FERROMAGNETIC DISTORTION OF A WAVE 



335 



^0 = -El 



Ao^ 



A,= 



4 



[(2 - ki')Ei - 2(1 - ki^)Ki'] 



ISirki 



-[8(2 - ki'){l - ki')K, - (16 - 16^i2 _|_ ^^4)£j 



(16) 



Here Ki and Ei are complete elliptic integrals of the first and second 

 kind, respectively, with modulus ki. 



The series (14) may be used to evaluate the variable permeability 

 anywhere on the loop, for upon substitution in equation (8) reference 

 to a particular branch is eliminated by the disappearance of the double 

 sign on the second term. 



The square of the maximum magnetizing force needed in the final 

 term of equation (8) comes directly from equation (13) ; to determine 

 the sign of this term at any instant remains the only problem. Inter- 

 pretation of (— 1)' seems simple when it is remembered to be positive 

 for decreasing h and negative for increasing h, and therefore an odd 

 function of time. The rate of change of the magnetizing force is 



dh 

 dt 



= — Pp sin pt — Qq sin qt, 



so it follows that 



(- !)'■ = + 1, sin pt + K sin qt > 0' 

 = — 1, sin ^/ + K sin qt < 0\ 



(17) 



The solution may be completed by expanding this quantity in a 

 Fourier's series : * 



00 00 



(- 1)'' = L Z ^m» sin (mp + nq)t. 



m=0 n=—ao 



(18) 



When m = the summation is to be extended over only ppsitive 

 values of n. With this convention the coefficients are 



Amn = 2^2 I 1 (~ 1)' sin {mx + ny)dx dy 



•7 — TT fJ —IT 



(19) 



or, with the use of equation (17), 



^mn={-\) ' ^,X dyl 



COS mx cos ny dx, (20) 



*W. R. Bennett, "New Results in the Calculation of Modulation Products," 

 B. S. T. J., Vol. 12, pp. 228-243, April, 1933. 



