﻿Highest 
  Wave 
  in 
  Deep 
  Water. 
  1055 
  

  

  The 
  function 
  F(<£) 
  must 
  be 
  real. 
  There 
  must 
  also 
  be 
  a 
  

   limit 
  to 
  the 
  form 
  of 
  F 
  owing 
  to 
  the 
  fact 
  that 
  x 
  must 
  be 
  real. 
  

   If 
  we 
  suppose 
  that 
  the 
  depth 
  of 
  the 
  fluid 
  is 
  infinite, 
  the 
  

   condition 
  to 
  be 
  satisfied 
  is 
  

  

  dz 
  1 
  T 
  . 
  

  

  ^— 
  =^= 
  - 
  , 
  when 
  ylr 
  = 
  — 
  go 
  . 
  

  

  aw 
  c 
  

  

  A 
  function 
  satisfying 
  these 
  conditions 
  is 
  that 
  given 
  by 
  the 
  

   equation 
  

  

  qiv 
  

  

  c 
  — 
  = 
  sm 
  

  

  for 
  which 
  

  

  »V¥*\/?('-¥). 
  

  

  nA(?-*) 
  + 
  2 
  :-»-\/i+-. 
  

  

  where 
  in 
  applying 
  equation 
  (1) 
  we 
  have 
  put 
  G 
  = 
  c 
  2 
  ; 
  and 
  the 
  

   equation 
  of 
  the 
  free 
  surface 
  is 
  

  

  • 
  B= 
  \/'Kf 
  -^) 
  + 
  f 
  sin 
  "V§- 
  • 
  • 
  

  

  (2) 
  

  

  Since 
  y 
  = 
  Y, 
  and 
  has 
  to 
  be 
  real, 
  the 
  greatest 
  value 
  of 
  y 
  is 
  

   c 
  2 
  /2g, 
  and 
  the 
  least 
  is 
  0. 
  Therefore 
  the 
  amplitude 
  of 
  the 
  

   wave 
  is 
  c 
  2 
  /2g, 
  which 
  is 
  the 
  value 
  of 
  y 
  at 
  the 
  crest. 
  

  

  The 
  wave-length 
  is 
  

  

  = 
  7*6 
  a, 
  nearly, 
  

   where 
  a 
  is 
  the 
  amplitude 
  

  

  The 
  velocity 
  is 
  given 
  by 
  

   2ttc 
  2 
  4:ira 
  

  

  = 
  1-64, 
  

  

  ! 
  + 
  -£- 
  

  

  Z7T 
  

  

  C 
  2 
  

  

  At 
  the 
  crest, 
  where 
  y= 
  s- 
  , 
  we 
  find 
  

  

  ^ 
  /o 
  

  

  T 
  = 
  +^3, 
  

  

  so 
  that 
  there 
  is 
  a 
  sharp 
  angle 
  of 
  120° 
  in 
  the 
  wave-profile 
  at 
  

   this 
  point. 
  

  

  