﻿ITS 
  Bucher 
  — 
  Origin 
  of 
  Ripples, 
  and 
  

  

  waves, 
  as 
  I 
  have 
  frequently 
  observed, 
  does 
  not 
  require 
  

   that 
  the 
  water 
  surface 
  should 
  be 
  previously 
  in 
  waves. 
  

   Therefore, 
  I 
  infer 
  that 
  in 
  deep 
  streams 
  sand- 
  waves 
  will 
  be 
  

   similarly 
  produced 
  when 
  the 
  velocity 
  reaches 
  the 
  point 
  at 
  

   which 
  the 
  lower 
  strata 
  of 
  water 
  become 
  heavily 
  charged 
  

   with 
  sand 
  in 
  eddying 
  suspension" 
  (1901, 
  p. 
  198). 
  

  

  The 
  mode 
  of 
  progression 
  which 
  is 
  to 
  be 
  expected 
  in 
  

   deeper 
  water 
  may 
  be 
  explained 
  with 
  the 
  help 
  of 
  fig. 
  5> 
  

   The 
  quantity 
  of 
  water 
  which, 
  in 
  the 
  unit 
  of 
  time, 
  passes 
  

   the 
  cross-section 
  A-B 
  between 
  the 
  level 
  of 
  uniform 
  flow 
  

   (iv-iv) 
  and 
  the 
  sand 
  surface, 
  (s-s) 
  is 
  represented 
  by 
  the 
  

   area 
  abed. 
  The 
  same 
  quantity, 
  when 
  passing 
  the 
  cross- 
  

   section 
  A'-B', 
  is 
  greatly 
  reduced 
  in 
  height 
  and 
  therefore 
  

   must 
  increase 
  its 
  horizontal 
  dimensions 
  (a'b'c'd'). 
  Since 
  

   it 
  has 
  to 
  pass 
  in 
  the 
  unit 
  of 
  time, 
  this 
  means 
  an 
  increase 
  

   of 
  velocity. 
  

  

  The 
  velocity, 
  therefore, 
  is 
  increased 
  from 
  the 
  trough 
  

   to 
  the 
  crest 
  and 
  decreased 
  from 
  crest 
  to 
  trough. 
  As 
  with 
  

   decreasing 
  velocity 
  sediment 
  is 
  thrown 
  down, 
  the 
  sand- 
  

   wave 
  must 
  move 
  down-stream. 
  It 
  seems 
  highly 
  probable 
  

   that 
  this 
  is 
  the 
  mode 
  of 
  progression 
  of 
  the 
  sand-waves 
  of 
  

   large 
  streams. 
  

  

  The 
  paroxysmal 
  appearance 
  and 
  disappearance 
  and 
  

   the 
  rhythmic 
  fluctuation 
  in 
  activity 
  of 
  the 
  regressional 
  

   sand-waves, 
  bear 
  all 
  characteristics 
  of 
  phenomena 
  of 
  

   interference 
  of 
  two 
  systems 
  of 
  waves, 
  the 
  sand-waves 
  

   below 
  and 
  some 
  system 
  of 
  water-waves 
  above 
  (Gilbert, 
  

   1914, 
  p. 
  244) 
  and 
  need 
  not 
  to 
  be 
  discussed 
  any 
  further 
  

   here. 
  

  

  We 
  may, 
  therefore, 
  describe 
  the 
  formation 
  of 
  sand- 
  

   waves 
  as 
  the 
  establishment 
  of 
  a 
  new 
  surface 
  of 
  contact, 
  

   offering 
  an 
  optimum 
  of 
  conditions 
  of 
  flow, 
  probably 
  a 
  

   minimum 
  of 
  friction. 
  As 
  above 
  a 
  certain 
  velocity 
  (first 
  

   critical 
  point) 
  the 
  smooth 
  surface 
  of 
  the 
  sediment 
  at 
  rest 
  

   proves 
  unstable, 
  so 
  here 
  above 
  a 
  certain 
  higher 
  velocity 
  

   (third 
  critical 
  point) 
  the 
  smooth 
  surface 
  of 
  the 
  sediment 
  

   in 
  motion 
  again 
  proves 
  unstable. 
  This- 
  time 
  the 
  sedi- 
  

   ment 
  yields 
  as 
  a 
  whole, 
  not 
  grain 
  by 
  grain, 
  as 
  in 
  the 
  case 
  

   of 
  current-ripples. 
  The 
  sediment 
  mixed 
  with 
  water, 
  

   over 
  which 
  the 
  water 
  mixed 
  with 
  sediment 
  flows, 
  is 
  itself 
  

   in 
  constant 
  motion 
  to 
  a 
  considerable 
  depth 
  ; 
  it 
  flows 
  too. 
  

   It 
  behaves 
  like 
  a 
  viscous 
  liquid. 
  If 
  we 
  consider 
  it 
  as 
  

   such, 
  sand-waves 
  may 
  be 
  explained 
  as 
  friction-waves 
  

   between 
  two 
  liquids 
  of 
  different 
  velocity. 
  

  

  