﻿"^'^-^Aiiij. 
  

  

  THE 
  

   LONDON, 
  EDINBURGH, 
  and 
  DUBLIN 
  

  

  PHILOSOPHICAL 
  MAGAZINE 
  

  

  AND 
  

  

  JOURNAL 
  OF 
  SC^NCj!; 
  

  

  [SIXTH 
  SERIEsl'fr! 
  

  

  JULY 
  1909. 
  

  

  I. 
  On 
  the 
  Instantaneous 
  Propagation 
  of 
  Distim 
  

  

  Dispersive 
  Medium^ 
  e.vempUjied 
  hy 
  Waves 
  on 
  Water 
  deep 
  

   and 
  shallow. 
  Bj/ 
  Lord 
  Rayleigh, 
  O.J/., 
  F.R.S* 
  

  

  THE 
  solution, 
  first 
  obtained 
  by 
  r'auchy 
  and 
  Poisson, 
  for 
  

   the 
  propagation 
  in 
  one 
  dimension 
  over 
  deep 
  water 
  of 
  

   the 
  waves 
  which 
  issue 
  from 
  a 
  concentrated 
  initial 
  disturbance 
  

   presents 
  peculiar 
  features. 
  One 
  form 
  of 
  it 
  may 
  be 
  written 
  

  

  "^ 
  ~ 
  iTxX-Zx 
  '6 
  . 
  5 
  \2j 
  3.5.7 
  . 
  9 
  V2.i7 
  '" 
  j 
  ^^ 
  

  

  where 
  rj 
  denotes 
  the 
  elevation 
  of 
  the 
  surface 
  at 
  time 
  t 
  and 
  

   place 
  .1', 
  dr);dt 
  being 
  initially 
  zero 
  throughout 
  and 
  tj 
  being 
  

   initially 
  zero 
  except 
  at 
  .v 
  = 
  0. 
  g 
  denotes, 
  as 
  usual, 
  the 
  

   acceleration 
  of 
  gravity. 
  So 
  far 
  as 
  general 
  mechanical 
  theory 
  

   is 
  concerned, 
  it 
  might 
  be 
  expected 
  that 
  under 
  these 
  initial 
  

   conditions 
  the 
  elevation 
  would 
  at 
  first 
  vary 
  as 
  t^ 
  ; 
  but 
  that 
  

   the 
  effect 
  should 
  commence 
  without 
  delay 
  at 
  all 
  points 
  seems 
  

   at 
  first 
  to 
  conflict 
  with 
  ideas 
  derived 
  from 
  the 
  more 
  familiar 
  

   theories 
  of 
  sonorous 
  and 
  luminous 
  undulations, 
  according 
  to 
  

   which 
  a 
  finite 
  time 
  must 
  elapse 
  before 
  any 
  effect 
  reaches 
  

   places 
  finitely 
  distant 
  from 
  the 
  source. 
  

  

  A 
  plausible 
  explanation 
  of 
  the 
  discrepancy 
  may 
  be 
  found 
  

   in 
  the 
  reflexion 
  that 
  trains 
  of 
  simple 
  waves 
  move 
  in 
  deep 
  

   water 
  with 
  velocities 
  proportional 
  to 
  the 
  square 
  roots 
  of 
  the 
  

   wave-lengths 
  and 
  thus 
  capable 
  of 
  assuming 
  infinite 
  values, 
  

   and 
  that 
  in 
  accordance 
  with 
  Fourier's 
  theorem 
  the 
  initial 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

   Phil. 
  Maq. 
  S. 
  6. 
  Vol. 
  IK 
  No. 
  103. 
  -Lihi 
  1909. 
  B 
  

  

  