Pressure Pulses in Liquid-Bubble Mixtures 



where 



M 



Ai(z) = - I cos zi + \ l,^ d^ 



is the Airy function of the first kind. The result (7) represents a waveform ad- 

 vancing as a whole at unit velocity (note p keeps a constant value at XT, i.e., 

 at X = Cgt), but with a length scale increasing with time like T''^^ (cf. T. B. 

 Benjamin and B. J. S. Barnard, "A Study of the Motion of a Cavity in a Rotating 

 Liquid," J. Fluid Mech. 19:193 (1964), p. 205). 



If \~'' is negligible, the approximation (5) to the dispersion relation is reli- 

 able for any waves whose dimensionless length is 0(1); and for waves traveling 

 in the positive X direction, it is equivalent to p and the other dependent varia- 

 bles satisfying 



3p dp 1 3^p 



BT ^ BX ^ ^ 3^ 



(8) 



We note, for example, that (7) is a solution of (8). Equation (8) may be regarded, 

 therefore, as the most general governing equation in the case of one-way propa- 

 gation by long waves of very small amplitude. The effects of dispersion are rep- 

 resented by the third term on the left side, which can be regarded as a first- 

 order correction to the "acoustic" wave equation. We go on to find a correspond- 

 ing correction for the effects of finite wave amplitude, and finally we combine 

 the two. 



LONG WAVES OF FINITE AMPLITUDE 



An exact nonlinear theory can be formulated very readily if one assumes 

 0(>v' ^) can be neglected and hence ignores the mechanism that causes frequency 

 dispersion (i.e., the lag between local mean pressure p and the bubble pressure 

 Pg). Thus one takes p= Pg, implying that the local net density p of the mixture 

 (which of course is less than the liquid density, say p^) is a function of p only. 

 The mixture can then be treated as a continuum, by the same method as is famil- 

 iar in elementary gas dynamics. 



The equation of motion and the equation of mass conservation are (in terms 

 of dimensional variables) 



J^, ,^],^-_ (9) 



, Bt Bx / Bx 



and 



If. v|^.p|^= 0. (10) 



c3t ox ox 



As is well known, these two equations may be transformed to yield two in char- 

 acteristic forms, thus 



131 



