﻿728 
  Flexural 
  Vibrations 
  of 
  Thin 
  Rods. 
  

  

  directly 
  as 
  the 
  square 
  root 
  of 
  the 
  wave-length 
  and 
  inversely 
  

   as 
  the 
  wave-length 
  respectively, 
  for 
  an 
  infinite 
  succession 
  of 
  

   regular 
  sinusoidal 
  waves. 
  It 
  is 
  interesting 
  to 
  observe 
  that 
  

   in 
  both 
  cases 
  the 
  wave-disturbance 
  is 
  ultimately 
  spread 
  

   throughout 
  the 
  entire 
  medium, 
  but 
  that 
  in 
  the 
  case 
  of 
  water- 
  

   waves 
  the 
  wave-length 
  of 
  the 
  disturbance 
  at 
  any 
  time 
  increases 
  

   continuously, 
  and 
  in 
  the 
  case 
  of 
  flexural-waves 
  it 
  diminishes 
  

   continuously, 
  as 
  we 
  pass 
  outwards 
  from 
  the 
  middle 
  point 
  of 
  

   the 
  initial 
  disturbance. 
  In 
  the 
  case 
  of 
  water-waA'es 
  also. 
  

   Lord 
  Kelvin's 
  investigations 
  show 
  that 
  each 
  individual 
  wave 
  

   lengthens 
  and 
  increases 
  in 
  speed 
  as 
  it 
  advances, 
  while 
  in 
  the 
  

   case 
  of 
  flexural-waves 
  each 
  wave 
  lengthens 
  and 
  diminishes 
  in 
  

   speed, 
  as 
  we 
  shall 
  see 
  by 
  equations 
  (15) 
  and 
  (16) 
  below 
  for 
  

   the 
  solution 
  we 
  have 
  chosen 
  for 
  illustration. 
  

  

  § 
  10. 
  At 
  a 
  moderate 
  time 
  after 
  the 
  motion 
  has 
  commenced 
  

  

  T 
  may 
  be 
  put 
  equal 
  to 
  — 
  in 
  equation 
  (12), 
  and 
  the 
  argument 
  

  

  xH 
  

   of 
  the 
  cosine 
  varies 
  then 
  only 
  with 
  j-ttTs' 
  ^^ 
  *^^^ 
  ^* 
  ^^ 
  ^^^^ 
  ^^^ 
  

  

  us 
  to 
  trace 
  the 
  outward 
  progress 
  of 
  any 
  particular 
  maximum 
  

   or 
  zero 
  of 
  displacement 
  along 
  the 
  bar. 
  Thus 
  the 
  position 
  of 
  

   any 
  zero 
  is 
  determined 
  by 
  an 
  equation 
  of 
  the 
  form 
  

  

  ^^^ 
  4n-f-3 
  

  

  j^.=c 
  = 
  -^-7r, 
  .... 
  (14) 
  

  

  and 
  the 
  velocity 
  of 
  the 
  zero 
  is 
  given 
  by 
  

  

  dx 
  _ 
  ^Khct—f 
  . 
  2Kbc_ 
  

  

  dt 
  " 
  2ur 
  ^ 
  X 
  ^^^^ 
  

  

  When 
  t 
  is 
  large, 
  so 
  that 
  we 
  may 
  put 
  rn2 
  = 
  -, 
  equation 
  (14) 
  

   enables 
  us 
  to 
  write 
  (15) 
  in 
  the 
  following 
  approximate 
  form, 
  

  

  dx 
  , 
  X 
  , 
  

  

  dt-It 
  (15') 
  

  

  It 
  can 
  easily 
  be 
  verified 
  that 
  when 
  t 
  is 
  very 
  small 
  and 
  a; 
  

   great 
  the 
  velocity 
  of 
  any 
  zero 
  is 
  given 
  by 
  

  

  S=^-l 
  (1*^) 
  

  

  In 
  each 
  case, 
  what 
  we 
  obtain 
  in 
  these 
  equations 
  is 
  also 
  the 
  

   wave-velocity 
  for 
  an 
  infinite 
  train 
  of 
  waves 
  of 
  wave-length 
  

   equal 
  to 
  that 
  maintaining 
  in 
  the 
  immediate 
  neighbourhood 
  

   of 
  the 
  point 
  x 
  at 
  time 
  t. 
  

  

  