﻿Decay 
  of 
  ^yal'cs 
  in 
  a 
  Canal. 
  485 
  

  

  considered 
  by 
  Dr. 
  Houstoun. 
  This 
  will 
  be 
  seen 
  by 
  comparing 
  

   the 
  third 
  cokiinn 
  in 
  Table 
  II. 
  with 
  the 
  sixth 
  column 
  in 
  the 
  

   table 
  on 
  page 
  159 
  o£ 
  his 
  paper. 
  

  

  The 
  period 
  equation 
  for 
  this 
  motion 
  is 
  given 
  by 
  Bassett*. 
  

   For 
  convenience 
  it 
  will 
  be 
  repeated 
  here 
  in 
  the 
  notation 
  of 
  

   this 
  paper. 
  

  

  = 
  0, 
  

  

  ni 
  cosh 
  mli^ 
  

  

  — 
  sinh 
  mil, 
  

  

  , 
  

  

  2vkma 
  

  

  — 
  ?7isinh 
  mh^ 
  

  

  cosh 
  mli^ 
  

  

  a 
  + 
  21- 
  V, 
  

  

  5/^- 
  

  

  — 
  h 
  sinh 
  kh 
  , 
  

  

  cosh 
  kh. 
  

  

  ■2k'v 
  , 
  

  

  rjk 
  

  

  k 
  cosh 
  kli 
  , 
  

  

  — 
  sinh 
  kh 
  , 
  

  

  , 
  

  

  a^^'lvk'^a 
  

  

  where 
  m- 
  = 
  k^ 
  + 
  a/v. 
  

  

  As 
  regards 
  .c 
  and 
  t 
  the 
  motion 
  is 
  of 
  the 
  type 
  e'^'^'^'^^. 
  

  

  In 
  the 
  approximation 
  it 
  is 
  assumed 
  that 
  mh 
  is 
  large^ 
  on 
  

   account 
  of 
  the 
  smallness 
  of 
  v, 
  and 
  with 
  this 
  assumption 
  the 
  

   equation 
  becomes 
  

  

  m 
  \ 
  a- 
  cosh 
  kJi+gk 
  sinh 
  kh 
  } 
  —k{a^ 
  sinh 
  kh 
  -{-gk 
  cosh 
  kh 
  } 
  

  

  IV 
  L 
  W'Ml 
  a/t 
  — 
  '/A. 
  Sill 
  

  

  -t-ilrvma 
  cosh 
  kh 
  =■ 
  0. 
  

   AVe 
  obtain 
  by 
  successive 
  approximations 
  

  

  a= 
  ± 
  {;//>■ 
  tanh 
  a}i- 
  1% 
  / 
  ^' 
  "^ 
  

  

  "■v 
  - 
  cosh^ 
  kh 
  sinh-^ 
  kh 
  

  

  -2F; 
  

  

  /rv(cosh^7i 
  + 
  sinh^ 
  /jA) 
  

   4 
  siulr 
  kh 
  cosh- 
  kh 
  

  

  The 
  additional 
  term 
  obtained 
  by 
  this 
  more 
  rigorous 
  

   method 
  of 
  approximation 
  is 
  negligible 
  compared 
  witli 
  2k-v 
  

   when 
  the 
  wave-length 
  is 
  small 
  ; 
  it 
  has 
  only 
  an 
  appreciable 
  

   value 
  when 
  the 
  wave-length 
  is 
  large 
  compared 
  with 
  the 
  

   depth. 
  This 
  is 
  probably 
  the 
  reason 
  why 
  it 
  was 
  omitted 
  

   from 
  Hough's 
  approximation. 
  

  

  The 
  influence 
  of 
  a 
  superposed 
  fluid 
  on 
  the 
  damping 
  of 
  

   long 
  leaves. 
  

  

  §3. 
  The 
  results 
  given 
  in 
  this 
  paragraph 
  have 
  little 
  bearing 
  

   on 
  the 
  problem 
  under 
  consideration 
  for 
  reasons 
  already 
  givei?, 
  

   but 
  it 
  may 
  be 
  as 
  well 
  to 
  record 
  the 
  solution 
  of 
  the 
  period 
  

   equation 
  to 
  a 
  second 
  approximation. 
  

  

  * 
  Treatise 
  on 
  Hydrodynamics, 
  vol. 
  ii. 
  p. 
  314. 
  

  

  