﻿of 
  Disturbance 
  in 
  a 
  Dispersive 
  Medium, 
  5 
  

  

  be 
  very 
  deep 
  in 
  relation 
  to 
  all 
  the 
  wave-lengths 
  concerned, 
  

   since 
  among 
  these 
  infinite 
  wave-lengths 
  are 
  included. 
  In 
  

   the 
  present 
  case 
  all 
  the 
  wave-velocities 
  of 
  simple 
  trains 
  are 
  

   finite, 
  and 
  thus 
  the 
  sudden 
  propagation 
  to 
  a 
  distance 
  is 
  

   independent 
  of 
  an 
  interpretation 
  earlier 
  suggested. 
  

   If 
  in 
  (12) 
  li 
  is 
  small 
  compared 
  with 
  a', 
  we 
  may 
  write 
  

  

  - 
  p 
  ■nx,2h 
  

  

  (13) 
  

  

  showing 
  that 
  when 
  x 
  is 
  relatively 
  great 
  the 
  elevation 
  in 
  its 
  

   earlier 
  stages, 
  though 
  finite, 
  is 
  on 
  a 
  greatly 
  diminished 
  

   scale. 
  

  

  As 
  regards 
  the 
  general 
  question, 
  I 
  do 
  not 
  think 
  that 
  the 
  

   instantaneous 
  propagation 
  of 
  disturbance 
  should 
  be 
  considered 
  

   paradoxical. 
  It 
  is 
  to 
  be 
  remembered 
  that 
  in 
  (9) 
  the 
  water 
  

   is 
  treated 
  as 
  incompressible^ 
  i. 
  e. 
  that 
  the 
  velocity 
  of 
  pro- 
  

   pagation 
  of 
  waves 
  of 
  expansion 
  is 
  regarded 
  as 
  infinite. 
  

  

  A 
  more 
  complete 
  treatment 
  of 
  the 
  case 
  of 
  finite 
  depth 
  on 
  

   the 
  basis 
  of 
  (2) 
  and 
  (9) 
  would 
  be 
  instructive, 
  even 
  if 
  limited 
  

   to 
  selected 
  values 
  of 
  the 
  ratio 
  x 
  : 
  li. 
  So 
  far 
  we 
  have 
  con- 
  

   sidered 
  only 
  the 
  first 
  stage. 
  When 
  h 
  is 
  small 
  compared 
  with 
  

   »r, 
  an 
  important 
  part 
  of 
  the 
  disturbance 
  arrives 
  under 
  the 
  

   law 
  applicable 
  to 
  shallow 
  water, 
  viz. 
  

  

  a^^{gh).k{l-llrlr). 
  . 
  . 
  .. 
  (U) 
  

  

  Writhig 
  (2) 
  in 
  the 
  form 
  

  

  1 
  C^ 
  if" 
  

  

  V 
  = 
  ^\ 
  cos 
  {at 
  — 
  kx)dk 
  + 
  -^~\ 
  cos(a't 
  + 
  kr) 
  dk, 
  (15) 
  

  

  we 
  see 
  that 
  the 
  first 
  integral 
  acquires 
  a 
  specially 
  enhanced 
  

   value 
  when 
  x 
  and 
  t 
  are 
  so 
  related 
  that 
  crt 
  — 
  kx 
  is 
  approximately 
  

   zero 
  for 
  small 
  values 
  of 
  k. 
  The 
  condition 
  is 
  of 
  course 
  

  

  t 
  = 
  ^'/^(gh), 
  (16) 
  

  

  and 
  the 
  important 
  part 
  of 
  the 
  integral 
  will 
  be 
  

  

  " 
  " 
  if 
  '"' 
  ^^ 
  '''''^'^'^^' 
  = 
  ^ 
  "^"^ 
  f 
  • 
  ^ 
  ^^^ 
  • 
  (x)'*' 
  <^^) 
  

  

  large 
  when 
  h 
  is 
  relatively 
  small, 
  but 
  still 
  finite. 
  

  

  For 
  larorer 
  values 
  of 
  t 
  the 
  most 
  important 
  part 
  of 
  the 
  first 
  

   integral 
  of 
  (15) 
  occurs 
  in 
  the 
  neighbourhood 
  of 
  such 
  values 
  

   of 
  k 
  as 
  make 
  ort 
  — 
  hv 
  stationary. 
  This 
  happens 
  when 
  k 
  has 
  a 
  

   value 
  X'o 
  such 
  that 
  

  

  jc-tdir/dko^O 
  (18) 
  

  

  