Pressure Pulses in Liquid -Bubble Mixtures 



Seeking solutions in the form of progressive waves 



exp i(ky - at ) 



we obtain the dispersion equation 



k 



a - . 



,1/2 



Waves are dispersed. The phase velocity is given by 



a 1 



k „ 1/ 2 ' 



(l + k2) 



whereas the group velocity is 



da; 

 dk 



1 



(l + k2) 



3/2 



(2.17) 



(2.18) 



(2.19) 



Only for very long waves (k ^ 0) the propagation is nondispersive, but otherwise 

 the group velocity is smaller than the phase velocity, indicating a normal dis- 

 persion pattern. 



For the present investigation it is convenient to apply a Laplace transform 

 to (2.16). Defining £ [f(t)] by 



00 



i:[f(t)] = I e-^* f(t) dt 



(2.20) 



we obtain from (2.16), using (2.9), (2.10), (2.14) and (2.15), 



Sy 



and 



cosh 



Up- Po) = -^ 



(8^+ 1) 



cosh 



S\ 



L(S^+ 1) 



cosh 



^(Pg- Po) 



Ap 



Sy 



L(S2+ 1) 



S(S2+ 1) 



cosh 



S\ 



(2.21) 



(2.22) 



L(S^+ 1) 

 The parameter x is the (dimensionless) thickness of the bubble- water mixture 



\ 



OJnh 



(2.23) 



119 



