﻿v- 
  " 
  a 
  

  

  Plane 
  Waves 
  of 
  Sound. 
  445 
  

  

  Moreover, 
  

  

  ~dv_ 
  T(t) 
  Q( 
  t 
  )T(t)-t-Y(t) 
  

   bf-2P(r) 
  i: 
  2a 
  

  

  3_u_ 
  Tffl) 
  Q(0)T{0)-0-Y(0) 
  

   "d 
  v 
  2F(0) 
  + 
  2a 
  

  

  whence 
  we 
  find 
  the 
  solution 
  of 
  equation 
  (1), 
  when 
  7 
  = 
  3, 
  i. 
  <?., 
  

  

  la 
  J 
  a* 
  2 
  a^ 
  z 
  ' 
  (1Uj 
  

  

  subject 
  to 
  the 
  conditions 
  (9), 
  in 
  the 
  form 
  

   + 
  f(§ 
  + 
  Q 
  2 
  T 
  2 
  -r-Y 
  2 
  )(Q 
  2 
  '- 
  a 
  ^|)rfr 
  

  

  + 
  i[«(r 
  1 
  +^)-V*](f-§- 
  QlTl 
  - 
  Q 
  ' 
  T 
  ' 
  +<,+T+Tl+ 
  ^ 
  

  

  « 
  = 
  (§+§-Q,T 
  1 
  +QiT 
  I 
  + 
  tf-r 
  + 
  Y 
  1 
  -T^y{a(^+.i)-Qi+Q 
  i 
  }, 
  

  

  where 
  the 
  subscript 
  1 
  means 
  that 
  the 
  variable 
  is 
  0, 
  and 
  the 
  

   subscript 
  2 
  that 
  the 
  variable 
  is 
  t. 
  We 
  also 
  have 
  

  

  g, 
  Sp 
  = 
  2V 
  {a(i- 
  + 
  i-)-Q 
  I 
  + 
  Q 
  2 
  }, 
  

  

  The 
  solution 
  evidently 
  satisfies 
  the 
  conditions 
  (9) 
  when 
  we 
  

   put 
  — 
  t 
  ( 
  = 
  X). 
  We 
  obtain 
  the 
  solution 
  corresponding 
  to 
  

   given 
  initial 
  values 
  of 
  the 
  displacement 
  and 
  velocity 
  by 
  

   putting 
  T 
  = 
  0. 
  Or 
  we 
  may 
  obtain 
  the 
  solution 
  corresponding 
  

  

  to 
  given 
  values 
  of 
  y 
  and 
  ~ 
  when 
  # 
  = 
  0, 
  by 
  putting 
  X= 
  0. 
  

  

  The 
  conditions 
  might 
  be 
  given 
  in 
  other 
  forms 
  ; 
  for 
  instance, 
  

   one 
  might 
  wish 
  to 
  find 
  the 
  solution 
  such 
  that 
  the 
  displace- 
  

   ment 
  \\as 
  a 
  given 
  function 
  of 
  x 
  when 
  t 
  = 
  0, 
  and 
  a 
  given 
  

  

  Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  26. 
  No. 
  153. 
  Sept. 
  1913. 
  2 
  H 
  

  

  