﻿nipple 
  Formation 
  in 
  Kundt^s 
  Tube 
  Experiment, 
  183 
  

  

  Now 
  

  

  J 
  5 
  cos^ 
  d 
  .de= 
  cos* 
  ^ 
  sin 
  (9 
  -(- 
  4 
  J 
  cos^ 
  6 
  . 
  t?^, 
  

   and 
  3 
  f 
  cos^ 
  d.de= 
  cos^ 
  ^ 
  . 
  sin 
  ^ 
  + 
  2 
  sin 
  d, 
  

  

  .'. 
  \ 
  fcos^ 
  ^(3-5 
  cos2 
  e)de= 
  - 
  igsin 
  (9(2+ 
  cos^i^ 
  + 
  Bcos'l?) 
  

  

  and 
  

  

  F 
  = 
  /z7ro-R' 
  

  

  .»<4^^,^(2 
  

  

  + 
  cos^ 
  6 
  + 
  3 
  cos 
  

  

  tan-l^ 
  

   Jo 
  

  

  Now 
  if 
  the 
  ripples 
  are 
  wide, 
  as 
  they 
  are 
  when 
  wide 
  tubes 
  

   are 
  used, 
  we 
  can 
  have 
  conditions 
  so 
  that 
  b 
  is 
  large 
  compared 
  

  

  with 
  a, 
  and 
  then 
  tan-^- 
  is 
  nearly 
  a 
  right 
  angle. 
  Then 
  we 
  

  

  can 
  neglect 
  cos^ 
  and 
  cos"^^, 
  and 
  Ave 
  find 
  

  

  which 
  we 
  can 
  put 
  approximately 
  

  

  2n7r(rU^w^ 
  

  

  tau-l 
  

  

  We 
  will 
  now 
  consider 
  three 
  consecutive 
  ripples 
  Aj, 
  Ag, 
  Aj 
  

  

  Fig. 
  4. 
  

  

  ^, 
  ^2 
  ^3 
  

  

  '^'""ef 
  

  

  (fig. 
  4) 
  so 
  arranged 
  that 
  Aj 
  is 
  distant 
  h 
  from 
  the 
  antinode, 
  A2 
  

   is 
  distant 
  a^^ 
  from 
  Ag, 
  and 
  A3, 
  ^23 
  from 
  Ag. 
  

   The 
  values 
  of 
  n 
  and 
  lu 
  at 
  Ai 
  are 
  n^ 
  and 
  u'l, 
  

  

  " 
  ?5 
  )^ 
  A3 
  ,, 
  ??3 
  ,, 
  li?3. 
  

  

  Then 
  the 
  forces 
  exerted 
  by 
  the 
  ripples 
  Ai 
  and 
  A3 
  on 
  a 
  

  

  