﻿62 
  L. 
  DeForest 
  — 
  Reflection 
  of 
  Hertzian 
  Waves. 
  

  

  8^ 
  

  

  IT 
  

  

  [.LS-1]=0 
  

  

  1 
  1 
  

  

  whence 
  v 
  •■= 
  /^-t^ 
  or 
  Lv 
  r= 
  -— 
  , 
  the 
  well-known 
  Thomson 
  form- 
  

  

  ula 
  for 
  the 
  velocity 
  of 
  the 
  propagation 
  of 
  the 
  wave. 
  

  

  Now 
  the 
  capacity 
  per 
  unit 
  length 
  of 
  the 
  normal 
  part 
  of 
  tlie 
  

   Lecher 
  parallel 
  wires 
  is 
  given"^ 
  very 
  closely 
  by 
  the 
  form 
  

  

  a 
  

   4 
  Log 
  nat 
  — 
  

  

  in 
  electro-static 
  units, 
  where 
  r 
  = 
  radius 
  of 
  the 
  wires, 
  d 
  their 
  

   distance 
  apart. 
  

  

  In 
  the 
  wires 
  used 
  r=-37^"° 
  ; 
  6?=l-7'°'. 
  Also 
  L=4 
  Los: 
  nat 
  — 
  

  

  in 
  electro-magnetic 
  units. 
  Thus 
  the 
  period 
  of 
  Lecher's 
  sys- 
  

   tem, 
  as 
  well 
  as 
  for 
  circular 
  or 
  rectangular 
  resonators, 
  is 
  inde- 
  

   pendent 
  of 
  r 
  or 
  d. 
  And 
  if 
  the 
  resistance 
  is 
  negligible 
  com- 
  

   pared 
  to 
  2?;L 
  (which 
  is 
  the 
  case 
  for 
  wires 
  greater 
  than 
  0"5""^ 
  

   diameter 
  or 
  thereabouts, 
  and 
  less 
  than 
  100°" 
  long), 
  the 
  velocit}^ 
  

   of 
  propagation 
  thereon 
  is 
  that 
  of 
  light. 
  Especially 
  at 
  the 
  

   point 
  of 
  reflection 
  the 
  resistance 
  R 
  may 
  be 
  neglected. 
  Equa- 
  

   tion 
  (I) 
  becomes 
  

  

  V 
  -^ 
  — 
  \- 
  -f- 
  and. 
  Its 
  solution 
  : 
  

  

  ay 
  at 
  

  

  n 
  1 
  

  

  where 
  ^r- 
  = 
  v 
  = 
  ^Vf^ 
  and 
  f 
  ; 
  /' 
  denote 
  any 
  functions 
  (in 
  our 
  

  

  ft 
  V 
  LS 
  "^ 
  1 
  ' 
  -^ 
  2 
  J 
  \ 
  

  

  case 
  sine 
  functions) 
  with 
  coefficients 
  for 
  the 
  various 
  ampli- 
  

   tudes, 
  and 
  a 
  third 
  term 
  in 
  the 
  parentheses 
  to 
  allow 
  for 
  change 
  

   of 
  phase. 
  So 
  for 
  the 
  original 
  and 
  reflected 
  waves, 
  where 
  x 
  = 
  

   write 
  : 
  

  

  V^ 
  = 
  <f>sm(^nt-^-^) 
  (2) 
  

  

  V,= 
  <^'sin(.^^-f 
  A.^ 
  

   where 
  8^ 
  is 
  the 
  angle 
  of 
  change 
  of 
  phase 
  in 
  reflection. 
  

  

  *e. 
  g. 
  Lord 
  Rayleigh, 
  Phil. 
  Mag., 
  August, 
  1897. 
  

  

  