Van Wijngaarden 



Pgi-Pc 



2 ^ 

 3AP 



which signal can be considered as the "forerunner" (see e.g., Brillouin (4)). 



For t > \ the contributions by k = to the integrals in (2.30) cancel just as 

 the integrals in (2.13) cancel for t' > h/c The total signal is, however, not 2Ap 

 for these values of t ' , as in (2.13), but 2Ap minus the contribution of the sta- 

 tionary point in the integrand of the second integral in (2.30). The arrival of the 

 total signal 2Ap is therefore not determined by the "sound" velocity c but by the 

 group velocity as given by (2.32). Writing 



J r"^" 



\k 



kt 



(l + k^) 



1/ 2 



dk = Im 



exp it 



(l + k^) 



dk 



(2.33) 



and evaluating the contribution of the point of stationary phase, we obtain for 



t > k 



Po = 2Ap 1 



(t/A.)- 



(67Tt)l''2 



(tA)2/3- 1 



1- (\/t)2/3 



3/2 7T 



(2.34) 



The expressions (2.31), (2.33), and (2.34) show how the pressure Pg in the 

 bubbles increases with time at the plate y' ^ 0. This forms an adequate descrip- 

 tion as long as t < 3a, since at that moment the "forerunner" associated with 

 the second term in (2.28) arrives at y' = 0. Schematically Pg as a function of t 

 is shown in Fig. 1. 



p 



Po+24p 

 n*Ap 



pit_y = X ; P g at y«3 



patp^ ot y«0 



imiuiMimnimi m 



Fig. 1 - Behavior of p and Pg as a function 

 of t and y = and y - k 



Next we consider the behavior of the pressure in the bubbles at the outer 

 boundary of the mixture. From the boundary conditions at y' = h and from (2.5) 

 it follows that at y ' = h 



Pg- Po = '^P (1 - COS OJg t') 



(2.35) 



122 



