﻿

  120 
  .,;.'.. 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  Let 
  us 
  now 
  plot 
  a 
  number 
  of 
  resonance 
  curves, 
  ench 
  of 
  

   the 
  type 
  shown 
  in 
  fig. 
  1, 
  but 
  with 
  their 
  central 
  ordinates 
  

   separated 
  and 
  distributed 
  according 
  to 
  the 
  natural 
  pitches 
  of 
  

   the 
  responding 
  system, 
  to 
  which 
  the 
  various 
  curves 
  corre- 
  

   spond. 
  As 
  previously 
  mentioned, 
  the 
  only 
  way 
  to 
  plot 
  

   musical 
  intervals 
  consistently 
  to 
  scale 
  is 
  to 
  measure 
  those 
  

   intervals 
  by 
  the 
  logarithms 
  of 
  their 
  frequency 
  ratios. 
  And 
  

   this 
  method 
  will 
  serve 
  for 
  the 
  abscissas 
  of 
  each 
  of 
  our 
  

   resonance 
  curves 
  as 
  well 
  as 
  for 
  their 
  distance 
  apart. 
  For, 
  

   as 
  we 
  saw 
  in 
  equations 
  (16) 
  and 
  (16 
  a), 
  the 
  mistiming 
  which 
  

   is 
  used 
  as 
  the 
  abscissa 
  is 
  nearly 
  proportional 
  to 
  the 
  logarithm 
  

   of 
  the 
  ratio 
  of 
  the 
  frequencies 
  characterizing 
  the 
  force 
  and 
  

   the 
  responding 
  system. 
  

  

  • 
  Fig. 
  3. 
  — 
  Harmonic 
  Series 
  of 
  Resonance 
  Curves. 
  

  

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  c 
  

  

  

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  '& 
  

  

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  2a 
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  iO 
  32. 
  

  

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  UP 
  fine/At 
  /»£*«- 
  

  

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  a/yd 
  M 
  I 
  STi//V//VfiS 
  

  

  

  

  

  

  In 
  laying 
  out 
  these 
  graphs, 
  we 
  have 
  again 
  supposed 
  that 
  

   the 
  damping 
  coefficient 
  k 
  is 
  constant 
  throughout. 
  The 
  

   pitches 
  of 
  best 
  resonance 
  for 
  the 
  several 
  tones 
  are 
  taken 
  in 
  

   the 
  harmonic 
  series 
  as 
  for 
  a 
  string 
  or 
  open 
  parallel 
  pipe. 
  

   The 
  corresponding 
  ranges 
  then 
  follow 
  inversely 
  as 
  those 
  

   frequencies, 
  the 
  values 
  of 
  the 
  sharpness 
  being 
  directly 
  as 
  

   the 
  frequencies. 
  Thus, 
  the 
  responses 
  are 
  in 
  each 
  case 
  fullest 
  

   at 
  the 
  precise 
  frequencies 
  natural 
  to 
  the 
  supposed 
  system, 
  

   and 
  fall 
  off 
  when 
  the 
  frequency 
  of 
  the 
  forcing 
  is 
  too 
  high 
  or 
  

   too 
  low. 
  But 
  the 
  degree 
  of 
  falling 
  off 
  for 
  a 
  given 
  mistuning 
  

   varies 
  with 
  the 
  natural 
  pitch 
  of 
  resonance, 
  being 
  small 
  for 
  

   the 
  lower 
  tones 
  and 
  larger 
  for 
  the 
  higher 
  ones. 
  In 
  other 
  

   words, 
  the 
  range 
  is 
  extensive 
  for 
  the 
  low 
  tone 
  , 
  but 
  the 
  

   sharpness 
  is 
  striking 
  for 
  the 
  high 
  ones. 
  Here 
  again 
  the 
  

   values 
  of 
  the 
  sharpness 
  of 
  resonance 
  are, 
  as 
  in 
  fig. 
  1, 
  

  

  10, 
  20, 
  30, 
  40, 
  50, 
  and 
  60. 
  

  

  Range 
  of 
  Resonance 
  nearly 
  proportional 
  to 
  Decrement. 
  — 
  

   By 
  equation 
  (7) 
  we 
  see 
  that 
  the 
  vibrations 
  natural 
  to 
  the 
  

   responding 
  system 
  are 
  proportional 
  to 
  

  

  e~ 
  kt 
  ' 
  2 
  sin 
  qt 
  ) 
  

  

  1 
  \ 
  (24) 
  

  

  where 
  ( 
  ] 
  = 
  }> 
  s/l~—k"lkp 
  2 
  ) 
  

  

  