﻿Z. 
  DeForest 
  — 
  Reflection 
  of 
  Hertzian 
  Waves. 
  63 
  

  

  But 
  we 
  must 
  consider 
  the 
  case 
  of 
  a 
  condenser, 
  of 
  capacity 
  S^ 
  

   between 
  the 
  ends 
  of 
  the 
  two 
  parallel 
  wires. 
  Then 
  if 
  Y 
  denote 
  

   the 
  resultant 
  potential-difference 
  in 
  the 
  wires 
  at 
  the 
  ends 
  from 
  

   incident 
  and 
  reflected 
  waves, 
  and 
  C 
  the 
  resultant 
  current, 
  we 
  

   have 
  the 
  condenser 
  equation 
  

  

  8V 
  C 
  ., 
  , 
  . 
  . 
  , 
  

  

  —r- 
  =: 
  — 
  ^ 
  if 
  there 
  is 
  no 
  transmitted 
  

  

  dt 
  b^ 
  

  

  wave 
  ; 
  and 
  J, 
  the 
  coefficient 
  of 
  reflection, 
  equals 
  1. 
  

  

  But 
  in 
  the 
  case 
  of 
  open 
  ends 
  we 
  must 
  suppose 
  a 
  part 
  of 
  the 
  

   energy 
  radiated 
  off, 
  either 
  in 
  expanding, 
  approximately 
  spher- 
  

   ical, 
  waves, 
  or 
  for 
  a 
  distance 
  as 
  a 
  beam, 
  more 
  or 
  less 
  plane 
  

   polarized. 
  The 
  amount 
  lost 
  will 
  be 
  greater 
  the 
  less 
  the 
  capacity 
  

   of 
  the 
  end 
  condenser. 
  

  

  But 
  for 
  our 
  purpose 
  it 
  is 
  unnecessary 
  to 
  determine 
  the 
  exact 
  

   course 
  or 
  nature 
  of 
  the 
  wave 
  after 
  leaving 
  the 
  wires. 
  We 
  

   may 
  consider 
  the 
  case 
  as 
  though 
  the 
  wires 
  were 
  continued 
  past 
  

   the 
  end 
  condenser, 
  by 
  a 
  pair 
  having 
  a 
  different 
  capacity 
  S3 
  per 
  

   unit 
  length. 
  

  

  Let 
  e^ 
  be 
  the 
  charge 
  on 
  unit 
  length 
  of 
  the 
  wires 
  due 
  to 
  the 
  

   incident 
  wave 
  ; 
  and 
  e^ 
  the 
  charge 
  per 
  unit 
  length 
  carried 
  off, 
  

   with 
  the 
  same 
  velocity 
  v. 
  

  

  Then 
  we 
  may 
  write 
  : 
  

  

  C3 
  = 
  e^v 
  =. 
  ce^v 
  =z 
  ge, 
  

   where 
  c 
  = 
  coeflicient 
  of 
  transmission, 
  and 
  g 
  = 
  cv. 
  

  

  C 
  Y 
  

  

  Now 
  e, 
  = 
  — 
  !- 
  and 
  C, 
  = 
  .— 
  

   V 
  ^ 
  Lv 
  

  

  . 
  • 
  . 
  C3 
  = 
  cC^ 
  = 
  cY^Sv 
  

   Corresponding 
  to 
  the 
  condenser 
  equation 
  we 
  now 
  have 
  

  

  ^ 
  _ 
  C,,_+_C3 
  _ 
  C 
  cY^ 
  

  

  dt 
  "" 
  S^ 
  ~ 
  S^ 
  "^ 
  S^Lv 
  

   g 
  

   Let 
  7 
  = 
  ^5 
  the 
  ratio 
  between 
  the 
  capacities 
  of 
  the 
  end 
  con- 
  

  

  o 
  

  

  denser 
  and 
  that 
  of 
  a 
  unit 
  length 
  of 
  the 
  line. 
  

  

  Substituting 
  in 
  this 
  the 
  values 
  given 
  for 
  C 
  and 
  V^, 
  and 
  devel- 
  

   oping 
  the 
  equation, 
  we 
  get, 
  letting 
  ^ 
  = 
  0, 
  

  

  n 
  (<^ 
  + 
  <^') 
  cos 
  ^ 
  = 
  - 
  ^ 
  Vl^"^ 
  '''' 
  i 
  (^) 
  

  

  