Van Wijngaarden 



By analogy, the outcome of such an investigation in the present instance is obvi- 

 ous: the "correction" terms in (8), rewritten as an equation for Q, and in (23) 

 are simply added. Thus we have 



BQ 3Q BQ 1 B^Q 



This third-order nonlinear equation is often called the Korteweg-de Vries equa- 

 tion (see G. B. Whitham, "Nonlinear Dispersive Waves," Proc. Roy. Soc. A283: 

 238 (1965), p. 252), and a great deal is now known about its solutions. Its occur- 

 rence here indicates the possibility of several interesting wave phenomena in a 

 liquid-bubble mixture that would be analogous to water-wave phenomena known 

 to be described approximately by the equation. In particular, there appears the 

 possibility of oscillatory shock waves (analogous to undular bores or hydraulic 

 jumps) and of solitary waves (i.e., nonper iodic waves of finite amplitude but 

 permanent form). Note that the solitary- wave solution of (24) is 



Q = a sech^ l|- aj \(X-CT) 



with 



a = 3(C- 1) . 

 * * * 



REPLY TO DISCUSSION 



L. van Wijngaarden 



(25) 



I wish to thank Dr. Brooke Benjamin very much for his comments. His 

 suggestion, substantiated by his analysis, that there is an analogy between long 

 water waves and waves in a liquid-bubble mixture is most useful, both from a 

 theoretical and a practical point of view. Both types of waves display indeed 

 frequency dispersion and amplitude dispersion. However, upon looking deeper 

 in this matter, it appears that whereas in the theory of water waves there are 

 three characteristic lengths, viz., amplitude, wavelength and depth, there are 

 four lengths here, namely wavelength, amplitude, bubble size, and inter-bubble 

 distance. In the theory of water waves there are consequently tivo dimension- 

 less parameters, one giving the ratio between amplitude and wavelength, the 

 other the ratio between depth and wavelength. In the case of bubble -fluid mix- 

 tures there are three parameters. If e.g., bubble size plays the role of depth, 

 then the third parameter is the ratio between bubble size and bubble distance. 

 It appears that only when this ratio is of order 1, the analogy holds strictly. 

 The above mentioned ratio is essentially the cubic root of the volume fraction 

 nVg of the gas. Now in some of Dr. Benjamin's equations this fraction is as- 

 sumed to be of order 1 (Eqs. (9) and (10)), whereas in others it is assumed to be 



134 



