﻿On 
  the 
  Pressure 
  of 
  Vibrations. 
  33$ 
  

  

  The 
  mean 
  upward 
  force 
  upon 
  B 
  is 
  accordingly 
  equal 
  to 
  the 
  

   mean 
  value 
  of 
  Y-f-?; 
  or 
  since 
  the 
  mean 
  value 
  of 
  Y 
  is 
  half 
  the 
  

   constant 
  total 
  energy 
  E 
  of 
  the 
  system, 
  we 
  conclude 
  that 
  the 
  

   mean 
  force 
  (L), 
  driving 
  B 
  upwards, 
  is 
  measured 
  by 
  -JE/£. 
  

   From 
  the 
  equation 
  

  

  L=£E/Z 
  (1) 
  

  

  it 
  is 
  easy 
  to 
  deduce 
  the 
  effect 
  o£ 
  a 
  slow 
  motion 
  upwards 
  of 
  

   the 
  ring. 
  The 
  work 
  obtained 
  at 
  B 
  must 
  be 
  at 
  the 
  expense 
  

   of 
  the 
  energy 
  of 
  the 
  system, 
  so 
  that 
  

  

  tfE 
  = 
  -LdZ=~iEdZ/Z. 
  

  

  By 
  integration 
  

  

  E=E 
  1 
  Z-i, 
  (2) 
  

  

  where 
  E 
  x 
  denotes 
  the 
  energy 
  corresponding 
  to 
  1=1. 
  From 
  

   (2) 
  we 
  see 
  that 
  by 
  withdrawing 
  the 
  ring 
  B 
  until 
  I 
  is 
  infi- 
  

   nitely 
  great, 
  the 
  whole 
  of 
  the 
  energy 
  of 
  vibration 
  may 
  be 
  

   abstracted 
  in 
  the 
  form 
  of 
  work 
  done 
  by 
  B, 
  and 
  this 
  by 
  a 
  

   uniform 
  motion 
  in 
  which 
  no 
  regard 
  is 
  paid 
  to 
  the 
  momentary 
  

   phase 
  of 
  the 
  vibration. 
  

  

  The 
  argument 
  is 
  nearly 
  the 
  same 
  for 
  the 
  case 
  of 
  a 
  stretched 
  

   string 
  vibrating 
  transversely 
  in 
  one 
  plane. 
  The 
  string 
  itself 
  

   may 
  be 
  supposed 
  to 
  be 
  unlimited, 
  while 
  the 
  vibrations 
  are 
  

   confined 
  by 
  two 
  rings 
  of 
  which 
  one 
  may 
  be 
  fixed 
  and 
  one 
  

   movable. 
  

  

  If 
  the 
  origin 
  of 
  x 
  be 
  at 
  one 
  end 
  of 
  a 
  string 
  of 
  length 
  I, 
  the 
  

   transverse 
  displacement 
  may 
  be 
  expressed 
  by 
  

  

  sin7r,2? 
  . 
  2itx 
  

  

  y 
  = 
  $ 
  l 
  — 
  j— 
  +<£ 
  2 
  sm^- 
  + 
  ..., 
  . 
  . 
  . 
  (3) 
  

  

  • 
  sinirx 
  ; 
  . 
  %ttx 
  ,., 
  

  

  y=<i>i 
  — 
  j 
  — 
  +</> 
  2 
  sm— 
  + 
  •••> 
  - 
  • 
  • 
  ( 
  4 
  ) 
  

  

  where 
  (/> 
  1? 
  <j> 
  2 
  , 
  ... 
  are 
  coefficients 
  depending 
  upon 
  the 
  time. 
  

   For 
  the 
  kinetic 
  and 
  potential 
  energies 
  we 
  have 
  respectively 
  

   (< 
  Theory 
  of 
  Sound/ 
  § 
  128) 
  

  

  T=i/lfM 
  v=iwiT^+;, 
  . 
  . 
  (5) 
  

  

  in 
  which 
  W 
  represents 
  the 
  constant 
  tension 
  and 
  p 
  the 
  longi- 
  

   tudinal 
  density 
  of 
  the 
  string. 
  For 
  each 
  kind 
  of 
  <p 
  the 
  sums 
  

   of 
  T 
  and 
  Y 
  remain 
  constant 
  during 
  the 
  vibration 
  ; 
  and 
  the 
  

   same 
  is 
  of 
  course 
  true 
  of 
  the 
  totals 
  given 
  in 
  (5). 
  

   From 
  (3) 
  

  

  dy 
  IT 
  ( 
  TTX 
  , 
  a 
  , 
  27TX 
  , 
  \ 
  

  

  , 
  i 
  = 
  7^ 
  iCOS 
  x 
  +2 
  ^ 
  oos 
  ~r 
  + 
  ---> 
  

  

  Z2 
  

  

  