240 BELL SYSTEM TECHNICAL JOURNAL 



^^ + ^^ = (6.6) 



dx dy 



This will be satisfied if we express the magnetic field in terms of a "stream 

 function", t 



H. = g (6.7) 



H.= -^ (6.8) 



dx 



IT can be identified as the z component of the vector potential (the vector 

 potential has no other components). 

 We will assume x to be of the form 



T = f (x, y)e'^' (6.9) 



Here w (x, y) is a function of x and y only, which specifies the field dis- 

 tribution in any x, y plane. 

 We can apply Maxwell's equations to obtain the electric fields 



dH^ dHy . ^ 



dy az 



Using (6.7) and (6.8), and replacing dififerentiation with respect to z by 

 multiplication by — F, we find 



£. = -S^ ^ (6.10) 



ue ox 



Similarly 



E.-'^'~ (6.11) 



coe dy 



We see that in an x, y plane, a plane perpendicular to the direction of propa- 

 gation, the field is given as the gradient of a scalar potential V 



V = (-yr/aje)7r (6.12) 



This is because we deal with transverse magnetic waves, that is, with waves 

 which have no longitudinal or z component of magnetic field. Thus, a closed 

 path in an x, y plane, which is normal to the direction of propagation, will 

 link no magnetic flux, and the integral of the electric field around such a 

 path will be zero. 



We can apply the curl relation and obtain E^ 



dHy dH^ . ^ 



dx dy 



(6.14) 



coe Xdx^ dy"^) 



