﻿(2) 
  

  

  Wave-Resistance 
  of 
  a 
  Ship. 
  109 
  

  

  and, 
  therefore, 
  with 
  (1) 
  

  

  d 
  A 
  - 
  v 
  l. 
  d 
  3 
  

   dz 
  " 
  g 
  dx 
  

  

  On 
  account 
  of 
  the 
  symmetry 
  of 
  the 
  ship 
  with 
  respect 
  to 
  the 
  

   median 
  plane 
  y 
  = 
  0, 
  we 
  have 
  dcj)/dy 
  = 
  when 
  y 
  = 
  0, 
  except 
  

   over 
  the 
  ship, 
  where, 
  if 
  rj 
  is 
  the 
  semi-breadth 
  at 
  (x, 
  z), 
  

  

  dcj) 
  drj 
  

  

  dy 
  ~ 
  dx 
  

  

  = 
  - 
  »/(*, 
  z) 
  (^y), 
  . 
  • 
  (3) 
  

  

  and 
  this 
  condition 
  is 
  taken 
  to 
  hold 
  at 
  the 
  plane 
  y 
  = 
  0, 
  instead 
  

   of 
  at 
  the 
  surface 
  of 
  the 
  ship, 
  the 
  justification 
  being 
  the 
  same 
  

   as 
  that 
  for 
  equation 
  (1). 
  Finally, 
  d(f>/dz 
  = 
  at 
  the 
  bottom, 
  

   z 
  = 
  h, 
  of 
  the 
  water. 
  

  

  We 
  now 
  consider 
  the 
  solution 
  for 
  $, 
  in 
  the 
  part 
  of 
  the 
  

   water 
  where 
  y 
  is 
  positive, 
  with 
  the 
  given 
  boundary 
  conditions 
  

   at 
  z 
  = 
  0, 
  z=h, 
  y 
  = 
  0. 
  

  

  The 
  typical 
  term 
  in 
  the 
  solution 
  is 
  

  

  a 
  cos 
  ?i 
  (z—h) 
  cos 
  (mx 
  + 
  a) 
  cos 
  (py 
  + 
  /3), 
  

  

  where 
  m 
  2 
  + 
  n 
  2 
  +p 
  2 
  = 
  0. 
  Here 
  m 
  must 
  be 
  taken 
  real 
  as 
  the 
  

   water 
  extends 
  from 
  x= 
  — 
  oo 
  to#=-t-<x>; 
  n 
  and 
  p 
  maybe 
  

   either 
  real 
  or 
  imaginary, 
  but 
  if 
  p 
  is 
  imaginary 
  [ 
  = 
  ?//] 
  the 
  

   last 
  factor 
  must 
  take 
  the 
  form 
  e~ 
  p 
  'v. 
  

  

  This 
  term 
  satisfies 
  d<j)/dz 
  = 
  at 
  z 
  = 
  h, 
  and 
  it 
  also 
  satisfies 
  

   equation 
  (2), 
  if 
  

  

  n 
  tann/*= 
  — 
  v 
  2 
  m 
  2 
  /g 
  (4) 
  

  

  This 
  equation 
  has 
  an 
  infinite 
  number 
  of 
  real 
  roots 
  and 
  one 
  

   pure 
  imaginary 
  root 
  given 
  by 
  

  

  nf 
  tanh 
  n'li 
  = 
  v 
  2 
  m 
  2 
  jg^ 
  \n 
  = 
  in'~\ 
  . 
  

  

  We 
  shall 
  see 
  that 
  the 
  imaginary 
  root 
  is 
  alone 
  responsible 
  

   for 
  the 
  wave-making 
  resistance. 
  As 
  for 
  p 
  it 
  is 
  always 
  

   imaginary 
  for 
  the 
  real 
  roots 
  of 
  n, 
  and 
  is 
  so 
  for 
  the 
  imaginary 
  

   root 
  if 
  ?7i>r/. 
  

  

  The 
  condition 
  (3) 
  will 
  now 
  require 
  the 
  expansion 
  of 
  the 
  

   given 
  function 
  / 
  (a?, 
  z) 
  in 
  the 
  form 
  

  

  22tfm» 
  cos 
  n(z 
  — 
  h) 
  cos 
  (mx 
  + 
  a), 
  

  

  where 
  the 
  summation 
  with 
  respect 
  to 
  m 
  will 
  take 
  the 
  form 
  of 
  

   an 
  integral. 
  

  

  Suppose 
  at 
  first 
  the 
  function 
  periodic 
  in 
  x 
  so 
  that 
  

  

  f(x+l,z)=/(x-l,z), 
  

  

  