﻿202 
  Bucher 
  — 
  Origin 
  of 
  Ripples, 
  and 
  

  

  when 
  established, 
  offers 
  a 
  minimum 
  of 
  friction. 
  Whether 
  

   the 
  surrounding 
  surface 
  be 
  flat 
  or 
  consists 
  itself 
  of 
  inde- 
  

   pendent 
  dunes, 
  all 
  fortuitous 
  changes 
  in 
  the 
  distribution 
  

   of 
  sand 
  must 
  gradually 
  lead 
  towards 
  the 
  formation 
  of 
  

   this 
  optimum 
  surface. 
  In 
  regions 
  with 
  uniform 
  winds 
  

   this 
  process 
  will 
  finally 
  produce 
  a 
  series 
  of 
  transverse 
  

   dunes. 
  That 
  it 
  will 
  create 
  new 
  dunes 
  on 
  level 
  sand 
  sur- 
  

   faces 
  is 
  shown 
  by 
  King's 
  observations 
  in 
  the 
  vicinity 
  of 
  

   Cairo. 
  69 
  Since 
  the 
  minute 
  shiftings 
  of 
  grain 
  after 
  grain 
  

   on 
  the 
  sand 
  surface 
  are 
  of 
  the 
  same 
  order 
  of 
  magnitude 
  

   on 
  the 
  dunes 
  as 
  on 
  the 
  ripples, 
  the 
  building 
  up 
  of 
  a 
  series 
  

   of 
  transverse 
  dunes 
  requires 
  correspondingly 
  more 
  time, 
  

   measured 
  by 
  years 
  instead 
  of 
  minutes. 
  

  

  Turning 
  back 
  to 
  our 
  general 
  discussion, 
  we 
  may 
  express 
  

   the 
  principle 
  underlying 
  the 
  formation 
  of 
  ripples 
  in 
  a 
  

   broader 
  way 
  still. 
  The 
  sinuous 
  surface 
  differs 
  from 
  a 
  

   level 
  one 
  in 
  substituting 
  a 
  rhythmic 
  variation 
  of 
  velocity 
  

   for 
  uniform 
  flow. 
  Since 
  my 
  attention 
  was 
  first 
  called 
  to 
  

   this 
  phase 
  of 
  the 
  question, 
  I 
  have 
  tried 
  on 
  many 
  occasions 
  

   to 
  find 
  an 
  indication 
  of 
  a 
  preexisting 
  rhythm 
  in 
  the 
  motion 
  

   of 
  wind 
  or 
  water 
  passing 
  over 
  an 
  immovable 
  surface. 
  I 
  

   watched 
  pieces 
  of 
  cork 
  blown 
  over 
  the 
  pavement 
  by 
  wind, 
  

   or 
  small 
  stones 
  or 
  sandgrains 
  rolled 
  by 
  water 
  under 
  a 
  

   variety 
  of 
  circumstances 
  without 
  being 
  ever 
  able 
  to 
  dis- 
  

   cover 
  any 
  indication 
  whatsoever 
  of 
  a 
  rhythmic 
  motion 
  

   of 
  these 
  bodies. 
  I, 
  therefore, 
  consider 
  the 
  rhythmic 
  flow 
  

   of 
  water 
  as 
  the 
  result, 
  not 
  as 
  the 
  cause 
  of 
  the 
  formation 
  

   of 
  ripples. 
  70 
  

  

  Analogous 
  undulating 
  contact 
  surfaces 
  of 
  least 
  resist- 
  

   ance 
  originating 
  under 
  very 
  different 
  conditions 
  are 
  not 
  

   uncommon 
  in 
  nature, 
  some 
  even 
  of 
  every 
  day 
  occurrence. 
  

  

  At 
  the 
  contract 
  of 
  air 
  and 
  water, 
  the 
  prototype 
  of 
  all 
  

   waves 
  has 
  given 
  the 
  clue 
  to 
  this 
  interpretation. 
  71 
  Layers 
  

   of 
  air 
  of 
  different 
  velocity 
  and 
  density 
  give 
  rise 
  to 
  air- 
  

   waves 
  72 
  of 
  large 
  dimensions, 
  often 
  rendered 
  visible 
  

   through 
  condensation 
  of 
  moisture 
  (mackerel 
  sky). 
  The 
  

  

  69 
  It 
  should 
  be 
  noted, 
  that 
  in 
  his 
  experiments, 
  described 
  in 
  the 
  same 
  paper, 
  

   there 
  was 
  no 
  tl 
  growing 
  up" 
  of 
  small 
  current-ripples, 
  but 
  the 
  larger 
  

   ' 
  ' 
  dunes, 
  ' 
  ' 
  caused 
  by 
  the 
  obstacle, 
  were 
  superimposed 
  on 
  the 
  rippled 
  surface 
  

   and 
  their 
  size 
  and 
  wave-length 
  was 
  determined 
  entirely 
  by 
  the 
  pre-existing 
  

   obstacle. 
  

  

  Baschin, 
  e. 
  g., 
  assumes 
  such 
  pre-existing 
  periodic 
  changes 
  of 
  the 
  pressure 
  

   of 
  wind 
  on 
  rigid 
  surfaces, 
  such 
  as 
  asphalt 
  pavements 
  (1899, 
  pp. 
  419-420). 
  

  

  71 
  A 
  brief 
  abstract 
  of 
  the 
  modern 
  theory 
  of 
  water 
  waves 
  based 
  on 
  Helm- 
  

   holtz's 
  work 
  already 
  referred 
  to, 
  is 
  found 
  in 
  Kruemmel, 
  O., 
  Handbuch 
  der 
  

   Ozeanographie, 
  vol. 
  2, 
  pp. 
  61 
  ff. 
  

  

  72 
  For 
  a 
  brief 
  general 
  discussion 
  see, 
  e. 
  g., 
  Baschin, 
  1899. 
  

  

  