19 



-S from (») 



° Urom (5) 

 and P ■= 3 p„ a 



Inserting these values in (22) 



n = p^ ^ { (1 + a) - 4 a x' } 



(23) 



In the early stdje o' the motion, which is now considered, the terms a ana a x' can De nejlocted in 

 comparison with unity, and then 



n = 



(») 



This is ri more useful formula than (l8). 



The pe,ik viluc- F! of the external pressure corresponds to r = r , and is 



n = p (le) (25) 



a ^ ' 



Equation (24) can now De written as 



r^' = "o yf 



(26) 



so that, using equation (8) 



(27) 



which shows how n varies with t in the early stages. 



The maximum amplitude of the rarefaction at a distant point is deducible from the more 

 accurate equation (23). From formula (lO) it appe?.rs that the maximum expansion of the bubole 

 corrc-sponds to x = x,, where x is the larjor positive root of 



(1 * a) X 



and is approximately x, = 

 equation (23) as 



hen a is very small. The corresponding value of is jiven by 



this variation o.inj superposed upon the exist inj pressure/*. Suostitutinj in the above r = x. r 



[°n2'i 



- 3 n, a^ 



(from 25). 



The maximum amplitude- of the rarefaction is therefore a fraction 3 a^^' of the initial positive 

 pressure pulse. 



Comparison ... 



