﻿Laminar 
  Motion 
  of 
  an 
  Inviscid 
  Fluid. 
  1005 
  

  

  Let 
  us 
  now 
  trace 
  the 
  course 
  of 
  v 
  as 
  a 
  function 
  of 
  y, 
  

   starting 
  from 
  the 
  wall 
  where 
  y 
  = 
  0, 
  v 
  = 
  ; 
  and 
  let 
  us 
  

   suppose 
  first 
  that 
  U' 
  is 
  everywhere 
  positive. 
  By 
  (11) 
  

   K 
  has 
  the 
  same 
  sign 
  as 
  (dv/dy) 
  , 
  that 
  is 
  the 
  same 
  sign 
  as 
  

   the 
  early 
  values 
  of 
  v. 
  Whether 
  this 
  sign 
  be 
  positive 
  or 
  

   negative, 
  v 
  as 
  determined 
  by 
  (10) 
  cannot 
  again 
  come 
  to 
  

   zero. 
  If, 
  for 
  example, 
  the 
  initial 
  values 
  of 
  v 
  are 
  positive, 
  

   both 
  (remaining) 
  terms 
  in 
  (10) 
  necessarily 
  continue 
  positive; 
  

   while 
  if 
  v 
  begins 
  by 
  being 
  negative, 
  it 
  must 
  remain 
  finitely 
  

   negative. 
  Similarly, 
  if 
  U' 
  be 
  everywhere 
  negative, 
  so 
  that 
  

   K 
  has 
  the 
  opposite 
  sign 
  to 
  that 
  of 
  the 
  early 
  values 
  of 
  v, 
  it 
  

   follows 
  that 
  v 
  cannot 
  again 
  come 
  to 
  zero. 
  No 
  solution 
  can 
  

   be 
  found 
  unless 
  TJ' 
  somewhere 
  vanishes, 
  that 
  is 
  unless 
  n 
  

   coincides 
  with 
  some 
  value 
  of 
  — 
  /cU. 
  

  

  In 
  the 
  above 
  argument 
  U', 
  and 
  therefore 
  also 
  n, 
  is 
  supposed 
  

   to 
  be 
  real, 
  but 
  the 
  formula 
  (10) 
  itself 
  applies 
  whether 
  n 
  be 
  

   real 
  or 
  complex. 
  It 
  is 
  of 
  special 
  value 
  when 
  k 
  is 
  very 
  small, 
  

   lhat 
  is 
  when 
  the 
  wave-length 
  along 
  x 
  of 
  the 
  disturbance 
  is 
  

   very 
  great 
  ; 
  for 
  it 
  then 
  gives 
  v 
  explicitly 
  in 
  the 
  form 
  

  

  "= 
  e 
  ( 
  d+ 
  «4W# 
  • 
  • 
  • 
  (12) 
  

  

  When 
  k 
  is 
  small, 
  but 
  not 
  so 
  small 
  as 
  to 
  justify 
  (12), 
  

   a 
  second 
  approximation 
  might 
  be 
  found 
  by 
  substituting 
  

   from 
  (12) 
  in 
  the 
  last 
  term 
  of 
  (10). 
  

  

  If 
  we 
  suppose 
  in 
  (12) 
  that 
  the 
  second 
  wall 
  is 
  situated 
  

   at 
  y 
  = 
  l, 
  n 
  is 
  determined 
  by 
  

  

  i 
  

  

  -0 
  (13) 
  

  

  (U 
  + 
  «/i) 
  

  

  The 
  integrals 
  (12), 
  (13) 
  must 
  not 
  be 
  taken 
  through 
  a 
  

   place 
  where 
  V 
  -t-njk 
  = 
  0, 
  as 
  appears 
  from 
  (8). 
  We 
  have 
  

   already 
  seen 
  that 
  any 
  value 
  of 
  n 
  for 
  which 
  this 
  can 
  occur 
  is 
  

   admissible. 
  But 
  (I'd) 
  shows 
  that 
  no 
  other 
  real 
  value 
  of 
  n 
  

   is 
  admissible 
  ; 
  and 
  it 
  serves 
  to 
  determine 
  any 
  complex 
  

   values 
  of 
  n. 
  

  

  In 
  (13) 
  suppose 
  (as 
  before) 
  that 
  njk 
  = 
  p 
  + 
  iq 
  ; 
  then 
  

   separating 
  the 
  real 
  and 
  imaginary 
  parts, 
  we 
  get 
  

  

  X 
  {(p+vy+q 
  2, 
  / 
  J 
  ' 
  jo{(p+vy+?y 
  ' 
  w 
  

  

  from 
  the 
  second 
  of 
  which 
  we 
  may 
  infer 
  that 
  if 
  q 
  be 
  finite, 
  

   p 
  + 
  ~U 
  must 
  change 
  sign, 
  as 
  we 
  have 
  already 
  seen 
  that 
  it 
  

  

  