﻿Interpretation 
  of 
  the 
  Bessel 
  Function 
  of 
  Zero 
  Order. 
  431 
  

  

  If 
  the 
  length 
  of 
  the 
  wire 
  is 
  infinite 
  (iii.) 
  and 
  (iv.) 
  will 
  be 
  

   non-existent, 
  but 
  it 
  is 
  important 
  to 
  note 
  that 
  the 
  resultant 
  

   waves 
  will 
  still 
  contain 
  a 
  term 
  involving 
  reflected 
  waves, 
  and 
  

   arising 
  from 
  (ii.). 
  This 
  is 
  the 
  important 
  point 
  in 
  which 
  the 
  

   waves 
  propagated 
  along 
  such 
  a 
  wire 
  differ 
  from 
  those 
  along 
  

   a 
  uniform 
  wire. 
  

  

  It 
  will 
  now 
  be 
  shown 
  that 
  proceeding 
  on 
  this 
  assumption, 
  

   by 
  the 
  use 
  of 
  the 
  ordinary 
  methods 
  for 
  obtaining 
  the 
  re- 
  

   fraction 
  and 
  reflexion 
  of 
  simple 
  sine 
  waves, 
  an 
  equation 
  of 
  

   the 
  form 
  of 
  VIII. 
  will 
  again 
  be 
  reached, 
  showing 
  that 
  we 
  

   have 
  obtained 
  a 
  correct 
  representation 
  (within 
  the 
  limits 
  

   of 
  the 
  theory) 
  of 
  what 
  is 
  actually 
  occurring 
  in 
  the 
  wire 
  at 
  

   any 
  instant. 
  

  

  Consider 
  a 
  sine 
  wave 
  of 
  voltage 
  travelling 
  along 
  the 
  axis 
  

   of 
  x 
  in 
  the 
  positive 
  direction. 
  

  

  It 
  will 
  be 
  of 
  the 
  form 
  

  

  y 
  = 
  A 
  e 
  i(pt-ax) 
  ( 
  rea 
  i 
  p 
  ar 
  t). 
  

  

  At 
  the 
  point 
  jj 
  = 
  X 
  let 
  L, 
  K, 
  R, 
  G 
  change 
  to 
  L', 
  K', 
  R', 
  G'. 
  

   The 
  refracted 
  wave 
  will 
  be 
  of 
  the 
  form 
  

  

  Y=A 
  1 
  e 
  i 
  ^ 
  t 
  ~ 
  a 
  ' 
  x 
  ^ 
  (real 
  part), 
  

  

  and 
  reflected 
  wave 
  will 
  be 
  

  

  y 
  = 
  A> 
  e 
  i( 
  P 
  t+ax) 
  ( 
  rea 
  j 
  p 
  ar 
  t). 
  

  

  At 
  the 
  surface 
  of 
  separation 
  both 
  current 
  and 
  voltage 
  must 
  

   be 
  continuous 
  for 
  all 
  values 
  of 
  the 
  time 
  : 
  

  

  Ae- 
  iaX 
  + 
  A 
  / 
  e 
  iaX 
  =A 
  1 
  e~ 
  ia 
  ' 
  K 
  , 
  (1) 
  

  

  for 
  continuity 
  of 
  voltage. 
  

  

  Again 
  equations 
  VI., 
  p. 
  429, 
  must 
  also 
  be 
  satisfied, 
  

  

  do 
  

   .'. 
  — 
  j- 
  = 
  e^{Kip 
  + 
  G} 
  {Ae~ 
  icu: 
  + 
  A'e 
  iax 
  }, 
  

  

  c 
  = 
  — 
  ]Kip 
  + 
  G}{Ae- 
  iax 
  -A'e 
  iax 
  \ 
  ; 
  

  

  .*. 
  for 
  continuity 
  of 
  current 
  

  

  Kje+G 
  / 
  Ae 
  _ 
  t 
  ,x_ 
  AV 
  «x) 
  = 
  K^+G' 
  A 
  « 
  ( 
  (2) 
  

  

  a 
  I 
  J 
  a' 
  

  

  Also 
  

  

  (Kip 
  + 
  G) 
  (Lip 
  + 
  R) 
  __ 
  (K'ip 
  + 
  G') 
  (L'ip 
  + 
  B!) 
  _ 
  _ 
  1 
  m 
  

   a 
  2 
  a' 
  2 
  " 
  ' 
  l 
  ; 
  

  

  Hence, 
  in 
  general, 
  we 
  have 
  equations 
  (1), 
  (2), 
  and 
  (3) 
  to 
  

   determine 
  A', 
  A 
  1? 
  and 
  a. 
  

  

  2 
  G2 
  

  

  