﻿Plane 
  Waves 
  of 
  Sound. 
  447 
  

  

  And 
  after 
  redaction 
  we 
  find, 
  as 
  a 
  second 
  order 
  approximation 
  

   to 
  the 
  solution 
  of 
  equation 
  (10), 
  such 
  that 
  

  

  y 
  — 
  x 
  + 
  Y(x), 
  q 
  = 
  0, 
  when 
  £ 
  = 
  0, 
  

  

  y 
  = 
  x+±[Y(x+at) 
  + 
  Y{x-at)~] 
  

  

  -i[Y(x 
  + 
  at)-Y(x-at~\[Y'(x 
  + 
  at)-Y'(x-at)] 
  

  

  -iat[Y" 
  2 
  {x 
  + 
  at)-Y'*{x-at)]. 
  

  

  The 
  solution 
  in 
  this 
  form 
  is 
  very 
  easily 
  verified. 
  And 
  in 
  

   the 
  process 
  of 
  verification 
  we 
  see 
  that 
  the 
  second-order 
  

   approximation 
  corresponding 
  to 
  the 
  conditions 
  

  

  y 
  = 
  x, 
  q 
  — 
  aZ'(x), 
  when 
  £ 
  = 
  0, 
  

   is 
  

  

  y 
  = 
  x 
  + 
  ±[Z(x+at) 
  — 
  Z(x—at)] 
  

  

  + 
  i[Z(x 
  + 
  at)-Z(,v-at)][Z'(.c 
  + 
  at)-Z'Lv-at)] 
  

   — 
  lat[Z"\x 
  + 
  at) 
  -Z' 
  2 
  (x 
  — 
  at)]. 
  

  

  This 
  solution 
  might, 
  of 
  course, 
  have 
  been 
  obtained 
  in 
  the 
  

   same 
  way 
  as 
  the 
  other 
  by 
  assuming 
  Y 
  = 
  0. 
  

  

  We 
  note 
  that 
  the 
  last 
  two 
  results 
  are 
  true 
  for 
  the 
  general 
  

  

  equation 
  (I), 
  provided 
  that 
  we 
  substitute 
  -■. 
  for 
  £ 
  in 
  the 
  

  

  four 
  places 
  in 
  which 
  it 
  occurs. 
  This 
  follows 
  by 
  verification. 
  

  

  We 
  proceed 
  to 
  obtain 
  the 
  second 
  order 
  approximation 
  to 
  

   that 
  solution 
  of 
  equation 
  (1), 
  which 
  is 
  such 
  that 
  

  

  y 
  = 
  x 
  + 
  Y(x), 
  q 
  — 
  aZ! 
  {x), 
  when 
  t 
  = 
  0. 
  

  

  We 
  write 
  Y 
  x 
  for 
  Y(x 
  + 
  at), 
  Y 
  2 
  for 
  Y(x 
  — 
  at), 
  &c, 
  and 
  

   assume 
  

  

  y 
  = 
  iK+ 
  i(Y 
  1+ 
  Y 
  2 
  )-^+l(Y 
  1 
  -Y 
  2 
  )(Y 
  1 
  '-Y 
  2 
  0- 
  7 
  -±l 
  a 
  «(Y 
  1 
  ' 
  2 
  -Y^ 
  

  

  16 
  v 
  ' 
  2/v 
  ' 
  "' 
  16 
  

  

  '■) 
  

  

  + 
  i(Z 
  1 
  -Z 
  2 
  ) 
  + 
  7 
  -±i 
  (Z 
  1 
  -Z 
  2 
  )(Z 
  1 
  '-Z 
  a 
  ')-?±-Vz 
  1 
  ' 
  2 
  -Z 
  2 
  ' 
  

  

  The 
  first-order 
  terms 
  in 
  ~ 
  and 
  =-^ 
  are 
  

   M 
  = 
  i(Yi' 
  + 
  Y/)+KZ/-Z 
  2 
  ') 
  

  

  p= 
  itt«CB 
  r 
  i 
  ,/ 
  +Y 
  1 
  / 
  o+J« 
  , 
  (Zi' 
  , 
  -z 
  1 
  ") 
  ; 
  

  

  2H2 
  

  

  + 
  <j>(sc 
  + 
  at, 
  x—at). 
  

  

  