Van Wijngaarden 



The parameter ^ expresses the ratio between the period of free oscillations 

 of a bubble and the time for a disturbance to travel over a distance h with ve- 

 locity c . With help of (1.1) and (1.2) k may be written as 



^. ^ i^.n-R,')'"'. (2.24) 



For a gas volume fraction 10"^ andh = io~^ m we obtain X ^ 10^^^ for Rq = 10"^ m 

 and X % 10'''^ for R,, - I0"''m. These figures pertain for example to a cloud of 

 cavitation bubbles, as considered in Van Wijngaarden (2). Here also we shall 

 consider values of \ of about 10^ or 10^. 



The transforms (2.21) and (2.22) cannot be inverted as they stand, but suf- 

 ficient information for the present purpose can be derived from these expres- 

 sions. For small times, we obtain allowing s to tend to infinity 



and 



^ Apt^ cosh y (2.25) 



Po ^ 2 cosh k 



cosh y 



'^-Po^^P^^iin"^*^- (2.26) 



Pressure changes occur instantaneously, because in the present approximation 

 we have considered the fluid as incompressible. Form (2.17) we infer that 

 whereas for real k, a is real also, k is only real for cr< i. Frequencies higher 

 than cr= 1 are strongly absorbed, since from (2.17) it follows that 



For cr » 1, k = ±i, which explains the nature of the signal for small values of t. 

 For later reference we note that at y = (2.25) may be written as 



Pg-Po %Apt2e-^(l-e-2^+...)- (2.27) 



In order to investigate the development of p„ - Pg at y = further we write the 

 transform of this quantity (as given by (2.22)) as 



2Ap y 



/ . exp 



S(l + S^) 



(2m + 1 ) \S 



,1/2 

 (1+S2) 



(2.28) 



The first term (m= 0), as is shown in the Appendix, can be inverted exactly, 

 which results in 



120 



