﻿490 
  

  

  The 
  Decay 
  of 
  Waves 
  in 
  a 
  Canal. 
  

  

  and 
  falls 
  at 
  the 
  sides 
  that 
  there 
  must 
  be 
  either 
  slipping 
  

   at 
  the 
  sides_, 
  or 
  else 
  there 
  must 
  exist 
  surfaces 
  of 
  slip 
  near 
  the 
  

   sides 
  in 
  the 
  liquid, 
  most 
  probably 
  both 
  coexist. 
  This 
  fact 
  shows 
  

   that 
  we 
  have 
  been 
  unable 
  to 
  analyse 
  the 
  motion 
  correctly. 
  

   The 
  sides 
  must 
  influence 
  the 
  damping 
  of 
  the 
  motion 
  in 
  any 
  

   actual 
  case, 
  and 
  this 
  has 
  been 
  neglected 
  in 
  the 
  analysis, 
  the 
  

   sides 
  having 
  been 
  taken 
  as 
  smooth. 
  It 
  will 
  be 
  seen 
  from 
  

   what 
  has 
  been 
  said 
  above, 
  thai 
  we 
  do 
  not 
  know 
  sufficient 
  

   about 
  the 
  modes 
  in 
  which 
  the 
  water 
  actually 
  vibrates 
  to 
  say 
  

   that 
  one 
  predominates 
  to 
  such 
  an 
  extent 
  that 
  the 
  damping 
  

   calculated 
  for 
  that 
  mode 
  should 
  agree 
  with 
  the 
  observed 
  

   (c/. 
  page 
  158 
  of 
  Dr. 
  Houstoun^s 
  paper). 
  But 
  even 
  if 
  we 
  

   admit 
  that 
  part 
  of 
  the 
  observed 
  damping 
  may 
  be 
  due 
  to 
  the 
  

   existence 
  of 
  higher 
  modes 
  of 
  appreciable 
  amplitude, 
  we 
  have 
  

   only 
  found 
  an 
  explanation 
  in 
  the 
  case 
  of 
  the 
  depths 
  1, 
  2, 
  

   3 
  cms. 
  The 
  discrepancy 
  is 
  still 
  unexplained 
  for 
  the 
  depths 
  

   5, 
  7, 
  10 
  cms., 
  as 
  will 
  be 
  seen 
  from 
  Table 
  II. 
  Hence 
  the 
  only 
  

   rational 
  assumption 
  is 
  that 
  the 
  increase 
  of 
  damping 
  is 
  due 
  to 
  

   the 
  sides. 
  The 
  existence 
  after 
  a 
  time 
  of 
  a 
  surface 
  film 
  such 
  

   as 
  suggested 
  by 
  Dr. 
  Houstoun 
  would 
  only 
  increase 
  the 
  

   damping 
  over 
  those 
  values 
  tabulated 
  by 
  him 
  (compare 
  data 
  

   on 
  pages 
  159 
  and 
  160 
  of 
  his 
  paper). 
  

  

  Table 
  II. 
  exhibits 
  the 
  reciprocal 
  {k) 
  of 
  the 
  modulus 
  of 
  decay 
  

   for 
  the 
  same 
  cases 
  as 
  Table 
  I. 
  gives 
  the 
  periods. 
  

  

  Table 
  II. 
  

  

  Depth 
  

   in 
  cms. 
  

  

  1 
  

  

  Observed 
  

  

  

  p=0, 
  

   q 
  = 
  l. 
  

  

  q=l. 
  

  

  q=\. 
  

  

  q 
  = 
  'd. 
  

  

  1 
  

  

  •048 
  

  

  •0324 
  

  

  •0826 
  

  

  •0849 
  

  

  •0853 
  

  

  •1271 
  

  

  2 
  

  

  •032 
  

  

  •0183 
  

  

  •0481 
  

  

  •0463 
  

  

  •0472 
  

  

  •0521 
  

  

  3 
  

  

  •028 
  

  

  •0133 
  

  

  •0309 
  

  

  •0313 
  

  

  •0316 
  

  

  •0239 
  

  

  5 
  

  

  •023 
  

  

  •0090 
  

  

  •0165 
  

  

  •0168 
  

  

  •0163 
  

  

  •0075 
  

  

  7 
  

  

  •022 
  

  

  •0070 
  

  

  •0093 
  

  

  •0093 
  

  

  •0089 
  

  

  •0052 
  

  

  10 
  

  

  •Oil 
  

  

  •0053 
  

  

  •0040 
  

  

  •0040 
  

  

  •0037 
  

  

  •0047 
  

  

  With 
  the 
  last 
  row 
  in 
  this 
  table 
  may 
  be 
  compared 
  the 
  values 
  

   of 
  K 
  when 
  the 
  damping 
  effect 
  of 
  air 
  is 
  taken 
  into 
  account. 
  

   These 
  quantities 
  are 
  calculated 
  on 
  the 
  assumption 
  that 
  in 
  the 
  

   formulae 
  in 
  §3, 
  we 
  may 
  replace 
  k 
  by 
  m, 
  as 
  for 
  waves 
  in 
  a 
  

   single 
  liquid. 
  

  

  