﻿Wave-Resistance 
  of 
  a 
  Ship. 
  121 
  

  

  Now 
  if 
  v 
  >c, 
  i. 
  e. 
  if 
  the 
  velocity 
  of 
  the 
  ship 
  is 
  greater 
  than 
  

   that 
  of 
  the 
  free 
  wave, 
  the 
  equation 
  

  

  dif 
  ~ 
  c 
  2 
  dx 
  2 
  

   is 
  solved 
  in 
  the 
  form 
  

  

  ? 
  = 
  f(*+<y/^V), 
  .... 
  (io) 
  

  

  where 
  the 
  boundary 
  condition 
  gives 
  

  

  v/ 
  

  

  t^-e'w, 
  r 
  3 
  ^ 
  

  

  F 
  W=7^ 
  

  

  r=F^ 
  = 
  --± 
  = 
  ^ 
  (id 
  

  

  The 
  form 
  of 
  solution 
  (10) 
  is 
  employed 
  in 
  order 
  to 
  make 
  the 
  

   diverging 
  waves 
  trail 
  aft. 
  

  

  The 
  disturbance 
  therefore 
  consists 
  of 
  two 
  bands 
  at 
  an 
  

   angle 
  tan 
  -1 
  (c/ 
  s/v 
  2 
  — 
  c 
  2 
  ) 
  with 
  the 
  line 
  of 
  the 
  ship's 
  motion, 
  

   the 
  front 
  of 
  each 
  band 
  being 
  a 
  hump 
  above 
  the 
  mean 
  level 
  

   and 
  its 
  back 
  part 
  a 
  hollow, 
  which 
  is 
  similar 
  to 
  the 
  hump 
  if 
  

   the 
  ship 
  is 
  similarly 
  shaped 
  fore 
  and 
  aft, 
  

  

  The 
  resistance 
  (R) 
  is 
  given 
  by 
  

  

  R 
  = 
  

  

  -Zhfop^dx 
  

  

  = 
  

  

  ^ 
  ll 
  yjx 
  dx 
  

  

  = 
  

  

  * 
  vV-c 
  2 
  JvW 
  

  

  so 
  that 
  it 
  is 
  infinite 
  when 
  the 
  velocity 
  of 
  the 
  ship 
  is 
  equal 
  to 
  

   that 
  of 
  the 
  free 
  wave, 
  and 
  ultimately 
  varies 
  as 
  the 
  velocity. 
  

  

  If 
  v<c, 
  the 
  differential 
  equation 
  for 
  f 
  takes 
  the 
  potential 
  

   form 
  

  

  putting 
  

  

  dx* 
  2 
  ^ 
  dy'* 
  " 
  Uj 
  

  

  j 
  _ 
  s/P-v 
  2 
  

  

  v 
  = 
  y> 
  

  

  c 
  

   The 
  solution 
  is 
  now 
  

  

  1 
  r 
  2 
  

  

  bJS^^+c 
  

  

  T0 
  Vc 
  2 
  ' 
  

   PM. 
  1%. 
  S. 
  5. 
  Vol. 
  45. 
  No. 
  272. 
  Jan. 
  1898. 
  K 
  

  

  