﻿Pressure 
  of 
  Vibrations. 
  341 
  

  

  retains 
  a 
  constant 
  value 
  over 
  the 
  length 
  o£ 
  the 
  cylinder. 
  

   If 
  p 
  denote 
  the 
  pressure 
  that 
  would 
  prevail 
  throughout, 
  had 
  

   there 
  been 
  no 
  vibrations, 
  p—po 
  is 
  small 
  and 
  we 
  may 
  replace 
  

   (9) 
  by 
  

  

  ^{»^^> 
  + 
  iju^... 
  (10) 
  

  

  The 
  expression 
  (10) 
  has 
  accordingly 
  the 
  same 
  value 
  at 
  the 
  

   piston 
  as 
  for 
  the 
  mean 
  of 
  the 
  whole 
  column 
  of 
  length 
  L 
  Now 
  

   for 
  the 
  mean 
  of 
  the 
  whole 
  column 
  

  

  §(p—p 
  )dx=0; 
  

  

  and 
  thus 
  if 
  p 
  x 
  denote 
  the 
  value 
  of 
  p 
  at 
  the 
  piston 
  where 
  w 
  = 
  l 
  f 
  

  

  J\ 
  Pa 
  Po 
  2 
  J 
  

  

  — 
  ajj^'^+aj/^**'-- 
  (11) 
  

  

  It 
  is 
  not 
  difficult 
  to 
  prove 
  that 
  the 
  right-hand 
  member 
  of 
  

   (11) 
  vanishes. 
  Thus, 
  expressing 
  the 
  motion 
  in 
  terms 
  of 
  </>, 
  

   suppose 
  that 
  

  

  siTX 
  sirat 
  ,.,_. 
  

  

  cp 
  = 
  cos 
  -j- 
  cos 
  —j— 
  (12) 
  

  

  Then 
  

  

  p—p 
  = 
  p 
  d(j)ldt, 
  V 
  = 
  d<f>/dx; 
  

  

  and 
  since 
  p 
  = 
  d 
  2 
  p 
  , 
  we 
  get 
  

  

  and 
  this 
  vanishes 
  by 
  (12). 
  Accordingly 
  

  

  ^{Pi-Pa)dt= 
  ^- 
  p 
  p 
  f 
  dt 
  (13) 
  

  

  Again 
  by 
  (12) 
  

  

  j(f)>=?jj~(fH<. 
  

  

  so 
  that 
  

  

  \{pi-po)dt= 
  -Shpi-pofdxdt 
  = 
  P 
  j 
  Wwdxdt. 
  

  

  Now 
  pojjU 
  2 
  doc 
  dt 
  represents 
  twice 
  the 
  mean 
  total 
  kinetic 
  

   energy 
  of 
  the 
  vibrations 
  or, 
  what 
  is 
  the 
  same, 
  the 
  constant 
  

   total 
  energy 
  E 
  . 
  Thus 
  if 
  L 
  denote 
  the 
  mean 
  additional 
  force 
  

  

  