191 



ratio of the minimum radius, R^i„ , 

 to J?o. The values of k were ob- 

 tained by numerical integration of 

 Equation [35] for several values 

 of C; «m„/«o and fl„,„/ie„ were 

 found as values of Xj and i^ from 

 Equation [36]. The value y = 4/3 

 was used, in order to simplify the 

 calculation; a somewhat different 

 value would give nearly the same 

 curve. 



For large amplitudes, 

 perhaps where R„„/Rq Is greater 

 than 2.25, the formula given in 

 Equation [22] on page 48 of TMB 

 Report 480 (4) may be used 



T = 1.83 i?^„ \/J- 



[4a] 



1.0 

 0.8 



Rron 



\l * T. 



1.6 



1.8 



Rir 



./Ro 



Figure 1 - Curves referring to Undamped 



Oscillations of a Bubble or Globe 



of Gas under Water 



Rg is the radius when the gas pressure equals the 

 hydrostatic pressure, Rmtx is the maximuni radius, 

 '^min is the minimum radius, T„ is the period of 

 very small oscillations, T is the period of oscil- 

 lation having given value of Rj^x/ ^o' "^^ curves 

 are drawn for y = U/'i, but y makes little difference. 



For a gas globe or bubble in water this may be written 



I'+b] 



where p^ is the hydrostatic pressure in atmospheres and iJ^ax is in inches. 

 For a given mass of gas, i2^„ « ^/Pa'' hence T is proportional to Vp^^. In 

 sea water, 217 is replaced in Equation [4b] by 214. If use is made of the 

 value Just given for /?n^, , Equation [4b] becomes 



T =0.23 



W\ 



[t^c] 



These latter formulas may be used to estimate the time of collapse 

 of a bubble under suddenly applied steady pressure. If R^^^ represents the 

 initial radius of the bubble and p^, or p^ the suddenly applied pressure, the 

 time of collapse is T/2. The estimate should be of high accuracy if the ra- 

 tio of pressure increase exceeds 2.25* or about 25- 



If the amplitude Rm,x/^o ^^ very large, compressibility of the wa- 

 ter will play an important part. The direct effect of compressibility on the 

 period will be small, since the high-pressure phase of the motion occupies 

 only a very small part of the total period; but a loss of energy occurs by 

 acoustic radiation during the time of Intense contraction, so that each out- 

 ward swing is less in amplitude than the preceding inward swing. The period 



