680 - 2 - 



beiny equal roughly to the volute of water above the initial free surface. If the initial depth were 

 greater than this critical depth, tnen the bubble will break at less than the rraximum radius, but the 

 chances of the radius being appreciably different from the maximum are not large because the radial 

 velocity of the bubble is small near the maximum, and very large near the minimum. On the other hand, 

 the velocity of rise is much greater in the contract^x) phase, and this fact implies that there is not 

 negligible chance that the oubble will break surface in a contracted condition. The occasional "spouts" 

 observed by Hilliar and subsequent observers are interpreted as events of this type. Presumably explosions 

 leading to "spouts" will be particularly ineffecfrve in causing gravity waves, but the probability of 

 obtaining them is small, and we neglect them. 



If V is the TOximum volume of the bubble, then the equation of the dome on the free surface, before 

 it breaks, may be taken to be 



(2) 



where h is the initial depth of the charge. The crater may be- taken to be a cylinder, paraboloid or 

 hemisphere. A convenient form, however, is one of the same class as (2) namely, 



(,^(r) - -3Vh3/(h^. r') "' b) 



The intial surface i, is the sum of ^^j and (,^ (as shown in figure l) . The volume V of the bubble 

 and dome may be estimiited in the following way. Imagine a spherical hole of radius h to be created in a 

 sea of infinite extent, in such a way that the top of the sphere is just below the horizontal free surface. 

 The work done in displacing the water against hydrostatic plus atmospheric pressure is 4 7T h pg (h + Z)/3, 

 where Z is the he3d of water producing atmospheric pressure. Equate this work to 10% of the chemical 

 energy of the explosion. Thus 



Utt h^ pj (h + Z)/3 = 0.40 E (4) 



Thf reason why 40$ has been chosen is that the energy of an underwater explosion is known to be 

 partitioned roughly as follows 



(a) 30* wasted irreversibly in heating the water. 



(b) lOSJ in the pulsating motion of the bubble. 



(c) 30? radiated as a non-returning pressure pulse. 



Thus (u) is equivalent 'to saying that the shock wave and consequent cavitation, disintegration and 

 spray formation are all unimportant as far as the large scale coherent surface waves are concerned. 



It will be noticed that the qualitative discussion given above leaids to the conclusion that the 

 greatest waves will be obtained from a given weight of charge when the critical depth is roughly equal to 

 h. At greater depths, the bubble is likely to break surface with a volume less than the maximum volume, 

 while at greater depths, the volume of the bubble may again be somewhat restored, it will never equal that 

 given by (t) because the energy of the pulsating motion rapidly diminishes with the order of the pulsation. 

 At very great depths, the bubble has disintegrated by the time it reaches the surface, and our theory does 

 not apply to such cases. Experimental evidence in any case shows that little or no wavey system is started 

 by such explosions. 



If the energy released by the charge is 1000 cal/gm., the optimum wave system is obtained with the 

 charge, of weight w lb. exploded at depths feet shown in the table overleaf. 



