JFAVE PROPAGATION 543 



E,{x,y)-E,(x,0) = -io:(A,(x,y)-A,(x,0))-^(y{x,y)-Vo). (16) 



Here Ez(x,0) is the axial electric intensily at the surface of tiie gnnmd 

 plane (3' = 0), and 



A,(x,v)-A-Xx,{))= r Ih{x,\)dv. (17) 



V{x,y)— I'u is the difference in the scalar potential between the point 

 x,y and the ground, which is due to the charges on the wire and on 

 the surface of the ground. For convenience, it will be designated by V. 

 By means of (16) and the preceding formulas we get ^ 



(Vm'+^-m)^"^^''^'''^ cos x'^x 



dfx 



■^ 2a;/ log (p"/p')-|^F, 3;^0 



(18) 



where 



p' = V(h-yy-\-x^ 



= distance of point x,y from wire, 

 p" = V(h+yy+x'~ 



= distance of point x,y from image of wire. 



The first two terms on the right hand side of (18) represent the 



electric force due to the varying magnetic field; the term — — F 



9s 

 represents the axial electric intensity due to the charges on the surface 

 of the wire and the ground. If Q be the charge per unit length, V 

 is calculable by usual electrostatic methods on the assumption that 

 the surface of the wire and the surface of the ground are equipotential 

 surfaces, and their difference of potential is Q/C where C is the electro- 

 static capacity between wire and ground.* 



II 



By aid of the preceding analysis and formulas, we are now in a 

 position to derive the propagation constant, T, and characteristic 

 impedance, K, which characterize wave propagation along the system. 

 Let s denote the "internal" or "intrinsic" impedance of the wire per 



2 As a check on this foniuihi note that together with (14) it satisfies the condition 

 of continuity of E^ at y =0. 



" See "Wave Propagation Over Parallel Wires: The Proxiniit\- Eftp-t," Phil. Mas 

 \"ol. xli. Apr., 1921. 



