168 - 6 - 



The final justification of this assumption is that the results to which it leads are in 

 reasonable agreement with the exact solution of equation (13). Hence 



,n = /*'" a^ dt = t^ .J f til) d^) 



or using (l6) ** ^m (non-dimensional) (18) 



The constant k' may be obtained from any one step Dy step solution of the equation of motion 

 of the bubble (l3), and its value is about 0.70, leading to equation (5) 



0.70 a ^ 

 "I " c (non-dimensional) w) 



The minimum radius - a^ 



In the oeighbourhood of the minimum radius the term .^z in equation (13) is negligible, but 

 the term ^(g|) must be included. As shown in the last section g| may be written (when a is near 

 the minimum radius) 



_ g| = i2 (non-dimensional) (l9a) 



and the equation of motion of the bubble becomes: 



af / Ytt"? (1 - i:a~*) - -^ -5 (non-dimensional) (l9b) 



The minimum radius a^ is given by setting g| = 0, yielding 



a^Ml-caf^) = ^-2!- (non-dimensional) (7) 



The firessure wave produced by the coi lapsin^j bubble. 



During the period when the radius of the bubble is near its minimum value, pressures are set 

 up in the immediate neighbourhood of the bubble which give rise to a pressure pulse propagated outward 

 with thi' velocity of sound. Taylor has discussed the pressure distribution close to the bubble, 

 neglecting the compressibility of the water, and gives the following equation (l). If P is the 

 pressure (in lbs. /square inch above thi' normal hydrostatic pressure at the same depth) at a point 

 distant r (non-dimensional units) from the bubble, and if 6 is the angle between the radius vector 

 V and the vertical, then 



^ = -1 .^^ (a^ a) + ^ S^ (au + 5au) cos 9 * ^1^ (cos 8 - \ sin^ 6) 



-[-^(2)''a2 * (|)^aucos0 - ^(|)* uMcos^ * ^sin^^)] (20) 



(R.H.S. non-dimensional) 

 where a = =? . ^ ~ gr • ^"'^ " '^ '^^ vertical velocity in non-dimensional units. 



In thrs 



( l) "Vertical rsotion of a spherical bubble and the pressure surrounding it'. Equation (30). 



Taylor's notation has been modified to conform with the rest of this note. 



