150 - « - 



It win be seen from this expression and from iti) that tor small pulsations, the period 

 is TT &^</{p/f). 



The equation for b^lt). 



The equation (ll) for a is reduced to the nortrol form 0/ writing 



y = a^'^ ■>„, (») 



which gives 



Substituting fora from (12), and for a from (17) withy^ »/3, we obtain 



5i^ =|'!fil ; . F(x)y, (2») 



where 



F(x) = (»n * j) a-' - 3n (1 * ^) a"^ - -^ a"^ (30) 



The following properties of this function are easily verified. 



(i) F(x) is periodic in x, the period being the sane as that of a, and therefore ranging frum 



77 for small ot^ to approximately a^ for large 0^. 



(ii) F{0) = (n t ^) (1 - 0/P). 



(iii) F(T/2) = (n + |) fa. "' - 5(l"^/p). Hence, when the bubble reaches its maximum size, 

 F(x) is negative. When P/o is large, its value is approximately -3(n + -j) a^ . 



(iv) F(x) is zero for only one value of a Between 1 and a,,. When fl/o is large, this value 

 cf a is approximately (8n + l)/6n and thus lies between the narrow limits ij and 1^ for 

 all n. 



(v) F(x) tas rraxima when a = 1 and a = a^j^, and has a minimum when 



^a" + 5n (I* P) a- (8n + l) = 0. 



For large P/Q, this minimum occurs apprr.ximately when a = (Sn + l)/5n, and Its value Is 

 approximately n f 5 n 1 



(vi) In the case of small pulsations, on substituting from {ii) in (30) and neglecting terms 



beyond the first order, we have 



F(x) = (n + |) e cos 2x. (31) 



Most of the above properties are shown diagrammat Ically in Figure 1, which refers to the 

 two extreme cases, P/O large and P/O = 1 + e, £ small. *s P/Q decreases from large values the 

 two minima shown In the figures move together, eventua ly obliterating the maximum at a • a^j, and 

 tending to the simple cosine function (31) for scieII pulsations. 



Solu tion /o r n_' 1. Hot t on of bubbl e j n absence of extraneous f orces , 



The first hariTonlc term b,(t)S In (l) simply represents a displacement of the bubble, without 

 change of form and of amount b,(t5 along the axis of the harmonic. Hence the solution when n = 1 

 gives the nctlon of the bubble In the absence of extraneous forces. 



The sol.ition In this case Is easy, and does not require the preceding analysis. We have 

 from (l3) wi-th g = 0, (compare also lUa) 



bj = bj(0)/a^, (31) 



which 



