﻿4 
  Lord 
  Rayleigh 
  on 
  the 
  Instantaneous 
  Propagation 
  

  

  which 
  vanishes 
  when 
  ^ 
  = 
  0, 
  and 
  

  

  

  TT^ 
  = 
  1 
  d^ 
  COS 
  ci 
  COS 
  kx 
  ak, 
  

  

  dt' 
  ttJo 
  

  

  so 
  that 
  for 
  i 
  = 
  

  

  de 
  

  

  1 
  r^ 
  

  

  - 
  - 
  I 
  cr' 
  cos 
  k.v 
  dk. 
  .... 
  (8) 
  

  

  For 
  waves 
  on 
  water 
  of 
  depth 
  h 
  

  

  a^ 
  = 
  gk 
  tanh 
  /c/i, 
  . 
  . 
  . 
  . 
  (9) 
  

  

  passing 
  when 
  k 
  is 
  great 
  (\ 
  small) 
  into 
  the 
  simple 
  deep 
  water 
  

   formula 
  a^ 
  — 
  gk, 
  and 
  when 
  k 
  is 
  small 
  (\ 
  great) 
  into 
  the 
  

   shallow 
  water 
  formula 
  cr^ 
  — 
  gh 
  ¥^ 
  indicating 
  a 
  uniform 
  

   velocity 
  of 
  propagation 
  *. 
  

   When 
  a 
  < 
  IT, 
  we 
  have 
  t 
  

  

  J" 
  

  

  ^ax^^-ax 
  ^ 
  ^ 
  (^^c 
  — 
  e-i*^) 
  sin 
  ^a 
  

  

  enx-j^e"^ 
  ^^'' 
  '^^^' 
  '^'^ 
  ~ 
  ^^ 
  + 
  e- 
  + 
  2cosa' 
  

  

  J' 
  

  

  J' 
  

  

  If 
  we 
  differentiate 
  this 
  with 
  respect 
  to 
  c 
  and 
  then 
  make 
  a 
  = 
  7r, 
  

   to 
  which 
  there 
  seems 
  to 
  be 
  no 
  objection 
  on 
  the 
  understanding 
  

   postulated, 
  

  

  Adapting 
  (10) 
  to 
  our 
  present 
  purpose, 
  we 
  have 
  

  

  gj^2 
  ^nx/2h 
  J^ 
  g- 
  Jrx/2h 
  

  

  a-2 
  cos 
  k^ 
  dk 
  = 
  ~ 
  jj-^^ 
  ^xi2k_^.i 
  my 
  ' 
  • 
  (11) 
  

  

  whence 
  d^rj/dt^ 
  is 
  given 
  by 
  (8). 
  The 
  first 
  term 
  in 
  the 
  

   expression 
  lor 
  rj 
  is 
  acccordingly 
  

  

  _ 
  girt^ 
  cosh(^'/2A) 
  

   ^ 
  *" 
  8/i2 
  sinh2(7rer/'2/0 
  ^ 
  ^ 
  

  

  We 
  learn 
  from 
  (12) 
  that, 
  no 
  more 
  than 
  in 
  the 
  case 
  of 
  deep 
  

   water, 
  is 
  there 
  any 
  delay 
  in 
  the 
  commencement 
  of 
  disturb- 
  

   ance 
  at 
  a 
  finite 
  distance 
  x 
  from 
  the 
  source, 
  and 
  the 
  extension 
  

   is 
  not 
  without 
  importance, 
  seeing 
  that 
  in 
  truth 
  water 
  cannot 
  

  

  * 
  It 
  may 
  be 
  remarked 
  that 
  a 
  modified 
  formula, 
  viz. 
  

  

  agreeing 
  with 
  (9) 
  in 
  the 
  extreme 
  cases, 
  would 
  be 
  more 
  amenable 
  to 
  

   calculation. 
  

  

  t 
  De 
  Morgan's 
  ' 
  Differential 
  Calculus,' 
  p. 
  669. 
  

  

  