152 - e - 



The coefficient of y on R.H.S. Is positive in the range Ka < »/3, and Is negative for 

 a> H/i. Hence for large n, y increases exponentially in the range l<a< 4/3, and thereafter 

 oscillates. The interval between successive zeros, as n is varied, decreases like 1/vn. The 

 ratio of the maximum value of y and the value of y at a = I Is of the same order of magnitude as 



X = exp { / n. 



/(u o"* - 3 a-') dx } , 



where x is the value of x at the zero of the integrand. Replacing the integrand by an approximate 

 expression, obtained from 



a = 1 ♦ z 



x ■" (l ♦ 2z/3 + z^/5) / (Zz) (Lame's substitution) 



it will be found that 



X = exp { u /n/9 }. 

 If n = 10 the magnification X is 4.09; if n = too, X 's B5; and If n ■■ 1000, x Is 1,300,000. 



Solutt o n for small pulsations of the bubble . 



The solution of (29) in a simple form can be obtained for any value of n in the case of 

 SBBll pulsations. Although this theory is clearly not applicable to real bubbles because surface 

 tension, which would now be relatively important, is neglected, we give the theory because it may 

 indicate some features of the more general motion. 



From (26), (29) and (3l), the equation to determine y is now 



Sfl = in* i) e cos 2x (3») 



dx' ' 



which i's a degenerate form of Mathleu's equation. Since £ is small, we seek an approximate solution 

 in the form 



y = A + Bx + e f(x). 

 On substituting in the equation and retaining only first order terms we find 



f(x) = - 5 (n + -j) { (a + ex) cos X + e sin 2x }, 

 so that 



y = (A + 8x) { 1 - ^ (n + |) £ cos 2x } + jMn + ^) £ 8 sin 2x («0) 



Since y = a b , = const, y a'' , where a is given by (27); hence /9^ can be reduced to the 

 form 



/3j^ = (a' + B'x) { 1 - ^ (n - 1) e cos 2x } ♦ J (n + -1) £ B' sin 2x («l) 



This result may be checked when n = 1 from our previous calculation. We have 



/Sj = ^(1) a-3 = /5(l) { 1 - I £ (1 - cos 2x) }, 

 which on integration gives 



/3 = 1 + const. { (l - ^ £) X + ^ £ sin 2x }, 



which agrees with (uo) when n = 1. 



At any value of a, a measure of the non-sphericity associated with the harmonic S is afforded 

 by the ratio b^^^a; this divided by its initial value is equal to /S /a, which is found from (ti) 

 by dividing by a, as given by (27); this gives 



