﻿362 
  On 
  Wave-motion 
  along 
  a 
  Periodically 
  Loaded 
  Swing. 
  

  

  have 
  | 
  A 
  | 
  = 
  | 
  B 
  |, 
  i.e. 
  total 
  reflexion, 
  and 
  it 
  is 
  clear 
  from 
  (xvi.) 
  

   that 
  there 
  is 
  reversal 
  of 
  phase. 
  For 
  the 
  upper 
  limit 
  of 
  such 
  

   a 
  region 
  of 
  frequency, 
  yjr 
  = 
  s7r 
  and 
  C 
  = 
  (xvii.). 
  

  

  7. 
  Region 
  for 
  which 
  s*<l. 
  

   Here 
  we 
  shall 
  have 
  ,3 
  = 
  0. 
  

  

  A 
  2 
  

  

  sin' 
  

  

  2 
  

  

  But 
  

  

  sin 
  2 
  -*^ 
  

   cos 
  a 
  = 
  cos 
  "^ 
  

  

  B 
  2 
  

  

  (J 
  2 
  

  

  snr 
  

  

  

  2 
  sin 
  3- 
  «»« 
  

  

  sin 
  

  

  A 
  2 
  

   C 
  2 
  

  

  _ 
  /*V 
  

  

  4 
  sin 
  5 
  

  

  a-\-"^r 
  

  

  2 
  

  

  (j 
  2 
  

  

  sin 
  2 
  yjr 
  

   /jLyjr 
  sin 
  >|r, 
  

  

  = 
  fMyjr 
  sin 
  "»Jr, 
  

   4 
  sin 
  2 
  — 
  =r*- 
  

  

  (xviii.) 
  

  

  Here 
  equation 
  (xvi.) 
  shows 
  that 
  the 
  incident, 
  transmitted, 
  

   and 
  reflected 
  waves 
  will 
  be 
  in 
  different 
  phases. 
  

  

  For 
  the 
  lower 
  limit 
  of 
  such 
  a 
  region 
  both 
  yfr 
  and 
  a, 
  are 
  

   multiples 
  of 
  it 
  ; 
  further, 
  they 
  will 
  be 
  even 
  or 
  odd 
  together. 
  

  

  It 
  therefore 
  appears 
  that 
  sin 
  — 
  ^-and 
  sin 
  — 
  -^- 
  are 
  zero, 
  and 
  

  

  S7TV 
  

  

  C 
  = 
  0. 
  But 
  for 
  these 
  points 
  n= 
  —— 
  . 
  If 
  then 
  the 
  incident 
  

  

  L 
  

   wave 
  is 
  of 
  frequency 
  corresponding 
  to 
  any 
  one 
  of 
  the 
  natural 
  

   nodes 
  of 
  the 
  intervals 
  of 
  string, 
  the 
  masses 
  will 
  be 
  entirely 
  

   undisturbed. 
  

  

  The 
  case 
  of 
  ty 
  small 
  is 
  seen 
  to 
  be 
  exceptional 
  : 
  we 
  easily 
  

   find 
  that 
  

  

  A 
  2 
  

   C 
  2 
  

  

  F 
  

  

  {1- 
  y/l+ri 
  2 
  ' 
  

  

  B 
  2 
  

  

  C 
  2 
  

  

  F 
  

  

  U+ 
  Vi+M 
  2 
  * 
  

  

  8. 
  It 
  may 
  perhaps 
  be 
  allowed 
  that 
  the 
  phenomena 
  here 
  dis- 
  

   cussed 
  have 
  some 
  mathematical 
  analogy 
  with 
  an 
  ideal 
  case 
  in 
  

   optics. 
  We 
  may 
  think 
  of 
  the 
  incidence 
  of 
  light 
  from 
  the 
  free 
  

   sether 
  upon 
  a 
  solid 
  of 
  periodic 
  structure. 
  Without 
  pressing 
  

   the 
  analogy, 
  we 
  will 
  recapitulate 
  our 
  results 
  in 
  optical 
  phrase^ 
  

   ology 
  ; 
  this 
  presentation 
  will 
  have 
  the 
  advantage 
  of 
  brevity. 
  

  

  Light 
  being 
  incident 
  upon 
  a 
  periodic 
  distribution 
  of 
  mole- 
  

   cules, 
  the 
  light 
  is 
  analysed 
  by 
  a 
  spectroscope 
  after 
  transmission 
  

   through 
  a 
  considerable 
  thickness. 
  We 
  shall 
  find 
  narrow 
  bright 
  

   bands, 
  their 
  lower 
  edges 
  ranged 
  harmonically 
  ; 
  each 
  band 
  will 
  

  

  