﻿726 
  

  

  Mr. 
  G. 
  Green 
  on 
  Flexural 
  

  

  infinite 
  number 
  of 
  waves 
  are 
  formed 
  along 
  the 
  bar, 
  with 
  

  

  amplitudes 
  diminishing 
  according 
  to 
  the 
  law 
  e 
  ^'^^'^^ 
  and 
  with 
  

   the 
  distances 
  from 
  zero 
  to 
  zero 
  becoming 
  shorter 
  and 
  shorter 
  

   as 
  we 
  pass 
  from 
  the 
  middle 
  toward 
  the 
  ends 
  of 
  the 
  bar. 
  The 
  

   zeros 
  come 
  into 
  existence 
  at 
  the 
  ends 
  of 
  the 
  bar, 
  and 
  begin 
  

   travelling 
  inwards 
  to 
  the 
  middle. 
  The 
  first 
  zero 
  formed 
  

   comes 
  almost 
  instantaneously 
  to 
  the 
  middle 
  region, 
  and 
  the 
  

   others 
  follow 
  it 
  in 
  their 
  order 
  of 
  formation. 
  The 
  inward 
  

   progress 
  of 
  the 
  zeros 
  soon 
  ceases, 
  the 
  first 
  zero 
  never 
  

   quite 
  reaching 
  the 
  centre; 
  in 
  a 
  short 
  time 
  they 
  begin 
  to 
  

   move 
  in 
  the 
  opposite 
  direction 
  and 
  continue 
  to 
  do 
  so 
  for 
  

   ever, 
  and 
  the 
  amplitudes 
  at 
  any 
  point 
  os 
  ultimately 
  fall 
  off 
  

  

  according 
  to 
  -i, 
  

  

  The 
  middle 
  point 
  of 
  the 
  rod 
  subsides 
  to 
  its 
  

  

  undisturbed 
  position 
  nonvibrationally, 
  while 
  the 
  distance 
  from 
  

   it 
  to 
  the 
  first 
  zero 
  on 
  either 
  side 
  continually 
  increases 
  after 
  a 
  

   certain 
  time, 
  being 
  given 
  by 
  the 
  equation 
  x^=-?)Kb'iTt. 
  

  

  § 
  8. 
  The 
  seven 
  curves 
  given 
  by 
  Lord 
  Kelvin, 
  as 
  space 
  

   curves 
  for 
  water-waves, 
  on 
  page 
  191 
  of 
  his 
  paper 
  (Proc. 
  

   Koy. 
  Soc. 
  Edin. 
  vol. 
  xxv. 
  Feb. 
  1901), 
  show 
  the 
  condition 
  of 
  

   things 
  in 
  our 
  present 
  problem 
  at 
  seven 
  different 
  points 
  near 
  

   the 
  middle 
  of 
  the 
  bar, 
  as 
  t 
  increases 
  from 
  to 
  cc 
  , 
  provided 
  

   the 
  curves 
  be 
  continued 
  to 
  meet 
  the 
  axis 
  of 
  t 
  asymptotically 
  

   at 
  infinity 
  (see 
  fig. 
  1 
  and 
  § 
  9 
  below). 
  These 
  curves 
  show 
  

  

  Fio-. 
  1. 
  

  

  AbscisBas 
  represent 
  space 
  for 
  water-waves, 
  time 
  for 
  flexural-waves, 
  

  

  that 
  points 
  very 
  near 
  the 
  origin 
  never 
  pass 
  through 
  their 
  

   initial 
  positions, 
  but 
  fall 
  back 
  nonvibrationally 
  to 
  them 
  ; 
  

   points 
  farther 
  from 
  the 
  middle 
  rise 
  slightly 
  and 
  then 
  behave 
  

   in 
  the 
  same 
  way. 
  At 
  points 
  along 
  the 
  bar 
  more 
  and 
  more 
  

  

  