﻿Two-Dimensional 
  Motion 
  of 
  Infinite 
  Liquid. 
  121 
  

  

  It 
  is 
  understood 
  that 
  the 
  boundary 
  in 
  the 
  z 
  plane 
  is 
  the 
  

   locus 
  corresponding 
  to 
  77 
  = 
  0. 
  

  

  3. 
  Field 
  of 
  flow 
  due 
  to 
  translation 
  of 
  the 
  boundary. 
  — 
  If 
  

   the 
  boundary 
  have 
  a 
  velocity 
  V 
  in 
  a 
  direction 
  making 
  an 
  

   angle 
  fi 
  with 
  the 
  axis 
  of 
  a?, 
  the 
  superposition 
  on 
  the 
  whole 
  

   system 
  of 
  such 
  uniform 
  velocity 
  as 
  brings 
  the 
  boundary 
  to 
  

   rest 
  gives 
  an 
  irrotational 
  fluid 
  motion 
  which 
  has 
  zero 
  normal 
  

   velocity 
  at 
  the 
  boundary 
  and 
  tends, 
  for 
  z 
  infinite, 
  to 
  flow 
  V 
  

   in 
  the 
  direction 
  fi+Tr. 
  If 
  this 
  motion 
  have 
  velocity 
  poten- 
  

   tial 
  cj) 
  and 
  stream-function 
  yfr 
  9 
  so 
  defined 
  that 
  the 
  velocity 
  is 
  

   the 
  upward 
  gradient 
  of 
  </>, 
  and 
  if 
  ic 
  = 
  <j> 
  + 
  iyjr, 
  w 
  must 
  tend, 
  

   for 
  z 
  infinite, 
  to 
  the 
  form 
  

  

  — 
  Y^exp( 
  — 
  i/x) 
  + 
  const., 
  . 
  . 
  . 
  . 
  (6) 
  

   or, 
  in 
  terms 
  of 
  f, 
  

  

  -Y 
  Ko 
  (i\/27r)exipi(r 
  /o 
  - 
  f 
  i-27ry\). 
  . 
  . 
  (7) 
  

  

  Now 
  if 
  

  

  u;= 
  _(Y 
  / 
  , 
  o 
  X/ 
  7r 
  ) 
  S 
  i 
  n 
  (27rf/\-7o 
  + 
  /i), 
  . 
  . 
  (8) 
  

  

  this 
  tends, 
  for 
  r\ 
  — 
  -> 
  + 
  co 
  , 
  to 
  the 
  form 
  (7) 
  ; 
  and 
  as 
  

  

  yjr= 
  - 
  (Yk 
  \Itt) 
  cos 
  (2tt£/A,— 
  Y 
  + 
  yu) 
  sinh 
  (27r 
  v 
  /\), 
  

  

  it 
  is 
  clear 
  that 
  yjr 
  is 
  zero 
  along 
  the 
  boundary 
  77 
  = 
  0. 
  The 
  

   corresponding 
  form 
  of 
  </> 
  shows 
  that 
  there 
  is 
  no 
  circulation 
  

   round 
  the 
  boundary 
  ; 
  and 
  w 
  is 
  free 
  from 
  infinities 
  in 
  the 
  

   relevant 
  region. 
  

  

  Hence 
  formula 
  (8) 
  specifies 
  that 
  irrotational 
  motion 
  

   past 
  the 
  fixed 
  boundary 
  whose 
  limit 
  form, 
  at 
  indefinitely 
  

   great 
  distance, 
  is 
  the 
  assigned 
  uniform 
  flow. 
  

  

  4. 
  The 
  impulse 
  of 
  4he 
  motion 
  due 
  to 
  translation. 
  — 
  Though 
  

   modern 
  speculation 
  tends 
  to 
  regard 
  wave-motion 
  as 
  the 
  

   preponderating 
  factor 
  in 
  suction 
  and 
  other 
  inertia 
  pheno- 
  

   mena 
  of 
  floating 
  bodies, 
  it 
  can 
  hardly 
  be 
  doubted 
  that, 
  in 
  

   the 
  case 
  of 
  a 
  submarine 
  at 
  least, 
  the 
  ordinary 
  inertia 
  coeffi- 
  

   cients 
  measure 
  approximately 
  the 
  resistance 
  to 
  quick 
  changes 
  

   of 
  velocity. 
  Thus 
  the 
  evaluation 
  of 
  the 
  impulse 
  (X, 
  Y) 
  of 
  

   the 
  combined 
  motion 
  of 
  solid 
  and 
  fluid, 
  when 
  the 
  solid 
  has 
  

   translatory 
  motion, 
  is 
  of 
  interest. 
  

  

  If 
  an 
  approximation 
  to 
  w 
  for 
  | 
  z 
  | 
  great, 
  closer 
  than 
  that 
  

   afforded 
  by 
  formula 
  (6), 
  be 
  

  

  w=-V~exp(-t» 
  + 
  C 
  + 
  D/*, 
  . 
  . 
  . 
  {6 
  a) 
  

  

  