GUIDED-WAVK I'KOPACiATIOX TIIHOIXJH (; YK(XMA(i.\KTK' MIODIA 585 



H at that point, but on values of // Ihroiiohout the vokime of the sample. 

 Therefore B, which is luoU + wt, ^vill likewise be a functional of H over 

 the whole sample. Fortunately it turns out that the spatial \'ariation of 

 H in a microwave structure is so much slower than that characteristic of 

 the "spin waves" to which V"A/ gives rise that this effect is finite negli- 

 gible at microwave frequencies. Only in th(> most immediate vicinity of 

 gyromagnetic resonance could such effects become significant. 



Thus, we shall regard Ht simply as the sum of the dc and ac magnetic 

 fields, ^0 + U, ai^d correspondingly M as the sum of the dc magnetiza- 

 tion (directed along Ho in a saturated sample when anisotropy is neg- 

 lected) plus an ac part ?«. Ecjuation (2) must now be solved for m in 

 terms of H. It is a non-linear ecjuation, whose solution m will depend on 

 H non-linearly, as will B. Even if m could be determined in this way, 

 ]\Iaxwell's eciuations would become non-linear, and hope of their solu- 

 tion remote. It is therefore necessary, and in the great majority of ap- 

 plications also quite sufficient, to assume that the ac quantities in (2) 

 are so small that their products can be neglected and only linear terms 

 taken into account. The terms m and H may now be assumed to vary 

 as exp jo)t. 



Under these circumstances, (2) becomes 



"^-i = 7([m X ^o] + [Mo X H]) 

 at 



ay 



'Wo 



([l/o X [m X ^o]] + [Mo X [Mo X H]]), 



and is easily solved for m in terms of H, and of the dc quantities Hq , Mo 

 which we shall assume to point in the ^-direction. Each of the components 

 mx , rny is a linear function of both H^ and Hy and when they are sub- 

 stituted in the components of the equation B = ^iqH -{- m, lead to ex- 

 pressions of the form (1) for B in terms of H: 



(3) 



Mow 



