Pressure Pulses in Liquid-Bubble Mixtures 



DISCUSSION 



T. Brooke Benjamin* 



Institute of Geophysics and Planetary Physics 



University of California 



La Jolla, California 



Dr. van Wijngaarden's paper introduces several interesting new ideas, in 

 particular that a suspension of gas bubbles in liquid is a dispersive medium for 

 the propagation of pressure waves and that nonlinear effects may compete with 

 the effects of frequency dispersion. I noticed a fairly complete analogy between 

 this problem and the problem of nonlinear dispersion of long water waves, about 

 which an extensive body of theory is available, and I shall now outline an alter- 

 native treatment which brings out this helpful connection. The analysis applies 

 only to waves whose length is considerably greater than the characteristic 

 length hA = Cg/wg, where c^ is the sound velocity given by (1.2) in the paper 

 and tjg is the bubble resonance frequency given by (1.1). But within this limita- 

 tion a full account of essential wave behavior can be made quite easily. 



INFINITESIMAL LONG WAVES 



Let X and t denote dimensional distance and time, and define correspond- 

 ing dimensionless variables 



X = x/h ; T = Cgt/h . (1) 



Here h is merely understood to be some overall length typical of the physical 

 system, but the specific meaning given to h in the paper is of course included in 

 this more general definition. In terms of X and T, the linearized Eq. (2.16) 

 becomes 



32p B2p J 34p 



BX^ 3t2 \2 Bx^BT^ 

 where 



(2) 



is a parameter representing the relative importance of dispersion. For long 

 waves, as now in question, we have \" ^ « l. In Eq. (2), p is the local mean 

 pressure in the liquid-bubble mixture, but clearly in the linearized approxima- 

 tion every other dependent variable will satisfy the same equation. 



*On leave from the University of Cambridge. 



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