th about 6 feet. ' A similar charge on the surface produces much 

 smaller wsves. Using the experimental fact that the waves from a 1 oz. chargt appear to be. about thr 

 same whether the explosion is on the surface, or a foot or two below, it follows from the laws of scaling 

 that the surface explosion of 2000 tons will produce waves about five times smaller than those from the 

 explosion at 285 feet depth. 



This report discusses the wave systems produced by explosions above, on or under the surface 

 of water. A reasonably accurate solution is obtained for explosions above or on the surface, and an 

 exact law of scaling is discovered for all depths of water. In the case of underwater explosions, the 

 treatrrent is only approximate, because of the large number of conflicting factors such as pulsations of 

 the bubble, cavitation near the free surface, instability of the surface leading to spray formation, etc. 

 Laws of scaling are suggested in this case, and they are probably good enough to make fair predictions on 

 a full scale explosion, using inforration obtained from model small scale charges. 



The mechanism proposed for the formation of a wave system from a surface explosion is that at time 

 t = the free surface over a certjin area receives a downward impulse. This impulse has radial symmetry, 

 and is assumed to be a known function of radius I (r). Unfortunately, no experimental measurements are 

 available for constructing the function l(r). All that is known is that a distances r 5 30 charge radii, 

 the positive and negative phases of the pressure pulse are of equal area, within the experimental error. 

 At smaller values of r, of course, the positive phase must exceed the negative phase, and the difference in 

 the two areas is the function l(r). Two cases may be considered 



l(r) = Ig if r-5: a, l(r) = if r>a (Equation la) 



i(r) = K^ (l-r ^/a^) if r ^ a, l(r) =0 if r5a (Equation lb) 



The value assumed for a is the radius of the flame zone, say about 30 charge radii. 



When an explosion occurs underwater, the gaseous bubble at first increases very rapidly in size. 

 It continues to grow, but less and less quickly, until its radius is 10 - 30 times the initial radius, 

 depending on the depth. During this expanding phase, the centre of the bubble probably rises, but not 

 very much, the exact amount depending on how near the bubble is to the bottom and to the free surface. 

 Provided the bubble reaches its maximum possible size before breaking through the free surface, we have, 

 at the instant of maximum size, a free surface which is shaped in the conventional dome form, while below 

 the dome is the gas bubble. The contracting phase now sets in and the gas bubble begins to rise more 

 rapidly. If the initTal depth was chosen correctly, the conditions are now critical, the water film at 

 the top of the bubble breaks, and the free surface has the form of a volcano, the volume of the crater 



being 



