﻿160 
  Bucher 
  — 
  Origin 
  of 
  Ripples, 
  and 
  

  

  lines 
  must 
  represent 
  an 
  optimum 
  of 
  conditions 
  of 
  flow, 
  

   probably 
  a 
  minimum 
  of 
  friction, 
  as 
  any 
  form 
  of 
  contact 
  

   between 
  the 
  bed 
  and 
  the 
  liquid 
  other 
  than 
  that 
  of 
  current- 
  

   ripples 
  proves 
  to 
  be 
  unstable, 
  and 
  always 
  tends 
  toward 
  

   the 
  establishment 
  of 
  the 
  latter. 
  

  

  That 
  we 
  are, 
  in 
  all 
  probability, 
  dealing" 
  with 
  a 
  minimum 
  

   problem, 
  is 
  also 
  indicated 
  by 
  the 
  fact 
  that, 
  under 
  condi- 
  

   tions 
  not 
  yet 
  specified, 
  instead 
  of 
  the 
  normal 
  parallel, 
  

   straight 
  ripples, 
  rhomboid 
  ripples 
  make 
  their 
  appear- 
  

   ance, 
  offering 
  a 
  pattern 
  which 
  recurs 
  frequently 
  in 
  the 
  

   solution 
  of 
  minimum 
  problems 
  (cf. 
  Engels, 
  p. 
  675). 
  

  

  That 
  the 
  ripple 
  form 
  really 
  offers 
  a 
  mechanical 
  advan- 
  

   tage 
  over 
  the 
  even 
  surface 
  is 
  directly 
  shown 
  by 
  Gilbert's 
  

   observation, 
  that 
  transportation 
  reaches 
  a 
  maximum 
  with 
  

   the 
  definite 
  establishment 
  of 
  ripples. 
  

  

  The 
  formation 
  of 
  current-ripples 
  must 
  be 
  due 
  to 
  the 
  

   fact 
  that 
  any 
  accidental 
  shifting 
  of 
  sandgrains 
  into 
  a 
  

   position 
  favorable 
  to 
  the 
  establishment 
  of 
  vortices 
  of 
  

   suitable 
  size 
  and 
  spacing, 
  is 
  liable 
  to 
  be 
  permanent, 
  while 
  

   any 
  unfavorable 
  changes 
  put 
  the 
  sandgrains 
  into 
  unstable 
  

   positions 
  from 
  which 
  they 
  will 
  be 
  moved 
  again. 
  

  

  The 
  process 
  is 
  initiated 
  by 
  the 
  small 
  vortices 
  setting 
  

   up 
  in 
  the 
  small 
  depressions 
  left 
  by 
  such 
  grains 
  as, 
  owing 
  

   to 
  their 
  size, 
  shape, 
  position, 
  or 
  other 
  cause, 
  are 
  the 
  first 
  

   to 
  be 
  moved 
  (King. 
  p. 
  204). 
  As 
  the 
  small 
  vortices 
  

   coalesce 
  and 
  rapidly 
  grow 
  up 
  to 
  their 
  final 
  size, 
  an 
  equi- 
  

   librium 
  is 
  established 
  between 
  the 
  currents 
  of 
  different 
  

   velocities 
  within 
  and 
  without 
  the 
  vortical 
  layer 
  which 
  

   separates 
  the 
  sediment 
  below 
  and 
  the 
  lowest 
  level 
  of 
  

   uniform 
  flow 
  above. 
  

  

  The 
  comparison 
  of 
  these 
  vortices 
  with 
  ' 
  ' 
  a 
  kind 
  of 
  ball- 
  

   bearing 
  to 
  allow 
  the 
  upper 
  and 
  more 
  rapidly 
  moving 
  

   layer 
  ... 
  to 
  pass 
  along 
  more 
  easily" 
  is 
  quite 
  appro- 
  

   priate 
  (King. 
  p. 
  205). 
  

  

  The 
  distribution 
  of 
  velocities 
  necessary 
  for 
  this 
  equi- 
  

   librium 
  alone 
  determines 
  the 
  size 
  and 
  spacing 
  of 
  the 
  

   vortices, 
  and 
  this, 
  in 
  turn, 
  the 
  size 
  and 
  shape 
  of 
  the 
  cur- 
  

   rent-ripples. 
  

  

  The 
  distribution 
  of 
  velocities 
  is 
  determined 
  by: 
  (a) 
  

   the 
  velocities 
  existing 
  at 
  the 
  upper 
  and 
  lower 
  boundary 
  

   of 
  the 
  vortical 
  layer, 
  and 
  (b) 
  the 
  viscosity 
  of 
  the 
  liquid. 
  

  

  The 
  velocity 
  at 
  the 
  upper 
  boundary 
  of 
  the 
  vortical 
  

   layer 
  is 
  a 
  function 
  of 
  the 
  mean 
  velocity 
  of 
  the 
  stream. 
  

   It 
  probably 
  affects 
  chiefly 
  the 
  height 
  of 
  the 
  vortices. 
  An 
  

  

  