51 219 



dR 



where 4 denotes the change of a quantity during the time t , . Thus 



If dR/dt = Initially, the velocity produced by the impulsive variation of 

 p„ is, therefore. 



dR 



d 



T^-Tr/^^'' f5i 



EFFECT OF MANY SMALL BUBBLES ON LONG PLANE 

 SINUSOIDAL WAVES OF A SMALL AMPLITUDE 



In order to deal accurately with the effect of bubbles upon waves 

 of pressure, it is necessary to make full allowance for the compressibility 

 of the water. The analysis then becomes very difficult unless it is re- 

 stricted to very small variations of pressure, so that acoustic theory can 

 be employed. This restriction will now be made. Even so, only the case of 

 sinusoidal waves can be handled readily; waves of other forms may then be 

 treated if necessary, with the help of Fourier analysis. 



It will be assumed that the spacing of the bubbles, although large 

 relatively to their diameter, is small relatively to the wave length of the 

 wave, either in the bubbly water or in homogeneous water. This assumption 

 will be taken to imply, in particular, that the average pressure in the wa- 

 ter at any instant is sensibly the same as the pressure at points midway 

 between the bubbles, and also that the local pressure field around each bub- 

 ble is sensibly the same as it would be if this field were exactly spherical- 

 ly symmetrical and had a value at infinity equal to the actual mean pressure 

 between the bubbles. 



Let there be n bubbles per unit volume, all having radius Ro when 

 In equilibrium under the hydrostatic pressure p^. Let i denote distance in 

 the direction of propagation of the waves. 



The equations of propagation are easily obtained -in the usual way, 

 by considering an element of volume having the form of a cylinder of length 

 dx and of unit cross-sectional area. Let v denote the average particle ve- 

 locity of the water in the direction of x. Then the volume of the mixture of 

 bubbles and ^water that is in the element at any instant increases during the 

 next Interval d( by 



— — dx dt 

 ox 



