LINEAR AND NONLINEAR DISPERSION 

 OF PRESSURE PULSES IN LIQUID- 

 BUBBLE MIXTURES 



L. van Wijngaarden* 

 Netherlands Ship Model Basin 

 Wageningen, The Netherlands 



1. INTRODUCTION 



Thirty years ago Minnaert (1) derived a formula for the oscillation fre- 

 quency Wg of an air bubble in water: 



In (1.1) Rq is the bubble radius, p the density of water and Pq the ambient 

 pressure. 



When many small bubbles are present in a cubic meter of water, n say, it 

 is equally well known that the mixture can be considered as a homogeneous fluid 

 with mass density nearly equal to the water density and with compressibility 

 arising from the gas content. For a small gas volume in a unit of volume of the 

 mixture, the velocity with which infinitesimal disturbances of pressure propa- 

 gate through the mixture (the sound velocity) is given by 



^° ^ (1.2) 



TTnp Rq^ 



Equation (1.2) follows from considerations in which the bubble character of the 

 air content is not taken into account — in other words for wavelengths which are 

 large with respect to the bubble radius. How do waves with shorter wavelengths 

 propagate? This question is considered here. 



Furthermore, it is well known that for finite pressure variations, growth 

 and collapse of a bubble are highly nonlinear. High pressures occurring during 



♦Present address: Technische Hogeschool Twente, Afdeling der Worktuigbouwk- 

 unde, Postbus 217, Enschede, The Netherlands. 



115 



