﻿Related 
  Sedimentary 
  Surface 
  Forms. 
  179 
  

  

  In 
  1889, 
  Helmholtz 
  gave 
  the 
  mathematical 
  evidence 
  31 
  

   that 
  in 
  case 
  of 
  two 
  liquids 
  passing 
  each 
  other 
  with 
  differ- 
  

   ent 
  velocities, 
  the 
  flat 
  surface 
  represents 
  an 
  unstable 
  

   equilibrium, 
  as 
  the 
  amount 
  of 
  energy 
  used 
  up 
  along 
  the 
  

   contact 
  is 
  greater 
  with 
  a 
  flat 
  than 
  with 
  certain 
  wave-like 
  

   surfaces. 
  He 
  at 
  once 
  applied 
  this 
  theory 
  to 
  water-waves 
  

   and 
  to 
  waves 
  within 
  the 
  atmosphere, 
  the 
  existence 
  of 
  

   which 
  he 
  had 
  inferred 
  from 
  observations 
  which 
  were 
  the 
  

   immediate 
  cause 
  of 
  these 
  investigations. 
  32 
  

  

  Of 
  the 
  laws 
  derived 
  from 
  his 
  formulas, 
  the 
  following 
  

   are 
  of 
  importance 
  for 
  the 
  application 
  of 
  this 
  theory 
  to 
  

   sand- 
  waves 
  : 
  

  

  (1.) 
  The 
  linear 
  dimensions 
  of 
  such 
  waves 
  increase 
  as 
  

   the 
  square 
  of 
  the 
  velocities 
  of 
  the 
  two 
  media; 
  that 
  is, 
  

   with 
  doubled 
  velocity 
  of 
  wind, 
  water 
  waves 
  will 
  increase 
  

   four 
  times 
  their 
  original 
  size. 
  

  

  (2.) 
  The 
  smaller 
  the 
  difference 
  between 
  the 
  densities 
  

   of 
  the 
  two 
  liquids, 
  the 
  greater 
  is 
  the 
  wave-length. 
  For 
  

   instance, 
  a 
  wind 
  velocity 
  of 
  10 
  miles 
  per 
  second 
  would 
  

   at 
  0° 
  C. 
  produce 
  water 
  waves 
  of 
  0-8 
  m. 
  length, 
  while 
  

   between 
  two 
  layers 
  of 
  air 
  at 
  0° 
  and 
  10°, 
  it 
  would 
  form 
  

   waves 
  of 
  900 
  m. 
  length. 
  33 
  Water 
  waves 
  of 
  5-10 
  m. 
  length 
  

   would 
  correspond 
  to 
  air 
  waves 
  of 
  15-30 
  km. 
  length. 
  

  

  In 
  view 
  of 
  the 
  incompleteness 
  of 
  the 
  data 
  on 
  hand 
  and 
  

   of 
  the 
  great 
  complexity 
  of 
  the 
  process 
  in 
  rivers, 
  we 
  can 
  

   not 
  hope 
  at 
  present 
  to 
  use 
  these 
  deductions 
  to 
  test 
  this 
  

   interpretation 
  of 
  the 
  origin 
  of 
  sand-waves. 
  

  

  At 
  best 
  we 
  might 
  see 
  if 
  the 
  order 
  of 
  magnitude 
  of 
  the 
  

   waves 
  observed 
  in 
  nature 
  is 
  consistent 
  with 
  it. 
  

  

  At 
  high 
  water 
  stage, 
  Hider 
  found 
  near 
  Lake 
  Provi- 
  

   dence 
  (La.), 
  on 
  the 
  up-stream 
  side 
  of 
  a 
  sandbar, 
  sand- 
  

   waves 
  of 
  13 
  m. 
  length 
  formed 
  at 
  a 
  depth 
  of 
  11 
  m. 
  ; 
  1600 
  

   ft. 
  across 
  the 
  channel, 
  near 
  the 
  line 
  of 
  the 
  swiftest 
  cur- 
  

   rent, 
  the 
  sand-waves 
  measured 
  228 
  m. 
  from 
  crest 
  to 
  crest 
  

   at 
  a 
  depth 
  of 
  27 
  m. 
  Unfortunately 
  no 
  measurements 
  of 
  

   discharge 
  or 
  velocity 
  accompany 
  his 
  report. 
  In 
  order 
  

   to 
  obtain 
  some 
  idea 
  concerning 
  the 
  range 
  of 
  velocities 
  

   found 
  in 
  the 
  same 
  river 
  at 
  different 
  depths 
  and 
  stages, 
  

  

  S1 
  Wissenschaf 
  tliche 
  Abhandlungen, 
  vol. 
  3, 
  pp. 
  315-322, 
  1889; 
  cf. 
  Rey- 
  

   nold's 
  experiments, 
  Phil. 
  Trans. 
  Roy. 
  Soc, 
  London, 
  vol. 
  174, 
  pp. 
  943-944, 
  

   1883 
  

  

  32 
  1886, 
  ibid., 
  vol. 
  3, 
  p. 
  287. 
  

   33 
  d. 
  for 
  air 
  at 
  0° 
  C. 
  —0001293. 
  

   " 
  " 
  tl 
  10° 
  C. 
  —0-001247. 
  

   " 
  water 
  " 
  0° 
  C. 
  —1000. 
  

  

  