﻿236 
  Lord 
  Rayleigh 
  : 
  Acoustical 
  Jotes. 
  

  

  given 
  to 
  the 
  neighbourhood 
  oi* 
  loops 
  rather 
  than 
  to 
  that 
  of: 
  

   nodes. 
  

  

  The 
  effect 
  of 
  a 
  small 
  variation 
  of 
  density 
  upon 
  the 
  period 
  

   is 
  the 
  same 
  whether 
  it 
  occur 
  at 
  a 
  distance 
  x 
  from 
  one 
  end 
  

   of 
  the 
  string, 
  or 
  at 
  an 
  equal 
  distance 
  from 
  the 
  other 
  end. 
  

   The 
  mean 
  variation 
  at 
  points 
  equidistant 
  from 
  the 
  centre 
  

   is 
  all 
  that 
  we 
  need 
  regard, 
  and 
  thus 
  no 
  generality 
  will 
  be 
  

   lost 
  if 
  we 
  suppose 
  that 
  the 
  density 
  remains 
  symmetrically 
  

   distributed 
  with 
  respect 
  to 
  the 
  centre. 
  Thus 
  we 
  may 
  write 
  

  

  - 
  U2p 
  \l 
  + 
  u 
  r 
  ), 
  (3) 
  

  

  where 
  

  

  •r 
  - 
  r 
  2 
  Ti 
  

  

  2C 
  ¥ 
  $pfi 
  2*rx\. 
  ... 
  

  

  a 
  r 
  = 
  T 
  \ 
  — 
  ( 
  1 
  -cos— 
  y— 
  ) 
  dx. 
  ... 
  (4) 
  

  

  In 
  this 
  equation 
  hp 
  may 
  be 
  expanded 
  from 
  to 
  ^l 
  in 
  the 
  

   series 
  

  

  So 
  . 
  . 
  %7TX 
  A 
  2irrx 
  /eN 
  

  

  -£ 
  = 
  A 
  + 
  A 
  lC 
  os-r-+ 
  ... 
  +A..COS— 
  p- 
  + 
  ... 
  , 
  (o) 
  

  

  where 
  

  

  A 
  2f* 
  l 
  O>, 
  ^ 
  ±C 
  hl 
  *P 
  Z'*™S 
  fc 
  ,x 
  

  

  A 
  = 
  7 
  I 
  -t-dx, 
  A 
  r 
  = 
  yl 
  ^-cos 
  — 
  j 
  ;— 
  ax. 
  . 
  (6, 
  /) 
  

  

  l 
  Jo 
  Po 
  Uo 
  Po 
  l 
  

  

  Accordingly, 
  

  

  « 
  r 
  = 
  A 
  —£A, 
  (8) 
  

  

  This 
  equation, 
  as 
  it 
  stands, 
  gives 
  the 
  changes 
  in 
  period 
  in 
  

   terms 
  of 
  the 
  changes 
  of 
  density 
  supposed 
  to 
  be 
  known. 
  And 
  

   it 
  shows 
  conversely 
  that 
  a 
  variation 
  of 
  density 
  may 
  always 
  

   be 
  found 
  which 
  will 
  give 
  prescribed 
  arbitrary 
  displacements 
  

   to 
  all 
  the 
  periods. 
  This 
  is 
  a 
  point 
  of 
  some 
  interest. 
  

  

  In 
  order 
  to 
  secure 
  a 
  reasonable 
  continuity 
  in 
  the 
  density, 
  

   it 
  is 
  necessary 
  to 
  suppose 
  that 
  a 
  1? 
  a 
  2 
  » 
  • 
  • 
  • 
  are 
  so 
  prescribed 
  

   that 
  a. 
  r 
  assumes 
  ultimately 
  a 
  constant 
  value 
  when 
  r 
  is 
  in- 
  

   creased 
  indefinitely. 
  If 
  this 
  condition 
  be 
  satisfied, 
  we 
  may 
  

   take 
  A 
  = 
  a 
  oo 
  , 
  and 
  then 
  A 
  r 
  tends 
  to 
  zero 
  as 
  r 
  increases. 
  

  

  As 
  a 
  simple 
  example, 
  suppose 
  that 
  it 
  be 
  required 
  so 
  to 
  

   vary 
  the 
  density 
  of 
  a 
  string 
  that, 
  while 
  the 
  pitch 
  of 
  the 
  

   fundamental 
  tone 
  is 
  displaced, 
  all 
  other 
  tones 
  shall 
  remain 
  

   unaltered. 
  The 
  conditions 
  give 
  

  

  a 
  a 
  = 
  a 
  8 
  = 
  u 
  4 
  = 
  . 
  . 
  . 
  a 
  = 
  0, 
  

   Accordingly, 
  

  

  A 
  = 
  A 
  2 
  = 
  A 
  3 
  = 
  ... 
  =0, 
  Ax=-2ce, 
  

   Thus 
  by 
  (5) 
  

  

  8/>/ 
  /[ 
  , 
  =-2« 
  1 
  cos(27ra?/0 
  (?)" 
  

  

  