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  XVIII. 
  Acoustical 
  JS^otes.—VIU. 
  

  

  Partial 
  Tones 
  of 
  Stretched 
  Strings 
  of 
  Variable 
  Density 
  — 
  Maintenance 
  

   of 
  Vibrations 
  by 
  Impact 
  of 
  Water 
  Drops 
  — 
  Discrimination 
  between 
  

   Sounds 
  from 
  directly 
  in 
  front 
  and 
  directly 
  behind 
  the 
  Observer 
  — 
  The 
  

   Acousticon 
  — 
  Pitch 
  of 
  Sibilants 
  — 
  Telephones. 
  

  

  By 
  Lord 
  Rayleigh, 
  O.M., 
  Pres.R.S* 
  

  

  Partial 
  Tones 
  of 
  Stretched 
  Strings 
  of 
  Variable 
  Density. 
  

  

  IF 
  p 
  be 
  the 
  longitudinal 
  density 
  of 
  a 
  string 
  o£ 
  length 
  I 
  

   stretched 
  with 
  tension 
  T 
  1? 
  the 
  periodic 
  time 
  (7>) 
  or! 
  the 
  

   rth 
  component 
  vibration 
  is 
  given 
  by 
  

  

  T 
  r 
  ~^I\ 
  (0) 
  

  

  In 
  w 
  Theory 
  of 
  Sound,' 
  2nd 
  ed. 
  § 
  140, 
  an 
  approximate 
  

   theory 
  is 
  given 
  applicable 
  when 
  the 
  density, 
  in 
  place 
  of 
  

   being 
  strictly 
  uniform, 
  has 
  the 
  value 
  p 
  + 
  Sp 
  where 
  Bp 
  is 
  

   relatively 
  small. 
  We 
  have 
  

  

  " 
  If 
  the 
  irregularity 
  take 
  the 
  form 
  of 
  a 
  small 
  load 
  of 
  mass 
  

   m 
  at 
  the 
  point 
  x 
  — 
  b^ 
  the 
  formula 
  may 
  be 
  written 
  

  

  These 
  values 
  of 
  r 
  r 
  are 
  correct 
  so 
  far 
  as 
  the 
  first 
  power 
  of 
  

   the 
  small 
  quantities 
  Sp 
  and 
  ??i, 
  and 
  give 
  the 
  means 
  of 
  calcu- 
  

   lating 
  a 
  correction 
  for 
  such 
  slight 
  departures 
  from 
  uniformity 
  

   as 
  must 
  always 
  occur 
  in 
  practice. 
  

  

  As 
  might 
  be 
  expected, 
  the 
  effect 
  of 
  a 
  small 
  load 
  vanishes 
  

   at 
  nodes, 
  and 
  rises 
  to 
  a 
  maximum 
  at 
  the 
  points 
  midway 
  

   between 
  consecutive 
  nodes. 
  When 
  it 
  is 
  desired 
  merely 
  to 
  

   make 
  a 
  rough 
  estimate 
  of 
  the 
  effective 
  density 
  of 
  a 
  nearly 
  

   uniform 
  string, 
  the 
  formula 
  indicates 
  that 
  attention 
  is 
  to 
  be 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  "J" 
  The 
  r 
  2 
  in 
  the 
  denominator 
  was 
  inadvertently 
  omitted. 
  It 
  may 
  be 
  

   worthy 
  of 
  note 
  that 
  the 
  method 
  is 
  not 
  limited 
  to 
  the 
  case 
  of 
  density 
  

   originally 
  uniform. 
  In 
  general, 
  if 
  Y 
  be 
  the 
  normal 
  function 
  corresponding 
  

   to 
  a 
  variable 
  density 
  p, 
  r 
  2 
  is 
  altered 
  in 
  the 
  ratio 
  

  

  h 
  P 
  Y 
  2 
  dx 
  

  

  1+ 
  C 
  

  

  ) 
  P 
  Y*dx 
  

  

  R2 
  

  

  _4/ 
  

  

  