﻿Decay 
  of 
  Waves 
  in 
  a 
  Canal. 
  489 
  

  

  result 
  could 
  doubtless 
  be 
  obtained 
  much 
  more 
  simply 
  by 
  

   generalizing 
  Dr. 
  Houstoun's 
  method. 
  But 
  Dr. 
  Houstoun, 
  

   although 
  he 
  allows 
  no 
  motion 
  over 
  the 
  sides 
  o£ 
  the 
  box, 
  yet 
  

   obtains 
  the 
  same 
  period 
  and 
  damping 
  as 
  for 
  waves 
  propagated 
  

   in 
  one 
  direction. 
  Now, 
  in 
  the 
  case 
  of 
  wave-motion 
  in 
  the 
  

  

  2 
  2 
  2 
  2 
  

  

  rectangular 
  trough 
  which 
  he 
  used, 
  nr= 
  ^—^ 
  + 
  j^-* 
  ^^^ 
  

  

  waves 
  of 
  the 
  tvpe 
  sin^^^ 
  — 
  ^sin^— 
  ^> 
  where 
  a 
  = 
  152'4: 
  cms., 
  

   -" 
  ^ 
  a 
  b 
  

  

  5 
  = 
  20*3 
  cms. 
  Hence, 
  for 
  the 
  mode 
  p 
  = 
  l, 
  ^' 
  = 
  1 
  which 
  he 
  

  

  takes 
  as 
  the 
  fundamental 
  one 
  in 
  his 
  solution, 
  ?7i^=7r^/6^, 
  

  

  approximately. 
  This 
  leads 
  to 
  quite 
  different 
  results. 
  

  

  The 
  Table 
  below 
  gives 
  the 
  periods 
  of 
  the 
  different 
  modes 
  

  

  and 
  for 
  the 
  same 
  depths 
  as 
  given 
  by 
  Dr. 
  Houstoun. 
  

  

  Table 
  I. 
  

  

  Depth 
  

  

  in 
  cms. 
  

  

  Observed 
  

   period. 
  

  

  P=y 
  

   g=0. 
  

  

  P=0, 
  

   q=l. 
  

  

  1 
  

  

  sees. 
  

   10^1 
  

  

  sees. 
  

   10-25 
  

  

  sees. 
  

   1-323 
  

  

  2 
  

  

  712 
  

  

  7-02 
  

  

  •938 
  

  

  3 
  

  

  5 
  80 
  

  

  5-68 
  

  

  •779 
  

  

  5 
  

  

  4-49 
  

  

  4-39 
  

  

  •634 
  

  

  7 
  

  

  3-68 
  

  

  3-71 
  

  

  •573 
  

  

  10 
  

  

  312 
  

  

  3-11 
  

  

  534 
  

  

  P=l, 
  

   q=l. 
  

  

  i> 
  = 
  3, 
  

  

  q=3. 
  

  

  sees. 
  

  

  r3io 
  

  

  sees. 
  

   1-229 
  

  

  sees. 
  

   •451 
  

  

  •929 
  

  

  •872 
  

  

  •345 
  

  

  •771 
  

  

  •725 
  

  

  •313 
  

  

  •629 
  

  

  •596 
  

  

  •297 
  

  

  •569 
  

  

  •549 
  

  

  •295 
  

  

  •531 
  

  

  •509 
  

  

  •294 
  

  

  In 
  this 
  table 
  the 
  observed 
  periods 
  are 
  those 
  given 
  on 
  

   page 
  159 
  of 
  Dr. 
  Houstoun's 
  paper 
  ; 
  it 
  will 
  be 
  noticed 
  that 
  

   they 
  are 
  slightly 
  different 
  from 
  those 
  on 
  page 
  161^ 
  but 
  the 
  

   difference 
  is 
  quite 
  immaterial 
  for 
  our 
  purpose. 
  

  

  Thus 
  there 
  can 
  be 
  no 
  doubt 
  that 
  in 
  the 
  motion 
  experi- 
  

   mentally 
  observed 
  by 
  Dr. 
  Houstoun 
  there 
  existed 
  the 
  mode 
  

   ])=!, 
  q 
  = 
  0. 
  The 
  periods 
  of 
  the 
  other 
  modes 
  are 
  so 
  much 
  

   shorter 
  that 
  they 
  probably 
  passed 
  unnoticed. 
  

  

  Xow, 
  if 
  there 
  is 
  no 
  slipping 
  at 
  the 
  sides 
  and 
  there 
  are 
  no 
  

   surfaces 
  of 
  slip 
  in 
  the 
  liquid, 
  this 
  mode 
  cannot 
  enter 
  into 
  the 
  

   motion. 
  But 
  it 
  is 
  evident 
  from 
  the 
  fact 
  that 
  the 
  water 
  rises 
  

  

  * 
  Tills 
  difference 
  between 
  the 
  quantities 
  in 
  this 
  column 
  and 
  the 
  sum 
  

   of 
  those 
  in 
  the 
  3rd 
  and 
  5th 
  columns 
  on 
  p. 
  159 
  of 
  Houstoun's 
  paper 
  is 
  due 
  

   partly 
  to 
  the 
  tact 
  that 
  the 
  quantities 
  in 
  the 
  5th 
  column 
  are 
  sometimes 
  

   miscalculated. 
  

  

  FML 
  Mag, 
  S. 
  6. 
  Vol. 
  18. 
  Xo. 
  106. 
  Oct. 
  1009. 
  2 
  L 
  

  

  