153 



/Sn 



(A- ♦ 8'x) { i - I (n - 2) « COS 2x } ♦ ^ (n ♦ jM £ e- sin 2x (»2) 



Hence the perturbation associateO with 5^ may be regarded as a small oscillation of period T, 

 about a mean vdlue which Is continually increasing with t (or x). When n = 2, the amplitude 

 of this small oscillation is constant, Dut when n > 2 the amplitude also increases with t. The 

 greatsr the value of n, the greater is the amplitude of this oscillation. This is illustrated in 

 figure 3, where the graphs ot /S^/o- and yS^/o for two pulsations of the buBble are shown. 



When ,^ is Initially zero, the constant B" in (Ul) is zero, and the perturbation is simple 

 harmonic, the amplitude being proportional to (n - 2). 



The solution of the equation (3*) can also be obtained in the form (15). Adopting Hill's 

 method of solving Mathieu's equation, we seek a solution of (3') in the form 



\x f J, g2>'ix_ 



r = -co '' 



In which one coefficient, say c^, is arbitrary. 



Substituting in (3 9) and equating coefficients of like powers of e" to zero, we have 



^* 2ri)2 c,-^ (y 4)£ (c,.i*C^j) =■ 0, 



for all positive and negative intsgrel values of r. 



Eliminating the c's gives an infinite continuant equated to zero to determine X.. This 

 continuant is a limiting form of Hill's determinant (Whittaker and Watson, Modern Analysis, laZo, 

 p.uiS) and on neglecting terms of order. higher than the first, we find 



\ = ± \ ei yz Where 6 =-• ^.(n ♦ ^) e, 



and Ci= c_j = -^9 c^. c^ = c_2= ... = o(e^) . 



This leads to a real solution of the form 



[a- cos ^ + S- sin ^1 { 1 - i (n - 2) € cos 2x > (43) 



P. (.. .fix 



for small x this agrees with the result (4i) provided B'/B" = Oh/2, a small quantity of the first 

 order. The apparent difference in the two results for other values of x is due to the fact that 

 they are both .approximations to the first order In €, and it is possible for the ratio of the 

 arbitrary constants to have any order of smallness^ Thus if A''/B" is assumed to be of the first 

 order in e, a further term in £ would be required to be retained in the coefficient of e^'" In 

 the ab:ve solution, to ensure that all the terms retained are of the same order. This would give 

 a term corresponding to the sin 2x term in («l). 



Without going further into these refinements, it will be seen from (43) that, provided the 

 condition that fij (a - l) remains small is not violated, the perturbation can be regarded as a 

 quasi-periodic variation of period T in which both the mean value and the amplitude have a slow 

 periodic variation of period ( 2n * 1) 6 ^' 



Sol ution for tkc general case . 



In the general case where ()/P has any value < 1, and n is any positive integer, it is 

 convenient to derive an equation for /3 in terms of a, and solve this instead of dealing with the 

 equation (29) directly. This mc-thoJ is somewhat analogous to Lindemann's treatment of Itethelu's 

 equation. 



Since?, is periodic in t, b^ will be a multi-valued function of a with branch points at 

 a = a^ and a = a^. Denote the value of b^ in the half-period ^T<t<|TDyi5 , which Is 

 then a single-valued function of a. Thus b^^ ^ is the value of b^ during the first expansion, 

 ""n 2 ^""^ value during the first contraction and so on. 



Considering ,..., 



