﻿Wave-Resistance 
  of 
  a 
  Ship, 
  113 
  

  

  for 
  this 
  gives 
  

  

  -f- 
  = 
  — 
  vf 
  (x, 
  z) 
  = 
  — 
  v 
  j— 
  when 
  y 
  = 
  0. 
  

   dy 
  ax 
  

  

  In 
  this 
  expression 
  attention 
  must 
  be 
  called 
  to 
  the 
  factor 
  

   sin 
  {m(a? 
  — 
  f) 
  + 
  m 
  \Zm 
  2 
  vVg 
  2 
  — 
  ly} 
  

  

  in 
  the 
  second 
  integral. 
  This 
  form 
  is 
  not 
  required 
  to 
  satisfy 
  

   the 
  boundary 
  conditions 
  formulated 
  above 
  ; 
  and 
  it 
  is 
  evident 
  

   that 
  the 
  solution 
  is 
  to 
  a 
  certain 
  extent 
  indeterminate 
  with 
  

   those 
  conditions, 
  for 
  we 
  may 
  superpose 
  any 
  system 
  of 
  free 
  

   waves 
  symmetrical 
  with 
  respect 
  to 
  y 
  = 
  on 
  a 
  particular 
  

   solution 
  satisfying 
  them. 
  The 
  form 
  of 
  the 
  factor 
  in 
  question 
  

   is 
  chosen 
  in 
  order 
  to 
  make 
  the 
  elementary 
  diverging 
  waves 
  

   trail 
  aft; 
  in 
  other 
  words, 
  to 
  satisfy 
  the 
  condition 
  that 
  the 
  ship 
  

   advances 
  into 
  still 
  water. 
  

  

  Leaving 
  the 
  reduction 
  of 
  the 
  integrals 
  on 
  one 
  side, 
  for 
  the 
  

   present, 
  we 
  proceed 
  to 
  calculate 
  the 
  wave-resistance 
  (R), 
  

  

  Let 
  8p 
  be 
  the 
  increase 
  of 
  pressure 
  due 
  to 
  the 
  wave-dis- 
  

   turbance. 
  Then 
  , 
  

  

  R 
  = 
  — 
  2 
  (T 
  Sp 
  —dx 
  dz, 
  

  

  the 
  double 
  integral 
  extending 
  over 
  the 
  median 
  plane 
  of 
  the 
  

   ship. 
  Now 
  measuring 
  from 
  the 
  undisturbed 
  surface 
  

  

  p-7r+gpz-ipq 
  2 
  + 
  ipv 
  2 
  , 
  

  

  and 
  therefore 
  ^P 
  = 
  P 
  v 
  ^~j 
  (^PO 
  

  

  ax 
  

  

  so 
  that 
  R 
  = 
  — 
  2pv 
  (T-r* 
  -y-dxdz. 
  

  

  J 
  JJ 
  ax 
  ax 
  

  

  Substituting 
  the 
  value 
  of 
  d<j*/dx, 
  we 
  see 
  that 
  the 
  first 
  and 
  

   third 
  integrals 
  in 
  the 
  expression 
  for 
  <j> 
  add 
  nothing 
  to 
  the 
  

   resistance 
  because 
  

  

  JjJJ/(f, 
  DA*,*) 
  smm(x-Qd£d£d*dz=0, 
  

  

  on 
  account 
  of 
  the 
  odd 
  factor 
  smm(x— 
  f), 
  

   and 
  hence 
  

  

  QO 
  CO 
  

  

  4p 
  v 
  4 
  

  

  irg 
  

  

  g/v% 
  —oo 
  

  

  g/v-2 
  — 
  oo 
  -co 
  

  

  wr 
  (p 
  i 
  P) 
  mHm 
  - 
  

  

  ^T(I* 
  + 
  J*)-^, 
  (6) 
  

  

  7TV* 
  } 
  x 
  V 
  T 
  ' 
  VX* 
  — 
  1 
  

  

  PM. 
  3%. 
  S. 
  5. 
  Vol. 
  45. 
  No. 
  272. 
  Jan. 
  1898. 
  I 
  

  

  