﻿Plane 
  Waves 
  of 
  Sound. 
  441 
  

  

  motion 
  approaches 
  the 
  stage 
  in 
  which 
  it 
  becomes 
  dis- 
  

   continuous 
  

  

  2. 
  We 
  take 
  the 
  equation 
  of 
  motion 
  in 
  Earnshaw's 
  form, 
  

   using 
  the 
  notation 
  of 
  equation 
  (4), 
  paragraph 
  249. 
  of 
  Lord 
  

   Payleiglr's 
  ' 
  Sound,' 
  namely, 
  

  

  \dJ 
  -dt* 
  a 
  ^ 
  v 
  } 
  

  

  where 
  a 
  2 
  =p 
  y/p 
  , 
  and 
  y 
  is 
  the 
  ratio 
  of 
  the 
  specific 
  heats. 
  

  

  The 
  general 
  solution 
  of 
  equation 
  (1) 
  may 
  be 
  put 
  in 
  various 
  

   forms, 
  of 
  which 
  one 
  is 
  the 
  envelope 
  of 
  the 
  plane 
  {x 
  } 
  t, 
  y 
  being 
  

   the 
  current 
  coordinates) 
  : 
  

  

  P- 
  

  

  i" 
  ( 
  A 
  T 
  / 
  2na 
  - 
  -t-±\ 
  

   + 
  q 
  t-y 
  = 
  ^t 
  i 
  [K^ 
  l 
  (~ 
  1 
  P 
  ') 
  

  

  + 
  b„J 
  _i_(~ 
  p' 
  y 
  ^)} 
  cos 
  ("'y+O, 
  

  

  where 
  p 
  and 
  q 
  are 
  considered 
  as 
  variable 
  parameters, 
  and 
  

   A„, 
  B„, 
  c 
  n 
  are 
  constants. 
  

  

  The 
  form 
  of 
  this 
  solution 
  shows 
  that 
  when 
  1/(7 
  — 
  1) 
  is 
  the 
  

   half 
  of 
  an 
  odd 
  integer, 
  e.g. 
  7 
  = 
  1"4, 
  the 
  solution 
  will 
  be 
  

   expressible 
  in 
  finite 
  terms. 
  

  

  We 
  proceed 
  to 
  derive 
  the 
  solution, 
  in 
  the 
  particular 
  case 
  

   mentioned, 
  in 
  a 
  form 
  which 
  will 
  be 
  more 
  immediately 
  useful. 
  

   First, 
  by 
  Legendre's 
  transformation 
  (The 
  Principle 
  of 
  

   Duality), 
  we 
  obtain 
  

  

  y+lBV_ 
  2 
  B 
  2 
  m 
  

  

  p 
  W 
  ~ 
  a 
  bg 
  2 
  ' 
  

  

  where 
  

  

  by 
  ~dy 
  

  

  p 
  = 
  —^ 
  q 
  = 
  g|, 
  u 
  = 
  px 
  + 
  qt 
  -y. 
  

  

  Putting 
  u=pv, 
  the 
  equation 
  becomes 
  

  

  

  and, 
  finally, 
  putting 
  

  

  we 
  have 
  

  

  2a 
  _rz? 
  2a 
  _r=2 
  

  

  S=— 
  lP 
  * 
  +g, 
  v-—^ 
  2 
  -<7, 
  

  

  ~d 
  2 
  v 
  3 
  — 
  7 
  d£ 
  ~dv 
  

  

  bfdv 
  2(7-1) 
  Z 
  + 
  V 
  

  

  (2) 
  

  

  