﻿of 
  Disturbance 
  in 
  a 
  Dispersive 
  Medium, 
  3 
  

  

  Hence 
  all 
  the 
  coefficients 
  in 
  the 
  expansion 
  (3) 
  of 
  -q 
  in 
  powers 
  

   of 
  t 
  vanish, 
  and 
  the 
  general 
  conclusion 
  that 
  the 
  elevation 
  

   commences 
  everywhere 
  without 
  delay 
  fails, 
  as 
  we 
  know 
  it 
  

   ouofht 
  to 
  do. 
  At 
  the 
  same 
  time 
  the 
  special 
  character 
  of 
  the 
  

   failing 
  case 
  becomes 
  apparent. 
  But 
  we 
  have 
  not 
  yet 
  

   examined 
  the 
  convergencv 
  of 
  the 
  series. 
  

  

  Apart 
  from 
  signs 
  and 
  from 
  the 
  cosine 
  factor 
  in 
  (4), 
  the 
  

   ratio 
  of 
  the 
  term 
  in 
  ^^"+2 
  to 
  that 
  in 
  ^-" 
  is 
  

  

  Y{2mn 
  + 
  2m-\-l) 
  

  

  {2n 
  + 
  l)(2n 
  + 
  2).v^-"' 
  Vijlma 
  + 
  1) 
  

  

  (6) 
  

  

  To 
  find 
  the 
  limit 
  of 
  (G) 
  as 
  n 
  becomes 
  infinite, 
  we 
  may 
  use 
  

   the 
  formula 
  

  

  The 
  second 
  fraction 
  in 
  (6) 
  has 
  thus 
  the 
  limiting 
  value 
  

   {2mY-"\n-\-l)-"\ 
  

   so 
  that 
  (G) 
  becomes 
  ultimately 
  

  

  ^^„(2m)n« 
  + 
  l)^"»-= 
  (7) 
  

  

  The 
  series 
  is 
  thus 
  certainly 
  convergent 
  if 
  m 
  be 
  less 
  than 
  

   unitv. 
  If 
  ?n 
  = 
  l, 
  (7) 
  reduces 
  to 
  t^jx'^, 
  and 
  convergence 
  is 
  not 
  

   assured 
  unless 
  ci' 
  > 
  i. 
  This 
  marks 
  the 
  moment 
  at 
  which 
  the 
  

   wave 
  propagated 
  with 
  velocity 
  1 
  reaches 
  the 
  point 
  x. 
  

  

  If 
  we 
  were 
  to 
  make 
  m 
  = 
  2^ 
  corresponding 
  to 
  the 
  pro- 
  

   pagation 
  of 
  flexural 
  disturbances 
  along 
  a 
  bar, 
  the 
  cosines 
  in 
  

   (4l\ 
  would 
  all 
  vanish, 
  but 
  the 
  lack 
  of 
  convergency 
  indicated 
  

   by 
  (7) 
  prohibits 
  the 
  conclusion 
  that 
  the 
  disturbance 
  under- 
  

   goes 
  a 
  finite 
  delay 
  in 
  reaching 
  the 
  point 
  x. 
  From 
  Fourier^s 
  

   special 
  solution 
  * 
  we 
  may 
  see 
  that 
  there 
  is 
  in 
  fact 
  no 
  such 
  

   delay. 
  

  

  It 
  would 
  be 
  of 
  great 
  interest 
  to 
  examine 
  the 
  influence 
  on 
  

   w^ater 
  waves 
  due 
  to 
  a 
  limitation 
  of 
  depth 
  (/i), 
  but 
  a 
  complete 
  

   solution 
  for 
  this 
  case 
  analogous 
  to 
  (1) 
  seems 
  hardly 
  

   practicable. 
  But 
  without 
  much 
  difficulty 
  we 
  may 
  obtain 
  

   the 
  first 
  term 
  in 
  the 
  series, 
  viz., 
  the 
  term 
  proportional 
  to 
  i-. 
  

  

  In 
  general 
  from 
  (2) 
  

  

  -, 
  = 
  I 
  (7 
  sm 
  at 
  cos 
  k-x 
  dk; 
  

  

  dt 
  ir\^ 
  

  

  * 
  'Theorv 
  of 
  Sound; 
  § 
  192. 
  

   ■ 
  B2 
  

  

  