354 SECONDARY PRESSURE WAVES 



When these difficulties are considered it should be no surprise that 

 the theoretical results for bubble pressures are qualitatively useful 

 rather than quantitatively reliable. A heavier burden is therefore 

 placed on experimental investigations in evaluating the phenomena and 

 it is unfortunate that the theoretical difficulties have their counterpart 

 in serious experimental problems of measurement and analysis. These 

 problems, which arise largely from the relatively small magnitude and 

 long duration of the secondary pulse, are too numerous and technical 

 to be adequately summarized here (see section 9.5). In spite of both 

 types of difficulty, theoretical and experimental, a fairly satisfactory 

 picture has been developed which makes it possible to predict at least 

 roughly what will happen in most circumstances of interest. 



9.1. The Generalized Form of Bernoulli's Equation 



The pressure P at any point in an incompressible fluid of density po 

 without viscosity is most conveniently found from the generalized form 

 of Bernoulli's equation derived in section 8.4, which is 



(9.1) -= -W + ^ -a + Fit) 



Po Ot 



In this equation, u is the particle velocity at the point, (p the velocity 

 potential, 12 the potential of external force on unit volume of the fluid, 

 and F(t) is an arbitrary function of time. The physical significance of 

 F{t) lies in the fact that the distribution of flow in an incompressible 

 fluid is unaffected by changes in time of absolute pressure at all points 

 in its interior. These would be produced, for example, by variations of 

 atmospheric pressure at the free surface, which are propagated instan- 

 taneously to all points in this approximation. In the applications we 

 shall make, such changes need not be considered, and the function F{t) 

 is therefore a constant, the value of which is specified if the pressure 

 and velocity are known at some point at any time. It is, for example, 

 determined by the hydrostatic pressure existing initially before the 

 motion of the gas sphere and surrounding flow is initiated. The term 

 d<p/dt, which would be zero for steady flow, gives the effect of velocity 

 changes. 



The Bernoulli equation (9.1) bears a close relation to the principle 

 of conservation of energy for any element of the fluid, and in its more 

 elementary forms is usually derived by energy considerations from the 

 work done by pressure differences and external forces in increasing 

 kinetic energy of flow. In the problems to which we shall apply the 

 equation it is necessary to include the conditions at boundaries of the 

 fluid. This is most easily accomplished by use of the velocity potential 

 which is calculated by standard methods of potential theory, as has been 



