148 



When \ Is pure)/ imaginary, will be the product of two periodic functions of different 

 periods; if it Is not purely inajinary, will ultimately increase to large values. *ctu?'ly 

 k is found to be small and purely imaginary for small pulsations of the Bubble, i.e. when P is 

 nearly equal to 0. for other values of p/Q It is complex, except v.iien n = 1, when it is zero. 

 Hence in general the perturbation is unstable. 



The calculation of the mean radius a(tl of the bubble. 



Before proceeding with the solution of (it) for C^, it is convenient to consider the 

 function a(t) which deternines the coefficients appearing in (l»). 



Writing 



a = a/a , x = "^ot/a^ where c^^ = P/p (l6) 



the equation (12) has a first integral of the form 



and a further integration gives 



, -/{iii^} f - fa. ^^ (^, 



'' 2 I / {a ^ - a ^ ♦ (y - I) P '(3{a ^ - 1)} 



Values of this integral for a small range of a have been given by Lanc(l) for the case 

 Q = and y = n/3. in this case the motion Is not, of course, periodic but a continually incraases 

 with t(or x). 



When Q is not zero, a is periodic; i.e. the bubble pulsates. The period T , measured in 

 terms of x, Is equal to twice the Interval In x between the two singularities of the integrand in 

 (le). One of these Is at a = 1, and the other is at a = a^, say, where a is the ratio of the 

 maximum radius to the minimum radius.. 



The ev=iliatlon of the integral in (18), when P/0 is large, h~s been discussed by 9uttsrwortr(.!) 

 Willis(3), and Herrlng(U), but their m3thods and formulae are only applicable to the accurate 

 determination of a as a function of t when a is near unity, and to the determination of a/a as a 

 function of (T-t) when a is nearaj^,. Their results are in fact equivalent to first approxi.TBtions 

 to the integral between 1 and a near the singularity a = 1, and to the integral between a and a^ 

 near the singularity a = a . 



An extension of thesi reults, which enables a(t) to be determined with any desired accuracy 

 over the whole period, is readily obtained. The analysis Is greatly simplifed when y = »/3, and 

 as this is probably a fair approximation to the actu?,l case, only this value of y will be consider;", 

 Most of the investigations referred tj above have also dealt only with this value of y. 



Wheny=u/3, the expression (18) can be written in the factorlstd form 



t r 



'^± ^ ^ ^ 1 , alj'Jl 



pa^ V~ /(a-1) v^{l-0 (a + o^*r?)/(3P)} 



(l») 



from which it will be seen that a^^^ is the positive root of 



a^ + a^ * a = 3P/l5. (25) 



Since P > Q, tUs root is greater than unity, and for large P/i), its value is approximately 

 (3P/q) . An expression for the root in the form of a power series, suitable for all P/0 > 1, 

 can be obtained by a simp'e i sration method; this gives 



A o 



»:?■ 



(21) 



