Pressure Pulses in Liquid-Bubble Mixtures 



4Ap 



Bl 



r sin 

 J k 



kk 



1 - cos 



kt 



L (l + k^) 



.dk 



(2.29) 



or 



(00 



Kk + 



kt 



(l + k^)' 



/vk - 



kt 



(l + k^) 



. dk 



(2.30) 



The integrands in (2.29) and (2.30) reduce for small k (long waves) to the cor- 

 responding expressions in (2.12) and (2.13). 



By expansion of the terms in braces in (2.29) a series convergent for all t 

 can be obtained. The first term is 



2Ap 



CO 



r kt^ sin 

 X 1 + k' 



Kk 



dk = Ap t 2 e" 



(2.31) 



in agreement with the first term in (2.27). Higher terms are of the type 

 ^2k ^k- 1 e"'^. Although these higher terms can be obtained relatively easy, the 

 result is not easy to derive conclusions from. For larger values of t we pro- 

 ceed therefore otherwise and use (2.30). 



At t % \ the main contribution to the first integral in (2.30) stems from the 

 neighborhood of k = for the values of k considered here. The first integral in 

 (2.30) can therefore at t % \ be replaced by 



00 



1 r 1 1 



-J i^ sin (\k + kt) dk = 2" 



The second integral has a similar contribution from k = but also an important 

 one from the stationary phase point where K/t equals the group velocity or with 

 help of (2.19) 



1 



(l + k2)- 



(2.32) 



A stationary phase exists only for t > k. At t = \ the stationary phase coincides 

 with k = 0. Therefore at t = \ the second integral in (2.30) is approximately 

 given by 



- r ^ sin ^kH dk = -J- . 



77 J k 2 6 



The total signal at t = \ is 



121 



