﻿2 
  Lord 
  Rayleigh 
  on 
  the 
  Instantaneous 
  Propagation 
  

  

  disturbance 
  in 
  question 
  must 
  actually 
  be 
  regarded 
  as 
  in- 
  

   cluding 
  trains 
  of 
  infinite 
  wave-length 
  and 
  velocity. 
  But 
  in 
  

   the 
  sequel 
  we 
  shall 
  find 
  reason 
  for 
  questioning 
  the 
  sufficiency 
  

   o£ 
  this 
  explanation. 
  

  

  In 
  the 
  deep 
  water 
  solution 
  the 
  speed 
  a 
  is 
  proportional 
  \o 
  

   M, 
  where 
  k 
  = 
  2'jr/X 
  and 
  X 
  is 
  the 
  wave-length. 
  We 
  will 
  now 
  

   suppose 
  more 
  generally 
  that 
  in 
  a 
  dispersive 
  medium 
  (T=k"'. 
  

   In 
  the 
  general 
  Fourier 
  solution 
  

  

  1 
  r* 
  

  

  7] 
  =— 
  \ 
  cos 
  at 
  . 
  cos 
  Lc 
  . 
  dk 
  . 
  . 
  . 
  . 
  (2) 
  

  

  a 
  is 
  to 
  be 
  given 
  the 
  above 
  value, 
  and 
  for 
  the 
  sake 
  of 
  con- 
  

   vergency 
  we 
  must 
  introduce 
  under 
  the 
  integral 
  sign 
  the 
  

   factor 
  e^^ 
  where 
  y 
  is 
  negative 
  and 
  ultimately 
  is 
  made 
  to 
  

   vanish. 
  In 
  Prof. 
  Lamb''s 
  exposition 
  * 
  of 
  the 
  deep 
  water 
  

   problem 
  the 
  velocity-potential 
  is 
  employed 
  as 
  an 
  inter- 
  

   mediary, 
  and 
  into 
  tliis 
  the 
  exponential 
  factor 
  enters 
  essentially. 
  

   All 
  ambiguity 
  is 
  thus 
  avoided 
  ; 
  but 
  on 
  the 
  other 
  hand 
  it 
  

   would 
  seem 
  that 
  it 
  should 
  not 
  be 
  necessary 
  to 
  go 
  behind 
  (2), 
  

   at 
  any 
  rate 
  if 
  this 
  expression 
  for 
  the 
  elevation 
  is 
  correctly 
  

   understood. 
  And 
  in 
  the 
  extended 
  problem, 
  where 
  nothing- 
  

   more 
  is 
  known 
  than 
  that 
  cr 
  cc 
  ^'", 
  we 
  have 
  no 
  choice. 
  

   Expanding 
  the 
  cosine 
  in 
  (2), 
  we 
  get 
  

  

  »/ 
  ^ 
  

  

  172 
  "^ 
  1.2.3.4 
  " 
  " 
  ) 
  

  

  cos 
  kx 
  dk, 
  (3) 
  

  

  I 
  

  

  in 
  which 
  on 
  the 
  above 
  understanding 
  

  

  ' 
  7 
  2>H7i 
  7 
  7/ 
  T(2mn 
  + 
  1) 
  cos 
  {mn 
  + 
  4)7r 
  , 
  . 
  

  

  P>-- 
  cos 
  kj! 
  dk 
  == 
  -^ 
  '; 
  '—. 
  . 
  (I) 
  

  

  '^ 
  

  

  When 
  ?i 
  — 
  0, 
  (4) 
  vanishes; 
  and 
  in 
  general 
  we 
  have 
  a 
  term 
  

   in 
  (3) 
  proportional 
  to 
  t^, 
  piedominating 
  when 
  t 
  is 
  small. 
  

  

  We 
  lall 
  back 
  at 
  once 
  upon 
  the 
  simple 
  case 
  of 
  waves 
  in 
  

   deep 
  water 
  by 
  making 
  m 
  = 
  ^. 
  Then 
  (4) 
  vanishes, 
  if 
  n 
  is 
  

   even 
  ; 
  and 
  if 
  n 
  is 
  odd, 
  

  

  r 
  A- 
  cos 
  Li' 
  dk 
  = 
  |ii 
  ( 
  - 
  1)^"+S 
  ... 
  (5) 
  

   Jo 
  

   from 
  which 
  (1) 
  follows 
  (5/=!). 
  In 
  this 
  case 
  the 
  series 
  is 
  

   convergent 
  for 
  all 
  finite 
  values 
  of 
  x 
  and 
  t. 
  

  

  It 
  may 
  be 
  of 
  interest 
  to 
  examine 
  the 
  application 
  to 
  aerial 
  

  

  waves 
  for 
  which 
  the 
  velocity 
  of 
  propagation 
  (a/k) 
  is 
  constant. 
  

  

  Here 
  m 
  = 
  l, 
  and 
  (4; 
  vanishes 
  for 
  all 
  (integral) 
  values 
  of 
  n. 
  

  

  * 
  Proc. 
  Lond. 
  Matli. 
  Soc. 
  ser. 
  2, 
  vol. 
  ii. 
  p. 
  373 
  (1904) 
  j 
  Hydrodynamics; 
  

  

  § 
  236. 
  

  

  