﻿Ripple 
  Formatloti 
  in 
  Kandt's 
  Tuhe 
  Experiment. 
  181 
  

  

  A 
  consequence 
  of 
  this 
  would 
  be 
  that 
  for 
  equilibrium 
  the 
  

   arrangement 
  of 
  the 
  particles 
  is 
  not 
  the 
  same 
  at 
  all 
  parts 
  of 
  

   the 
  tube 
  when 
  a 
  stationary 
  sound-wave 
  is 
  set 
  up. 
  We 
  may 
  

   find 
  how 
  the 
  distance 
  between 
  the 
  ripples 
  varies 
  at 
  different 
  

   parts 
  of 
  the 
  tube 
  in 
  the 
  following 
  way. 
  

  

  For 
  the 
  equilibrium 
  of 
  a 
  particle 
  in 
  one 
  ripple, 
  the 
  forces 
  

   of 
  repulsion 
  acting 
  on 
  it 
  from 
  all 
  the 
  particles 
  in 
  the 
  ripples 
  

   to 
  one 
  side 
  of 
  it, 
  must 
  equal 
  those 
  from 
  the 
  particles 
  to 
  the 
  

   other 
  side 
  of 
  it. 
  Now 
  as 
  the 
  forces 
  considered 
  vary 
  in- 
  

   versely 
  as 
  the 
  fourth 
  power 
  of 
  the 
  distance 
  between 
  the 
  

   particles, 
  we 
  need 
  only 
  consider 
  consecutive 
  ripples, 
  and 
  we 
  

   get 
  an 
  approximate 
  result 
  as 
  follows: 
  — 
  

  

  We 
  consider 
  three 
  consecutive 
  ripples, 
  a, 
  ?>, 
  c 
  (fiig. 
  1), 
  and 
  

   for 
  the 
  equilibrium 
  of 
  the 
  particles 
  in 
  the 
  middle 
  one 
  h, 
  we 
  

   calculate 
  the 
  total 
  force 
  of 
  repulsion 
  exerted 
  by 
  the 
  particles 
  

   in 
  each 
  of 
  the 
  ripples 
  a 
  and 
  c 
  on 
  a 
  particle 
  in 
  h^ 
  and 
  equate 
  

   these 
  forces. 
  

  

  Fig. 
  1. 
  

  

  ct 
  b 
  c 
  

  

  Fig. 
  2. 
  

  

  The 
  cross-section 
  of 
  a 
  ripple 
  perpendicular 
  to 
  the 
  axis 
  of 
  

   the 
  tube 
  is 
  a 
  segment 
  of 
  a 
  circle 
  (fig. 
  2). 
  We 
  get 
  an 
  approxi- 
  

   mation 
  to 
  the 
  result 
  if 
  we 
  consider 
  the 
  particles 
  in 
  this 
  cross- 
  

   section 
  as 
  uniformly 
  distributed 
  over 
  a 
  rectangle 
  whose 
  length 
  

   is 
  the 
  same 
  as 
  that 
  of 
  the 
  chord. 
  This 
  simplifies 
  some 
  inte- 
  

   grations 
  which 
  follow. 
  

  

  Now 
  consider 
  the 
  total 
  force 
  of 
  repulsion 
  exerted 
  by 
  

   one 
  whole 
  ripple 
  A 
  B 
  of 
  length 
  26 
  on 
  one 
  particle 
  at 
  0, 
  

   at 
  the 
  centre 
  of 
  the 
  next 
  ripple 
  (fig. 
  3, 
  p. 
  182). 
  Let 
  the 
  

   distance 
  apart 
  of 
  the 
  ripples 
  =a. 
  In 
  the 
  ripple 
  A 
  B, 
  suppose 
  

   there 
  are 
  n 
  particles 
  per 
  unit 
  length, 
  all 
  the 
  dust 
  particles 
  

   in 
  the 
  tube 
  being 
  supposed 
  to 
  be 
  of 
  the 
  same 
  size, 
  L 
  e., 
  

   of 
  radius 
  R. 
  These 
  particles 
  are 
  of 
  course 
  supposed 
  to 
  be 
  

   small. 
  

  

  