Van Wijngaarden 



and 





+ (c + v) — 



BxJ 



(c- v) — 



3t 3x 



(v+ f) = 



(V- f) = 



(11) 



(12) 



where 



and 



dpy/2 

 dpj 



f = f - dp = f — dp 



J P J pc 



(13) 



(14) 



Equations (11) and (12) show that the "Riemann invariants" v + f propagate with 

 constant values along the respective characteristics 



dx 



— — = V + c . 



dt 



The net density of the mixture is given very closely by the expression 



p = p^(l - nV) , (15) 



in which nV = (4/3) ■nn'R^ is the volume fraction occupied by the bubbles. Hence, 

 assuming adiabatic compression or expansion of the gas in the bubbles, we have 



p = p. 



1 - nV„ 



l/y 



(16) 



where the zero suffix refers to the undisturbed state of the mixture. (Here an 

 arbitrary value y of the ratio of specific heats is represented, rather than the 

 value 4/3 assumed in the paper.) Upon substitution of this relationship between 

 P and p, (13) gives 



(•)'+i)/2r 



(17) 



where 



-ypn 



(cf. (1.2) in the paper), and (14) gives 



37P„ 



^^/'wnRo' 



(18) 



132 



