-7- 169 



In this expression for the pressure the first term is the most important aad will determine 

 the pressure at points not too near the Ouoble. Considerini) here only this leading term, the pressure 

 at a point R feet from the Dubblc centre becomes 



IWRP - d u2 1, , , , „ . 



— TT 17 ' ' IR.H.S. non-dimensional (20a) 



pi dt 



Since the simplified eauations of motion (19) are valid near the time of the minimum radius, 

 when the pressure disturbance is mainly produced, (20a) may be evaluated to give: 



2 



1U«RP_ = 1 c — * I 1L (R.H.S. non-dimensional) (21) 



or 4 77 a TT 3 a 



15 7T a 



The maximum value of the pressure Pm is obtained by insertimj in (21) the value of the minimum 

 radius a, giv^^n by (7); hence 



?f?' - = 2 — . il - ica.'^) (R.H.S. non-dimensional) (8) 



O.H3« L 'in a^ 



It may be remarked that in a simibr way all the other terms in (20) may be evaluated as 

 functions of th-; radius a alone, using equations (19) and their value at the minimum radius computed. 



When the radius is a minimum, a and u vanish while u = - g| is given by (l9a). Inserting 

 the value a, for the minimum radius, and converting to real units, the maximum vertical velocity U^ 

 (feet/ second) is 



11.31L 



T ' ~) 



(R.H.S. non-dimensional) (ll) 



Owing to the rise of the bubble points situated some distance vertically atJOve the charge may 

 be quite close to the bubble when it is near its minimum radius, and the peak value of the pressure 

 would then require the computation of further terms in (20). 



The total positive impulse in the pressure wave is got by integrating (20a) with respect to 

 time between the two times for which a g| is a maximum (l). Conyers Herring has obtained the value 

 of this impulse using an equation of motion which is identical with (l3) with the vertical velocity 

 and the gas energy term ca~ neglected. Since the value of the impulse depends only on the maximum 

 value of a ^, which occurs when the bubble is comparatively large. Herring's neglect of the two 

 above mentioned factors will not cause much error. Herring's value of the total positive impulse 

 (lbs. second/ square inch units) may be converted into the following useful form, if the simple 



(9) 



where R is the distance in feet from the bubble centre. 



It can be shown that the duration D of this positive pulse is a constant fraction of the 

 period T. Since the times when a g| is a maximum occur when the radius is large, the approximation 

 involved in equation (17) may be used. Differentiating (l7) twice wi-th respect to time, and using in 

 equation (20a). 



mRP , d ,.2 da, . a^3 ^_ . ^ (R.H.S. norKdimensional) (22) 



A 



3t-x' 



(l) The assumption is made that the expansion taking place at the beginnlncj of the second 



oscillation is similar to the contraction taking place at the end of the first osclHatic 



