244 BELL SYSTEM TECHNICAL JOURNAL 



Again applying the relation 



dE^ dEy 



dy dz 



we obtain 



ft + ^= -(t' + ^Dt-J (6.34) 



We will now divide both tt and / into the spatial distributions charac- 

 teristic of the normal unforced modes. 

 Let 



J{x, y) = ^ JnTTnix, y) (6.35) 



n 



// J{x, y)Trn{x, y) dx dy 



Jn = (6.36) 



// [ftnix, y)] dx dy 



This expansion is possible because the 7r„'s are orthogonal. Let 



Tt = e~ ' zl CnTtn{x, y) (6.37) 



n 



Here there is no question of forward and backward waves; the forced ex- 

 citation has the same ^-distribution as the forcing current. 

 For the wth component, we have, from (6.16), 



dTrn{x,y) dirn{x,y) , i , ^2w / n 



dx^ dy- 



From (6.34) we must also have 

 ^ / d'TTnix, y) d^TTnix, y) 



(6.38) 



= — C„(r" +;So)7r„(x, y) — JnTTnix, y) 



Accordingly, we must have 



The overall stream function is thus 



7r = e-"E^#^" (6.41) 



n i n i 



From (6.33) and (6.34) we see that 



E, = ^ (r'' + fiDir (6.42) 



