﻿(67) 
  

  

  concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  Ill 
  

  

  where 
  k 
  = 
  27r/\ 
  and 
  B 
  is 
  some 
  multiplier, 
  which 
  may 
  be 
  

   complex. 
  The 
  time 
  factor 
  e 
  ikat 
  is 
  suppressed. 
  In 
  order 
  to 
  

   obtain 
  plane 
  waves 
  we 
  may 
  suppose 
  that 
  <l> 
  acts 
  uniformly 
  

   over 
  the 
  whole 
  slice 
  between 
  x 
  and 
  x-\-dx. 
  The 
  effect 
  may 
  

   be 
  calculated 
  as 
  in 
  a 
  well-known 
  optical 
  investigation. 
  If 
  

   p 
  2 
  =zr 
  2 
  —x 
  2 
  , 
  the 
  element 
  of 
  volume 
  is 
  2irpdpdx, 
  or 
  2irrdrdx 
  ; 
  

   and 
  for 
  the 
  plane 
  waves 
  

  

  (f> 
  = 
  BG>dx 
  Zire^dr^ 
  B 
  <S> 
  dm 
  * 
  . 
  . 
  

  

  Here 
  <l> 
  acts 
  at 
  x 
  = 
  0, 
  and 
  

  

  <p 
  = 
  -rr-<$dx, 
  <j> 
  =z'27raB<&dx. 
  . 
  . 
  (68) 
  

  

  Since 
  <f> 
  Q 
  must 
  be 
  in 
  the 
  same 
  phase 
  as 
  <J>, 
  it 
  follows 
  that 
  B 
  

   is 
  real. 
  

  

  We 
  have 
  now 
  to 
  consider 
  the 
  work 
  done 
  in 
  generating 
  the 
  

   plane 
  waves 
  per 
  unit 
  of 
  time 
  and 
  per 
  double 
  unit 
  area 
  of 
  

   wave-front. 
  For 
  this 
  we 
  have 
  

  

  ^-\ 
  3>9 
  ^=7raB(<^) 
  2 
  Mod 
  2 
  </>;. 
  . 
  . 
  (69) 
  

   t 
  Jo 
  

   or 
  since 
  by 
  (67) 
  

  

  Mod0= 
  27r 
  ^ 
  'Mod<S>, 
  .... 
  (70) 
  

  

  we 
  get 
  for 
  the 
  work 
  propagated 
  in 
  one 
  direction 
  per 
  unit 
  of 
  

   area 
  of 
  wave-front 
  

  

  &*** 
  (71) 
  

  

  Reverting 
  now 
  to 
  (66), 
  we 
  see 
  that 
  for 
  divergent 
  waves 
  

  

  Mod 
  2 
  </>=^ 
  2 
  Mod 
  2 
  (3>rfV), 
  

  

  or 
  

  

  terf 
  Mod 
  2 
  (/> 
  = 
  4ttB 
  2 
  Mod 
  2 
  (3> 
  dV) 
  . 
  

  

  Accordingly 
  by 
  (71), 
  since 
  at 
  a 
  sufficient 
  distance 
  the 
  dis- 
  

   tinction 
  between 
  plane 
  and 
  divergent 
  waves 
  disappears, 
  the 
  

   work 
  emitted 
  in 
  unit 
  time 
  by 
  a 
  point-source 
  <£> 
  dV 
  is 
  

  

  i& 
  2 
  aBMod 
  2 
  (<S>dV) 
  (72) 
  

  

  It 
  may 
  be 
  observed 
  that 
  in 
  order 
  to 
  preserve 
  a 
  better 
  corre- 
  

   spondence 
  between 
  " 
  force 
  " 
  and 
  " 
  coordinate 
  " 
  a 
  somewhat 
  

   different 
  interpretation 
  is 
  here 
  put 
  upon 
  <1> 
  from 
  that 
  adopted 
  

  

  