-3- a 



In spite of the agreement Between the incotnpressiBle theory and observation as illustrated 

 in Figure 1 it is important to attempt to find the effect of the finite compressibility of the 

 water Before we can conclude that it is safe to extrapolate by the inverse distance law to obtain 

 the pressures in the immediate neighbourhood of the charge. Lamb has stateo that a complete 

 solution including comprjssioil i ty appears hopeless. we know, however, that if we choose a distance 

 sufficiently large th^ ordinary laws for propagation of sound will hold and that the compressioi 1 i ty 

 of water diminishes for very high pressures. Tno following procedure should therefore give us 

 results nearer the truth than if complete incomprussioil ity is assumed. Suppose the gas bubble 

 surrounded oy an incompressible envelope so large that outside it the ordinary laws for the 

 propagation of sound may be taken to hold and let us attempt to find the form of the pressure wave 

 thrown off from this system. If a is the radius of the boundary between the incompressible and 

 compressible region, then at a point r > a the velocity potential is given Oy 



(^ = F (t --^-S )/r (5) 



the equation of pressure is 



p*^pu^ = -p§4 = -pF-/r (6) 



anfl the velocity is 



u = ^ = - f, - i, (7) 



In these equations c is the velocity of sound in water and p the nonral aensity. 

 within the region r < a since incompressibil ity is assumed 



4> = X (t)/r + li (t)/a (5a) 



p + ^po^ = -PX' {t)/r-p0' {t)/a (6a) 



u = -X M'r' (7a) 



The function i/i is possible since there is no restriction to a pressure variation independent 

 of r. must be zero in the outer region as p a;id u vanish at infinity. 



Since p and u are continuous at r = a, 



F- = X' + 0'. f'a/c * f = X- 



or i//= F-x = - F'Vc (8) 



By (6) and (8) since u is negligible when r > a the pressure at a may be written 



p = pc i/j/a^ (9) 



so that the function gives the form of the prtssure-tiirt curve in the outer medium. 



To determine tnese functions we must form the energy equation from the assumed law of 

 expansion of the gas bubble. Let the bubble have initial radius R and radius R at time t so 

 that by (7a) 



^ - - ^ (10) 



at r' 



To form the eni^rgy equation the work done by the bubble in expanding from R to R must 

 be equated to the energy stored in the medium Between R and a together with the energy that has 

 escaped from the surface a. Tne energy escaping from a 



