PHYSICAL REALITY OF ZENNECK'S SURFACE WAVE 39 



Let the incident electric field be 



77 . — - glut — ihi{x sin 6—z cos) r-nSi 6 

 77 . __ giu<— iAi(sin 6—z cos 6) gjj^ a 



The reflected field is 



£^r = — e'"'-'*'(^ «'" ^+^ '^''s «) cos <9-i?, 



El __ /jtoj^— iiiCx sin 6+z cos 9) gjj^ d'R 



where 



R = 



cos 6 — T^\ — T^ sin^ 



cos + tVi - t2 sin2 e ' 



Fig. 1 



Then the total field is 



E^ = e'"'-**i* ^'° * cos ^(g'" - Re-'"), 

 E, = e^"'-**i* «•'» * sin die'" + Re-'"), 



where tj = ^iz cos ^. 



E" ^ giwt— tftix sin 9 T COS 

 £- = gtwt-ii-ia: sin 9 2 sin I 



cos^ + tVi - T^sin^ d 



In order to see better the significance of these formulas it is neces- 

 sary to write d = ir/2 — 8 where 5 is small, say less than 20°, and 

 suppose that kiz cos 6 is small. Then we may use the expansions 



cos ^ = sin 6 = 5 - 8^/31 + - • • • , 

 sin = cos 5 = 1 - 5V2 !+-•••, 

 tan 77 = cos e-kiz{l + h^z^^S + • • •). 

 cos 77 = 1 - ki-z-8-12 + - •••• 



