﻿Vibratio7is 
  of 
  Thin 
  Rods, 
  725 
  

  

  § 
  5. 
  Solutions 
  for 
  vibrations 
  of 
  a 
  rod 
  of 
  finite 
  length 
  I 
  are 
  

   derived 
  directly 
  from 
  the 
  hydrodynamic 
  solutions 
  for 
  waves 
  

   in 
  a 
  canal 
  of 
  lenoth 
  / 
  by 
  simple 
  interchange 
  of 
  <r 
  and 
  t 
  ; 
  and 
  

   they 
  are 
  applicable 
  to 
  all 
  cases, 
  whether 
  the 
  rod 
  has 
  its 
  ends 
  

   free, 
  or 
  clamped, 
  or 
  " 
  supported/^ 
  In 
  particular, 
  the 
  normal 
  

   functions 
  are 
  easily 
  obtained 
  in 
  this 
  way. 
  

  

  In 
  the 
  case 
  of 
  an 
  infinite 
  rod, 
  all 
  mathematical 
  results 
  

   relating 
  to 
  surface 
  waves 
  in 
  a 
  canal 
  infinitely 
  long 
  and 
  in- 
  

   finitely 
  deep 
  become 
  immediately 
  useful 
  ; 
  space-curves 
  in 
  the 
  

   hydrodynamical 
  waves 
  problems 
  becoming 
  time-curves 
  for 
  

   the 
  flexural 
  waves, 
  and 
  vice 
  versa. 
  

  

  § 
  6. 
  In 
  this 
  connexion 
  it 
  may 
  be 
  useful 
  at 
  a 
  later 
  time 
  to 
  

   -examine 
  some 
  of 
  the 
  numerous 
  hydrodynamical 
  solutions 
  

   relating 
  to 
  surface 
  waves 
  and 
  groups 
  of 
  waves. 
  A 
  number 
  

   of 
  curves 
  are 
  shown 
  in 
  papers 
  on 
  Water- 
  Waves* 
  by 
  the 
  late 
  

   Lord 
  Kelvin, 
  illustrating 
  results 
  derived 
  from 
  particular 
  

   hydrodynamic 
  solutions 
  comprehended 
  in 
  the 
  following 
  

   general 
  expression, 
  given 
  in 
  his 
  last 
  Waves 
  paper 
  : 
  — 
  

  

  {RSI 
  or 
  {RD}^-^^^, 
  -7^ 
  

  

  ix) 
  

  

  g 
  ^z+ix) 
  . 
  

  

  In 
  this, 
  {RS} 
  denotes 
  a 
  realization 
  by 
  taking 
  half 
  the 
  sum 
  

   of 
  what 
  follows 
  it 
  with 
  +i; 
  {RDj- 
  denotes 
  a 
  realization 
  by 
  

   taking 
  the 
  difi'erence 
  of 
  what 
  follows 
  it 
  with 
  + 
  i 
  divided 
  by 
  

   '2i. 
  As 
  an 
  example 
  of 
  fiexural 
  waves 
  in 
  an 
  infinite 
  elastic 
  

   rod, 
  arising 
  from 
  a 
  given 
  initial 
  displacement, 
  we 
  may 
  take 
  

   the 
  solution 
  

  

  = 
  A 
  /-cosT-^— 
  ^rV^^^^l. 
  . 
  . 
  (12) 
  

  

  where 
  ll^^{z'' 
  + 
  t% 
  and 
  t= 
  tan-^^-^^") 
  j 
  

  

  In 
  what 
  follows, 
  z 
  is 
  taken 
  as 
  1 
  and 
  kJj 
  as 
  \ 
  in 
  order 
  to 
  allow 
  

   us 
  to 
  use 
  Lord 
  Kelvin's 
  hydrodynamical 
  results 
  in 
  our 
  present 
  

   problem. 
  

  

  § 
  7. 
  Taking 
  the 
  origin 
  of 
  coordinates 
  at 
  the 
  middle 
  of 
  the 
  

   bar, 
  the 
  initial 
  configuration 
  is 
  given 
  by 
  

  

  y 
  = 
  e 
  4»c6^ 
  (13) 
  

  

  Almost 
  immediately 
  after 
  the 
  commencement 
  of 
  motion 
  an 
  

  

  * 
  Proc. 
  Roy. 
  Soc. 
  Edin. 
  vol. 
  xxv. 
  Feb. 
  and 
  June 
  1904; 
  vol. 
  xxvi, 
  

   Oct. 
  1906. 
  

  

  