﻿and 
  

  

  Wave-Resistance 
  of 
  a 
  Ship. 
  

   and 
  the 
  corresponding 
  term 
  in 
  the 
  resistance 
  is 
  

  

  119 
  

  

  2pg*<ra'00' 
  _ 
  

  

  irv 
  

  

  e-^+ 
  z 
  ^ 
  2v2 
  [K 
  {g(z 
  + 
  z 
  / 
  )/2v' 
  2 
  }-K 
  } 
  { 
  9 
  (z 
  + 
  z 
  / 
  )/2v 
  <2 
  }]. 
  

  

  The 
  functions 
  K 
  do 
  not 
  appear 
  to 
  have 
  been 
  calculated, 
  

   but 
  their 
  general 
  course 
  is 
  known. 
  It 
  will 
  be 
  sufficient 
  at 
  

   present 
  to 
  refer 
  to 
  Gray 
  and 
  Mathews, 
  chap. 
  vii. 
  

  

  As 
  an 
  illustration 
  of 
  the 
  process 
  of 
  calculating 
  the 
  resist- 
  

   ance 
  of 
  a 
  given 
  ship 
  at 
  any 
  speed, 
  we 
  may 
  consider 
  one 
  of 
  

   simple 
  analytical 
  form 
  which 
  is 
  fairly 
  ship-shape 
  and 
  easily 
  

   realisable. 
  Experimental 
  confirmation 
  of 
  the 
  result 
  was 
  not 
  

   practicable, 
  and 
  the 
  matter 
  must 
  be 
  left 
  in 
  the 
  hands 
  of 
  those 
  

   who 
  have 
  the 
  necessary 
  apparatus 
  at 
  command. 
  

  

  Let 
  the 
  surface 
  of 
  the 
  ship 
  be 
  

  

  between 
  

  

  y= 
  -^(l 
  + 
  cos^Xl 
  + 
  cosfe), 
  

  

  x 
  = 
  ±7r/a, 
  

  

  z 
  = 
  and 
  ir/b 
  ; 
  

  

  Fisr. 
  2. 
  

  

  so 
  that, 
  for 
  y 
  positive. 
  

  

  dv 
  

  

  f(oc, 
  z)= 
  ~ 
  — 
  —ac 
  sin 
  ax{\ 
  -f 
  cos 
  bz) 
  . 
  

  

  dx 
  

  

  Here 
  I==0, 
  

  

  (V 
  

  

  J= 
  — 
  ac 
  1 
  

  

  Jo 
  

  

  Jn/a 
  

   sii 
  

   -it 
  fa 
  

  

  (1 
  + 
  cos 
  bz) 
  e~ 
  K2 
  * 
  /k 
  dz 
  1 
  sin 
  ax 
  sin 
  \xjkdx 
  

  

  where 
  k 
  — 
  v^/g; 
  and 
  therefore 
  

  

  2 
  sin 
  2 
  irXlka 
  d\ 
  

  

  R 
  = 
  M^ 
  a 
  ^k 
  h 
  \ 
  (2 
  A 
  4 
  + 
  b*k* 
  - 
  e 
  -*W 
  bk 
  tftfj 
  

  

  {\ 
  4 
  + 
  b 
  2 
  k*)%a?k 
  2 
  -\y 
  \2 
  </y<i-\ 
  

  

  which 
  is 
  best 
  calculated 
  by 
  mechanical 
  quadrature. 
  

   Suppose, 
  for 
  example, 
  in 
  foot-second 
  units 
  

  

  v 
  = 
  20 
  (velocity 
  of 
  ship), 
  

  

  2<7r/a 
  = 
  200 
  (length 
  of 
  ship), 
  

  

  7r/b 
  — 
  20 
  (depth 
  below 
  water-line), 
  

  

  Sc 
  = 
  32 
  (greatest 
  breadth), 
  

  

  