BELL SYSTEM TECHNICAL JOURNAL 357 



(a) Wave perturbation, Fig. 2 



Wq = Jsin2(:x: - 2) 



W. = ^^in2(. - 2) + 'l^^±l..s2{x - 2). 



Figure 2 exhibits rapid pulling of the successive approximate waves to- 

 ward the exact even though the interval has been chosen deliberately 

 unfavorable to the straight wave perturbation method [see example (c) 

 and Fig. 4 for the improved treatment]. 



(b) Linear perturbation, Fig. 3 



Lq = X — 2 



L^ = -r^ (16 - 16x + 4:^3 - x'') 



' 63 "^ 45 9 9 90 "^ 504* 



> 



Using Wq instead of Lo 



£i = i - I + ^sin 2{x - 2) + i cos 2{x - 2). 



(c) Preliminary transformation of variables, Fig. 4 

 Introduce 6 = ^x'^, y = d~^'^v 



and the modified equation is 



Then, using for simplicity ^3 = 1 



vo = 2-'V sin {d - Bo), do = 4^/2// 

 or 



Wo = (2xr''' sin Ux" - 2'") 



It will be seen that Wq is a very good approximation throughout the 

 range (2, 6). Adding Wi obtained from 



vi = — ^ [cos(<9 + do) (Si 2d - Si 200) - sin((? + (?o)(Ci2^ - Ci 2do)] 

 36V2 



the accurate curve y is reproduced. 



In Fig. 2 the third approximation could not be distinguished from the 

 accurate curve though numerically the values are not identical. For 

 purposes of comparison the table of numerical values (Table A) may be 

 found interesting. 



