﻿Vibrations 
  of 
  Thin 
  Rods. 
  727 
  

  

  distant 
  from 
  its 
  middle 
  point 
  the 
  disturbance 
  consists 
  of 
  a 
  

   larger 
  and 
  larger 
  number 
  of 
  waves 
  which 
  travel 
  inwards 
  

   past 
  the 
  point 
  considered, 
  before 
  ^=1, 
  and 
  after 
  that 
  recross 
  

   the 
  point 
  in 
  the 
  opposite 
  direction 
  one 
  by 
  one. 
  When 
  the 
  

   first 
  zero 
  recrosses 
  the 
  point 
  it 
  subsides 
  gradually 
  to 
  its 
  

   original 
  place 
  of 
  rest, 
  only 
  reaching 
  it, 
  however, 
  after 
  an 
  

   infinite 
  time. 
  The 
  successive 
  maxima 
  of 
  displacement 
  in- 
  

   crease 
  very 
  slowly 
  at 
  first, 
  then 
  more 
  quickly^ 
  and 
  then 
  

  

  diminish 
  finally 
  according 
  to 
  -i 
  as 
  already 
  stated. 
  

  

  § 
  9. 
  The 
  diagrams 
  of 
  figs. 
  1 
  and 
  2 
  are 
  taken 
  from 
  Lord 
  

  

  Fig. 
  2. 
  

  

  •2 
  

  

  Abscissas 
  represent 
  time 
  for 
  water-waves, 
  space 
  for 
  flexural-waves. 
  

  

  Kelvin^s 
  paper 
  referred 
  to 
  in 
  § 
  8, 
  and 
  they 
  are 
  reproduced 
  

   without 
  change 
  of 
  the 
  lettering 
  applicable 
  to 
  them 
  as 
  water- 
  

   wave 
  diagrams. 
  To 
  make 
  them 
  correspond 
  exactly 
  to 
  the 
  

   flexural-waves 
  problem 
  solved 
  by 
  equation 
  (12), 
  we 
  must 
  

   reduce 
  the 
  ordinates 
  in 
  both 
  figures 
  in 
  the 
  ratio 
  ^2 
  : 
  1 
  ; 
  

   then 
  in 
  fig. 
  1 
  replace 
  t 
  by 
  cV 
  on 
  each 
  of 
  the 
  seven 
  curves, 
  and 
  

   take 
  ordinates 
  as 
  representing 
  displacement 
  and 
  abscissas 
  as 
  

   representing 
  time. 
  As 
  a 
  water-waves 
  diagram 
  fig. 
  2 
  repre- 
  

   sents 
  the 
  vertical 
  displacement 
  of 
  the 
  water 
  at 
  point 
  .r= 
  2, 
  

   from 
  ^=eO 
  to 
  ^ 
  = 
  cc 
  ; 
  as 
  a 
  flexural-waves 
  diagram 
  it 
  represents 
  

   the 
  shape 
  of 
  the 
  right-hand 
  half 
  of 
  the 
  rod, 
  ,^' 
  = 
  to 
  x 
  = 
  co, 
  

   at 
  time 
  t 
  = 
  2, 
  corresponding 
  to 
  the 
  initial 
  configuration 
  given 
  

   by 
  equation 
  (13). 
  

  

  These 
  curves 
  are 
  useful 
  chiefly 
  as 
  illustrations 
  of 
  the 
  pro- 
  

   pagation 
  of 
  waves 
  in 
  dispersive 
  media 
  from 
  a 
  given 
  initial 
  

   disturbance 
  confined 
  in 
  the 
  main 
  to 
  the 
  neighbourhood 
  of 
  the 
  

   origin, 
  They 
  show 
  clearly 
  the 
  distinctive 
  features 
  of 
  wave- 
  

   propagation 
  in 
  the 
  two 
  cases 
  where 
  the 
  wave-velocity 
  varies 
  

  

  