214 26 



'3/> C x^ dx 



" ' - - X [35] 



2po ^'f 



Po "^ (Cx - X* - 3)2 



and 



[36] 



in terras of the minimum or maximum values of x, i, or Xg. For purposes of 

 numerical calculation, the substitution, x = x^{^ + u^) Is useful near i,, 

 and I = Xjil - v^) near Xj. 



When the amplitude of oscillation is large, a useful analytical 

 approximation for ( can be obtained which is valid near the time <i at which 

 the radius R takes on its minimum value i?,. When R is near i?,, x* is rela- 

 tively small and can be dropped without much error; if Xj^ is similarly 

 dropped in Equation [36], so that C = 3/x,, the integral in Equation [35]. 

 taken between the limits Xj and x, becomes 



/•^l/H =^'dx ^ xj, i /16 ^ 1,^^ 2 X2X 



J^ ^ 3 (^ - x,)i >^ ^15 15 X. 5 x,2^ 



It is convenient, also, to eliminate p and p^ by means of T^ as given by 

 Equation [3'+a]. Then Equation [35] gives, since x, = R^/R^, 



t - t 



.= fM|f^F^(i|-^.^4f) "T, 



This formula should hold well so long as x* is small as compared to 3, per- 

 haps up to R = Ro/2. 



PRESSURE IN THE WATER 



The pressure in incompressible water around a sphere of gas exe- 

 cuting radial oscillations is given by Equation [8] on page 46 of TMB Report 

 480 C^), provided r, is replaced by R and u, by dR/dt. This gives, with u 

 replaced by v for the particle velocity of the water, 



P = T[p. + h{ljf-Po]-^Pv^ + Po [38] 



for the pressure p at a distance r from the center of the bubble. Here p is 

 the density of water and p^ is the hydrosfatlc pressure. Or, If substitution 

 is made for p^ from Equation [26], for dR/dt from Equation [28], and for C\ 

 from Equation [29], 



[39] 

 Po R U 1 /^oN^n/^n^ / 1 \/^o\^''l 1 



