﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  I 
  

  

  owing. 
  

  

  123 
  

  

  It 
  is 
  easy 
  to 
  see 
  that 
  the 
  value 
  of 
  x 
  thus 
  obtained 
  is 
  

   proportional 
  to 
  the 
  sharpness 
  of 
  resonance 
  H 
  as 
  defined 
  in 
  

   equation 
  (21). 
  Thus, 
  from 
  equation 
  (24), 
  we 
  see 
  that 
  the 
  

   required 
  decrement 
  of 
  energy 
  is 
  given 
  by 
  

   1 
  

  

  10 
  

  

  = 
  e~ 
  k 
  \ 
  

  

  where 
  t 
  = 
  x 
  periods 
  as 
  if 
  no 
  friction 
  were 
  present, 
  so 
  tha 
  

  

  t 
  = 
  2irxlp. 
  

   Hence 
  «»»***= 
  10 
  =«*"*' 
  H 
  J 
  

  

  or 
  H 
  = 
  (27rlog 
  10 
  ^ 
  = 
  2-7299 
  t 
  r, 
  . 
  . 
  . 
  (30) 
  

  

  It 
  may 
  be 
  well 
  to 
  note 
  here 
  the 
  values 
  of 
  range 
  and 
  

   sharpness 
  in 
  the 
  present 
  definitions 
  for 
  the 
  numerical 
  cases 
  

   calculated 
  by 
  Helmholtz. 
  These 
  are 
  shown 
  in 
  Table 
  I. 
  

  

  Table 
  I. 
  

   Range 
  and 
  Sharpness 
  of 
  Resonance 
  for 
  Helmholtz's 
  Values, 
  

  

  Musical 
  intervals 
  

   of 
  mistiming 
  in- 
  

   volving 
  a 
  reduc- 
  

   tion 
  of 
  the 
  re- 
  

   sponsive 
  energy 
  

   to 
  one-tenth. 
  

  

  Number 
  (.r) 
  of 
  com- 
  

   plete 
  periods 
  in 
  

   which 
  the 
  energy 
  

   of 
  the 
  natural 
  

   vibration 
  sinks 
  

   to 
  one-tenth. 
  

  

  Sharpness 
  of 
  

  

  Resonance. 
  

  

  H= 
  ! 
  =2-7299*. 
  

  

  Range 
  of 
  

  

  Resonance. 
  

  

  H 
  p 
  

  

  Tones. 
  

  

  1/8 
  .... 
  

  

  38-00 
  

  

  103-8 
  

  

  0-00904 
  

  

  1/4 
  .... 
  

  

  19-00 
  

  

  51-9 
  

  

  001928 
  

  

  1/2 
  .... 
  

  

  9-50 
  

  

  25-9 
  

  

  0-0386 
  

  

  3/4 
  .... 
  

  

  6-33 
  

  

  17-3 
  

  

  0-0579 
  

  

  1 
  

  

  4-75 
  

  

  33-0 
  

  

  0-077 
  

  

  H 
  

  

  3-80 
  

  

  10-4 
  

  

  0-0904 
  

  

  \\ 
  

  

  3-17 
  

  

  8-7 
  

  

  0-115 
  

  

  If 
  

  

  2-71 
  

  

  74 
  

  

  0-135 
  

  

  2 
  

  

  237 
  

  

  6-5 
  

  

  0-155 
  

  

  Bjerknes 
  on 
  Electric 
  Resonance. 
  — 
  Some 
  years 
  ago 
  V. 
  

   Bjerknes 
  published 
  a 
  number 
  of 
  papers, 
  mathematical 
  and 
  

   experimental, 
  on 
  the 
  resonance 
  phenomena 
  presented 
  by 
  

   high-frequency 
  electrical 
  oscillations 
  *. 
  In 
  some 
  of 
  these 
  

  

  * 
  Of 
  these 
  it 
  must 
  suffice 
  to 
  mention 
  the 
  following: 
  — 
  Ann. 
  d. 
  I'hysik, 
  

   vol. 
  xliv. 
  pp. 
  74-91, 
  pp. 
  92-101, 
  1891; 
  vol. 
  xlvii. 
  pp. 
  09-70, 
  1892; 
  

   Bih. 
  Till. 
  K. 
  Sv. 
  Vct.-Akod. 
  Handl. 
  vol. 
  xix. 
  pp. 
  3-10, 
  1893 
  ; 
  pp. 
  3-22, 
  

   1894 
  ; 
  vol. 
  xx. 
  no. 
  4, 
  pp. 
  3-58, 
  no. 
  5, 
  pp. 
  3-44, 
  1895 
  ; 
  Ofversirjt 
  af 
  K. 
  

   Vetemk.-Akad. 
  Forhandlingar, 
  pp. 
  381-380, 
  1894. 
  

  

  