18 - « - 



(3) T varies inversely with the b/6 oower of the total external pressure. Shorter time 



intervals are therefore to Be expected if a charje is fired at any apprc-ciable depth in 



P ressure variations outside the bubble . 



Equation {?) shows how the pressure inside the Bubble varies with time. This variation 

 is not readily measured. What is frequently measured is the form of the pressure pulse at a 

 distant point outside the bubble. The form of this pulse is not the same as that of the internnl 

 pressure, and the nature of the difference is now considered. 



At any instant, t, the 2xp,=indinj bubble funct ions •:! s a source whos= strength S is equal to 

 the rite at which w^tjr is biinj forced outw'.rJs, i.e. 



3 = -i (1? r^) (15) 



dt ^ 



The velocity potential, '/i 3' ^ point distant a from the bubble is, by the usual formula, 



V = .^_ = _J^ i (lil r^) = i A (4) (16) 



'^ UTia 4wa:t^ adt^ 



No retarded potential is here involved as tne medium is assumed incompressible, and the velocity 

 of sound is therefore infinite. The pressure at the point a is, therefore, by the usual 

 hydrodynamic formula 



n = p 1^ * i ^v^ 



? t 2 



Where v is the radial velocity at a 

 = e £,d) . 1, Lr!r (17, 



dt^ ^ 2 '^ 



The seccnd term on the right hand side veries inversely with a , and therefore rapidly becomes 

 neglijible in comparison with the first term which varies inversely with a. Equation (l7) ther:;fore 

 simplifies in practice to 



= 2-4 (4^1 (18) 



=> dt' ^ 



This equation, tojether with the solution for r furnished oy equation (8), specifies the vj-lues of 

 n for any time t. Tnis, however, is not : convenient sulution, and a more useful result can be 

 obtaitied in the following way;- 



Equat ion (18) can be written 



r^ 'r' + 2 r r2 = il? (19) 



P 



Differentiation of the enerijy equation (3) jives 



r^ "r* + 2 r 1-2 = I (c -P ] (20) 



2 P 



Subtraction uf '20) from (l9) jives 



U - £ \ (p-D . 1 -^2 I (21) 



Substituting in (2l) the value for r jivtn by (o), ana takinj y= u/3 jivcs 



♦ ^fd.a) x-ax'*- t"j 1 



(22) 



