﻿486 
  Mr. 
  ^Y. 
  J. 
  Harrison 
  on 
  the 
  

  

  The 
  pericdof 
  the 
  motion 
  is 
  27r/(/3 
  — 
  7), 
  and 
  the 
  modulus 
  of 
  

   decay 
  is 
  ry-\ 
  where 
  

  

  /3 
  

  

  ^ 
  / 
  gJc{p-p') 
  sinhkh 
  \^ 
  

   \p 
  cosh 
  kh 
  -h 
  p^ 
  sinh 
  /l7< 
  J 
  ' 
  

  

  g'-{p-p')-h"^p 
  Q^ 
  

  

  2^2{/3 
  cosh 
  kh 
  + 
  p^ 
  sinh 
  M}^ 
  sinh^ 
  kh 
  {p 
  s/ 
  »-'' 
  + 
  P^V'^) 
  

  

  where 
  

  

  Q 
  =^{ 
  V 
  v'}^- 
  + 
  2pV 
  cosh^ 
  M 
  -{- 
  p'(v 
  + 
  v') 
  sinh 
  /:7i 
  cosh 
  ^A 
  

  

  + 
  p'(v'-v)sinh2 
  /.•//; 
  

  

  p 
  and 
  J/ 
  refer 
  to 
  the 
  lower 
  fluid 
  o£ 
  depth 
  A; 
  p' 
  and 
  v' 
  to 
  the 
  

   superposed 
  fluid. 
  

  

  I 
  have 
  calculated 
  the 
  solution 
  to 
  a 
  third 
  approximation, 
  

   but 
  the 
  result 
  is 
  too 
  complicated 
  to 
  be 
  given 
  here, 
  and 
  the 
  

   additional 
  effect 
  derived 
  from 
  it 
  is 
  negligible, 
  even 
  when 
  the 
  

   second 
  approximation 
  gives 
  an 
  appreciable 
  additional 
  effect. 
  

  

  Some 
  idea 
  of 
  the 
  damping 
  due 
  to 
  air 
  can 
  be 
  derived 
  from 
  

   a 
  comparison 
  of 
  Tables 
  II. 
  and 
  III. 
  

  

  Waves 
  in 
  a 
  rectangular 
  canal. 
  

  

  §4. 
  There 
  is 
  one 
  difficulty 
  which 
  Dr. 
  Houstoun 
  escapes 
  by 
  

   his 
  use 
  of 
  long 
  waves 
  ; 
  it 
  is 
  that 
  the 
  lapping 
  at 
  the 
  sides 
  of 
  

   the 
  box 
  is 
  virtually 
  neglected, 
  since 
  the 
  vertical 
  motion 
  is 
  

   not 
  considered. 
  If 
  the 
  problem 
  be 
  attacked 
  in 
  the 
  ordinary 
  

   way, 
  and 
  the 
  usual 
  conditions 
  of 
  no 
  motion 
  at 
  a 
  fixed 
  

   boundary 
  be 
  written 
  down, 
  it 
  will 
  be 
  found 
  that 
  the 
  equations 
  

   are 
  only 
  satisfied 
  by 
  a 
  state 
  of 
  no 
  motion 
  everywhere. 
  The 
  

   same 
  applies 
  to 
  motion 
  in 
  which 
  slipping 
  at 
  the 
  boundary 
  is 
  

   allowed, 
  but 
  which 
  is 
  resisted 
  by 
  a 
  traction 
  proportional 
  to 
  

   the 
  velocity. 
  Hence 
  to 
  discuss 
  the 
  problem 
  at 
  all 
  we 
  are 
  

   forced 
  to 
  adopt 
  the 
  assumption 
  of 
  a 
  canal 
  with 
  smooth 
  sides. 
  

   The 
  only 
  alternative 
  is 
  to 
  generalize 
  the 
  method 
  for 
  long 
  waves. 
  

  

  Suppose 
  the 
  canal 
  to 
  be 
  of 
  depth 
  A 
  and 
  breadth 
  h. 
  Take 
  

   the 
  origin 
  of 
  coordinates 
  in 
  the 
  undisturbed 
  surface 
  at 
  one 
  

   side, 
  the 
  axis 
  of 
  z 
  perpendicular 
  to 
  the 
  side 
  and 
  in 
  the 
  surface, 
  

   and 
  the 
  axis 
  of 
  ?/ 
  \ertically 
  upwards. 
  

  

  The 
  equations 
  of 
  motion 
  are 
  

  

  Dv 
  

   Dt 
  - 
  

  

  P 
  'd'C 
  

  

  -\-v^hi, 
  

  

  Dc 
  

  

  Dt"^ 
  

  

  1 
  -dp 
  

  

  P 
  '^!/ 
  

  

  —g 
  + 
  v\7hK 
  

  

  Div 
  

   Dt 
  - 
  

  

  I'dp 
  

  

  "rV^hc, 
  

  

  