﻿Interpretation 
  of 
  the 
  Bessel 
  Function 
  of 
  Zero 
  Order, 
  433 
  

  

  consequently 
  gives 
  the 
  change 
  of 
  phase 
  as 
  well 
  as 
  the 
  ampli- 
  

   tude. 
  Hence 
  for 
  a 
  series 
  of 
  waves 
  the 
  sum 
  of 
  the 
  A's 
  will 
  

   give 
  the 
  true 
  amplitude 
  and 
  phase 
  position 
  of 
  the 
  resultant 
  

   wave. 
  

  

  Now 
  to 
  consider 
  the 
  reflected 
  wave, 
  i. 
  e. 
  the 
  main 
  wave 
  

   travelling 
  towards 
  the 
  origin. 
  

  

  In 
  the 
  portion 
  of 
  the 
  wire 
  beyond 
  the 
  change 
  the 
  incident 
  

   wave 
  is 
  of 
  the 
  form 
  

  

  J$ 
  e 
  Hpt+ax) 
  

  

  the 
  reflected 
  wave 
  of 
  the 
  form 
  

  

  and 
  the 
  refracted 
  wave 
  of 
  form 
  

  

  Constructing 
  the 
  equations 
  corresponding 
  to 
  (4) 
  and 
  (5) 
  

   we 
  have 
  

  

  Be 
  iaX 
  + 
  B'e- 
  iaX 
  = 
  B 
  1 
  e 
  iaX 
  , 
  (8) 
  

  

  Be 
  iaX 
  -B'e~ 
  iaX 
  = 
  B^ 
  x 
  

  

  X 
  + 
  dX 
  

  

  Solving 
  for 
  B' 
  and 
  B 
  x 
  in 
  terms 
  of 
  B, 
  we 
  get 
  

  

  dX 
  

  

  - 
  • 
  (9) 
  

  

  B' 
  = 
  Be 
  2iaX 
  

  

  2X 
  

  

  (10) 
  

  

  B 
  1 
  = 
  B(l+g) 
  (11) 
  

  

  Now 
  combine 
  the 
  results 
  of 
  equations 
  (6), 
  (7), 
  (10), 
  (11), 
  

   tabulating 
  for 
  clearness 
  as 
  follows. 
  

  

  

  X 
  - 
  

  

  = 
  X 
  

  

  

  Portion 
  of 
  wire 
  corresponding 
  1 
  

  

  to 
  

  

  Portion 
  of 
  wire 
  corresponding 
  

  

  to 
  

  

  K 
  = 
  KX. 
  

  

  

  K=K(X+riX). 
  

  

  

  Original 
  incident 
  wave 
  A 
  

   Reflected 
  wave 
  -Ae~ 
  2iaX< 
  ^ 
  

  

  -> 
  

   <- 
  

  

  Refracted 
  wave 
  A 
  [1— 
  -y^\ 
  

   Incident 
  wave 
  B 
  

  

  <- 
  

  

  Refracted 
  wave 
  B 
  /i_j-^M 
  

  

  <- 
  

  

  Reflected 
  wave 
  Be 
  2iaX 
  ^ 
  

  

  "> 
  

  

  giving 
  

   A 
  -> 
  

  

  

  giving 
  

   <- 
  B 
  

  

  

  B4-- 
  d 
  *-Ae- 
  2laxdX 
  

   ^ 
  2X 
  e 
  2X 
  

  

  4- 
  

  

  A.(l-g) 
  + 
  B^g-^-A' 
  

  

  say. 
  

  

  