258 -12- 



circumstances will be investigated. 



On the other hand, all these 'outside influences 

 do affect the bubble in the stage when it is not small. 

 Consequently, in contracting to its minimum size, the 

 bubble and surrounding water have a linear momentum which 

 remains practically constant while the bubble passes through 

 the minimum size stage when the outside influences can be 

 neglected. The pressure pulse produced by the bubble depends 

 on the linear momentum acquired by the bubble. The dependence 

 of the peak pressure on this linear momentum is the main objec- 

 tive of this investigation and is represented in figure 6- 



For the derivation of this result, the following assump- 

 tions are made: 



1. The water is an ideal incompressible fluid. 



2. The bubble remains spherical in shape. 



3. The gas inside the bubble is in thermal 

 equilibrium at each instant and follows the 

 adiabatic law. 



These assumptions are reasonable for the major portion of 

 the period of pulsation, and are violated only in the very 

 short time interval when the radius of the bubble is small. 

 The violations are in the nature of corrections to the theory, 

 and will not materially affect the principle of stabilization. 

 We shall not enter into a discussion of the assumptions, but 

 merely refer to [4]« 



2* The Energy Equation . 



Let A be the radius of the bubble at any time, B 

 the vertical distance of its center from some horizontal 

 level, and P the hydrostatic pressure of the water at the 

 center of the bubble. The motion of the bubble is described 

 by specifying A, B as functions of the time T» 



