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  LIX. 
  On 
  the 
  Wolf-note 
  in 
  Bowed 
  Stringed 
  Instruments. 
  By 
  

   C. 
  V. 
  Raman, 
  M.A., 
  Sir 
  Tarahiath 
  Palit 
  Professor 
  of 
  

   Physics 
  in 
  the 
  Calcutta 
  University*. 
  

  

  1. 
  TN 
  the 
  Phil. 
  Mag. 
  for 
  June 
  1917 
  (page 
  536), 
  Mr. 
  J. 
  W. 
  

   A 
  Giltay 
  has 
  questioned 
  the 
  correctness 
  of 
  the 
  remark. 
  

   made 
  by 
  me 
  in 
  the 
  Phil. 
  Mag. 
  for 
  Oct. 
  1916 
  (page 
  394), 
  that 
  

   the 
  explanation 
  of 
  the 
  effect 
  of 
  a 
  " 
  mute 
  " 
  on 
  the 
  tone 
  of 
  

   bowed 
  stringed 
  instruments 
  is 
  chiefly 
  to 
  be 
  sought 
  for 
  in 
  the 
  

   lowering 
  of 
  the 
  frequencies 
  of 
  resonance 
  of 
  the 
  instrument 
  

   produced 
  by 
  the 
  loading 
  of 
  the 
  bridge. 
  

  

  2. 
  Before 
  replying 
  to 
  the 
  specific 
  issues 
  raised 
  by 
  

   Mr. 
  Giltay, 
  I 
  may 
  be 
  permitted 
  to 
  point 
  out 
  that 
  the 
  view 
  

   of 
  the 
  action 
  of 
  the 
  mute 
  suggested 
  by 
  me 
  rests 
  upon 
  the 
  

   secure 
  foundation 
  of 
  mathematical 
  analysis. 
  The 
  effect 
  of 
  

   adding 
  inertia 
  to 
  any 
  part 
  of 
  a 
  dynamical 
  system 
  has 
  been 
  

   considered 
  by 
  Lord 
  Rayleigh, 
  Routh 
  and 
  others, 
  and 
  it 
  has 
  

   been 
  shown 
  that 
  the 
  natural 
  frequencies 
  as 
  altered 
  by 
  the 
  

   addition 
  of 
  the 
  load 
  are 
  given 
  by 
  the 
  roots 
  of 
  the 
  equation 
  

   (seeRouth's 
  'Advanced 
  Rigid 
  Dynamics,'' 
  Section 
  76) 
  

  

  (N^— 
  h 
  2 
  )(N 
  2 
  »-w 
  2 
  ) 
  &c., 
  -*n 
  2 
  (n 
  2 
  2 
  -n*)(n 
  3 
  2 
  -n 
  2 
  ) 
  &c. 
  = 
  0, 
  

  

  In 
  the 
  above, 
  N 
  l9 
  N 
  2 
  , 
  &c. 
  are 
  the 
  frequencies 
  before 
  the 
  

   addition 
  of 
  load, 
  n^ 
  n 
  2 
  , 
  rc 
  3 
  , 
  &c. 
  are 
  the 
  limiting 
  values 
  of 
  the 
  

   frequencies 
  attained 
  when 
  the 
  load 
  becomes 
  infinitely 
  large, 
  

   and 
  x 
  is 
  a 
  positive 
  quantity 
  proportionate 
  to 
  the 
  added 
  inertia. 
  

  

  [?2 
  ( 
  = 
  0, 
  n 
  2 
  >N 
  1? 
  ft 
  3 
  >N 
  2 
  , 
  &c, 
  according 
  to 
  the 
  theorem 
  

   due 
  to 
  Routh]. 
  

  

  The 
  forced 
  vibration 
  due 
  to 
  a 
  periodic 
  force 
  of 
  frequency 
  n 
  

   (assumed 
  to 
  act 
  on 
  the 
  system 
  at 
  the 
  point 
  at 
  which 
  the 
  

   load 
  is 
  fixed) 
  also 
  depends 
  on 
  the 
  magnitude 
  of 
  the 
  ex- 
  

   pression 
  on 
  the 
  left-hand 
  side 
  of 
  the 
  preceding 
  equation, 
  

   being 
  in 
  fact 
  inversely 
  proportional 
  to 
  it 
  except 
  in 
  the 
  

   immediate 
  neighbourhood 
  of 
  the 
  frequencies 
  of 
  resonance. 
  

   The 
  expression 
  may, 
  for 
  convenience, 
  be 
  written 
  in 
  the 
  form 
  

   (p 
  — 
  aq). 
  Assuming 
  that 
  the 
  frequency 
  n 
  of 
  the 
  impressed 
  

   force 
  lies 
  between 
  two 
  of 
  the 
  natural 
  frequencies, 
  say 
  N 
  x 
  

   and 
  N 
  2 
  , 
  of 
  the 
  system 
  without 
  any 
  load, 
  the 
  effect 
  of 
  the 
  load 
  

   on 
  the 
  forced 
  vibration 
  evidently 
  depends 
  on 
  whether 
  p 
  and 
  q 
  

   are 
  of 
  the 
  same 
  or 
  of 
  opposite 
  sign. 
  If 
  n 
  be 
  less 
  than 
  w 
  2 
  , 
  

   they 
  are 
  of 
  opposite 
  signs, 
  while 
  if 
  n 
  be 
  greater 
  than 
  w 
  2 
  , 
  they 
  

   are 
  of 
  the 
  same 
  sign. 
  In 
  the 
  former 
  case, 
  the 
  load 
  decreases 
  

   the 
  amplitude 
  of 
  the 
  forced 
  vibration 
  throughout. 
  In 
  the 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

   Phil, 
  Mag. 
  S. 
  6. 
  Vol. 
  35. 
  No. 
  210. 
  June 
  1918, 
  2 
  M 
  

  

  