366 SECONDARY PRESSURE WAVES 



The negative value of the derivative requires that P decrease with in- 

 creasing a*, and the peak pressure therefore occurs for the smallest value 

 of bubble radius a* which is attained. 



The peak pressure realized under any conditions can be determined 

 from Eq. (9.18) from the minimum value of radius a and the transla- 

 tional momentum s at this time. The factors influencing its value be- 

 come more evident if its pressure is expressed in terms of the radial 

 velocity da/dt and vertical velocity U at the time of minimum radius. 

 Substitution in Eq. (9.18) from Eq. (9.17) gives 





r [_2 \dt* ) 4: \dt*) 8a*i^/4j 

 and restoring dimensional variables (see section 8.9) gives 



(9.19) P-P. = ^[ip„ (IJ + i pM^ + kV-y'j 



The first two terms in this equation represent the kinetic energies asso- 

 ciated with radial and translational motion of the bubble, and thus 

 contribute what might be called hydrodynamic pressures resulting from 

 these two types of flow. The last term is simply the internal pressure 

 Pa in the gas sphere of specific volume V. 



The dynamic pressure terms in Eq. (9.19) decrease for smaller values 

 of the minimum radius, da/dt being zero at the minimum, and the term 

 resulting from compression of the gaseous products increases. The 

 optimum value of pressure therefore depends on the relative importance 

 of gas pressure and dynamic pressure from translation. The principle 

 of stabilization, as formulated by Friedman and Shiffman, is simply the 

 statement that the optimum value of peak pressure in the surrounding 

 water is obtained for very nearly the conditions which maximize the gas 

 pressure. These occur for initial conditions such that the gas sphere 

 has no vertical velocity at the time of greatest contraction and the 

 radius at this time has its smallest value. 



At the time of peak pressure, the radial velocity da*/dt* is zero and 

 it is convenient in establishing the principle of stabilization to express 

 the peak pressure Pm in terms of the momentum factor s which is pro- 

 portional to the vertical velocity U* of the bubble center. Setting 

 da*/dt* = in Eq. (9.17) gives 



(9.20) ^~ = - — 



It is convenient to express this relation in terms of the fraction E{d) of 

 the total energy Y which is in the form of internal energy of the gas at 



