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mine from the bottom. For example. If the mine is placed 

 either 17 ft. or 29 ft. from the bottom, the peak pressure 

 is only one-half of the maximum. 



If one is interested in the damage to objects on the 

 surface of the water, it is necessary to take into account 

 the migration of the gas bubble. Again considering the above 

 example, the peak pressure at the surface directly overhead 

 is 15.6 atmospheres (excluding the reflection from the bottom) 

 when the mine is placed 21.0 ft. from the bottom. For any 

 other location of the mine, this value is to be multiplied 

 by the factor whose graph is drawn in figure 4. The graph 

 still exhibits a considerable sharpness, but if one places 

 the mine somewhat higher than 21.0 ft. from the sea bed, the 

 effect on the surface would be improved slightly. This is 

 due to the upward migration of the bubble. Note, however, 

 that if the mine is placed too high, say 27 ft. from the 

 bottom, the favorable effect of the upward migration is 

 sharply counterbalanced by the resulting weakness of the 

 secondary pulse. 



Part II. Mathematical Study of the Secondary 

 Pressure Pulse 



1. Introduction . 



High pressure pulses are produced only when the size 

 of the gas bubble is near its minimum. During this time, 

 the buoyant force due to gravity is small. Likewise, the 

 proximity of rigid walls or free surfaces will not materially 

 affect the motion, since their influence depends on the ratio 

 of the bubble radius to the distance from the bubble, and 

 Is small if this ratio is small. Therefore, during the stage 

 of minimum size, the bubble can be considered as iimnersed 

 in an infinite body of water and subject to no outside forces. 

 The pressure pulse produced by the bubble under these 



