﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing, 
  115 
  

  

  When 
  the 
  harmonic 
  impressed 
  forcing 
  is 
  of 
  sustained 
  

   amplitude, 
  we 
  have 
  h 
  = 
  0, 
  and 
  so 
  may 
  obtain 
  from 
  (9) 
  and 
  

   (10) 
  the 
  simplified 
  expressions 
  for 
  the 
  corresponding 
  values 
  

   of 
  the 
  kinetic 
  energy 
  T 
  and 
  response 
  N. 
  The 
  response 
  under 
  

   sustained 
  forcing 
  may 
  accordingly 
  be 
  written 
  

  

  - 
  do 
  

  

  e-# 
  

  

  tf 
  + 
  & 
  

  

  Or, 
  calling 
  the 
  quantity 
  in 
  brackets 
  the 
  mistiming 
  and 
  

   denoting 
  it 
  by 
  M, 
  we 
  obtain 
  the 
  compact 
  expression 
  

  

  where 
  

  

  M 
  =±e-9 
  ™ 
  

  

  It 
  may 
  be 
  noted 
  further 
  that 
  if 
  the 
  forcing 
  is 
  damped 
  in 
  

   such 
  wise 
  that 
  h 
  = 
  k, 
  then 
  again 
  the 
  same 
  simplification 
  of 
  the 
  

   response 
  occurs 
  as 
  if 
  h 
  were 
  zero 
  ! 
  This 
  is 
  seen 
  on 
  reference 
  

   to 
  equation 
  (10). 
  

  

  Again, 
  if 
  the 
  damping 
  expressed 
  by 
  h 
  is 
  known 
  to 
  be 
  very 
  

   small, 
  then 
  equations 
  (11) 
  and 
  (12) 
  may 
  be 
  held 
  as 
  close 
  

   approximations 
  to 
  the 
  phenomena 
  occurring. 
  

  

  Referring 
  then 
  to 
  the 
  value 
  of 
  the 
  response 
  stated 
  in 
  

   equations 
  (12) 
  and 
  (1H) 
  as 
  the 
  expressions 
  with 
  which 
  we 
  are 
  

   mainly 
  concerned, 
  we 
  note 
  at 
  this 
  stage 
  that 
  for 
  any 
  system 
  

   with 
  given 
  constant 
  damping 
  coefficient 
  (&/2) 
  but 
  capable 
  of 
  

   vibrations 
  of 
  various 
  frequencies 
  ( 
  />/27r 
  and 
  multiples 
  of 
  this), 
  

   then 
  for 
  X 
  constant 
  the 
  mistiming 
  M 
  varies 
  inversely 
  as 
  the 
  

   frequency. 
  That 
  is, 
  the 
  mis 
  tunings 
  allowable 
  in 
  eliciting 
  

   the 
  various 
  responses 
  of 
  different 
  frequencies 
  in 
  the 
  harmonic 
  

   series 
  vary 
  inversely 
  as 
  the 
  frequencies 
  of 
  the 
  different 
  tones 
  

   which 
  it 
  is 
  sought 
  to 
  elicit. 
  

  

  This 
  applies 
  to 
  the 
  various 
  possible 
  musical 
  tones 
  of 
  a 
  

   string 
  or 
  pipe, 
  or 
  to 
  the 
  electrical 
  oscillations 
  possible 
  on 
  

   certain 
  circuits. 
  We 
  may 
  write 
  this 
  relation 
  in 
  the 
  form 
  

  

  Mp= 
  constant 
  for 
  both 
  N 
  and 
  k 
  constant. 
  . 
  (14) 
  

  

  It 
  is 
  obvious 
  from 
  (12) 
  that 
  N 
  reaches 
  its 
  maximum 
  value, 
  

   N 
  say, 
  for 
  M=0, 
  when 
  

  

  N 
  =p 
  (15) 
  

  

  Thus, 
  for 
  any 
  given 
  constant 
  value 
  of 
  k, 
  all 
  curves 
  for 
  

   different 
  p's, 
  but 
  with 
  corresponding 
  values 
  of 
  M 
  and 
  N 
  as 
  

  

  12 
  

  

  