﻿[ 
  i«o 
  ] 
  

  

  XIX. 
  Note 
  on 
  Konig's 
  Theory 
  of 
  the 
  Ripple 
  Formation 
  in 
  

   Kundt's 
  Tube 
  Experiment. 
  By 
  J. 
  Robinson, 
  B.Sc, 
  

   Pemberton 
  Fellow 
  of 
  the 
  Armstrong 
  College^ 
  University 
  of 
  

   Durham*. 
  

  

  EXPERIMENTAL 
  evidence 
  f 
  has 
  been 
  given 
  by 
  different 
  

   investigators 
  in 
  support 
  o£ 
  Konig's 
  theory 
  of 
  the 
  ripple 
  

   formation 
  in 
  a 
  Kundt's 
  tube. 
  A 
  point 
  will 
  here 
  be 
  brought 
  

   forward 
  which 
  at 
  all 
  events 
  is 
  not 
  in 
  opposition 
  to 
  this 
  

   theory. 
  

  

  Konig 
  X 
  showed 
  that 
  when 
  two 
  spheres 
  of 
  the 
  same 
  

   diameter 
  are 
  in 
  a 
  tube 
  in 
  which 
  stationary 
  sound-waves 
  are 
  

   set 
  up, 
  there 
  is 
  a 
  force 
  of 
  repulsion 
  between 
  them 
  in 
  the 
  

   direction 
  of 
  the 
  axis 
  of 
  the 
  tube 
  of 
  the 
  magnitude, 
  

  

  — 
  ^ 
  ^ 
  — 
  ^cos 
  ^ 
  (3 
  — 
  5 
  cos^ 
  6)y 
  

  

  ■where 
  cr 
  = 
  density 
  of 
  the 
  gas 
  in 
  the 
  tube, 
  

   R 
  = 
  radius 
  of 
  each 
  sphere, 
  

  

  ro 
  = 
  distance 
  between 
  the 
  centres 
  of 
  the 
  spheres, 
  

   ^ 
  = 
  angle 
  between 
  the 
  line 
  joining 
  the 
  centres 
  of 
  the 
  

   spheres 
  and 
  the 
  axis 
  of 
  the 
  tube, 
  

   WQ=-27rvao, 
  where 
  rr^ 
  is 
  the 
  amplitude 
  of 
  vibration 
  of 
  

   one 
  of 
  the 
  spheres, 
  and 
  v 
  is 
  the 
  frequency 
  of 
  

   the 
  sound-wave. 
  

  

  Now 
  this 
  force 
  of 
  repulsion 
  between 
  the 
  spheres 
  depends 
  

   on 
  Wq^ 
  which 
  varies 
  along 
  the 
  tube 
  when 
  stationary 
  waves 
  

   are 
  set 
  up. 
  If 
  we 
  consider 
  that 
  the 
  amplitude 
  is 
  aQ 
  at 
  an 
  

   antinode, 
  it 
  is 
  zero 
  at 
  a 
  node, 
  and 
  at 
  a 
  distance 
  x 
  from 
  the 
  

   antinode 
  it 
  is 
  

  

  TT 
  X 
  

  

  a 
  = 
  «ocos 
  — 
  .J 
  , 
  

  

  where 
  2Z 
  is 
  the 
  distance 
  between 
  two 
  consecutive 
  nodes. 
  

  

  Therefore 
  in 
  the 
  expression 
  for 
  the 
  force 
  exerted 
  by 
  one 
  

   sphere 
  on 
  the 
  other, 
  we 
  must 
  substitute 
  w 
  for 
  Wq 
  where 
  

  

  w 
  = 
  ivq 
  cos 
  n" 
  • 
  7 
  > 
  

  

  ioq 
  being 
  the 
  value 
  of 
  27rva 
  at 
  the 
  antinode^ 
  and 
  w 
  its 
  value 
  

   at 
  a 
  distance 
  a; 
  from 
  it. 
  

  

  * 
  Commuuicated 
  bv 
  Prof. 
  H. 
  Stroud, 
  M.A., 
  D.Sc. 
  

   t 
  VV. 
  Koni-, 
  Wied. 
  Ann. 
  xlii. 
  pp. 
  363, 
  549 
  (1891) 
  ; 
  S. 
  R 
  Cook, 
  

   Phil. 
  Mag. 
  May 
  1902, 
  p. 
  471 
  ; 
  J. 
  Robinson, 
  Fhi/s 
  Zeit. 
  1908, 
  p. 
  809. 
  

   + 
  W. 
  Konig, 
  loc. 
  cit. 
  

  

  