﻿Plane 
  Waves 
  of 
  Sound. 
  443 
  

  

  t 
  , 
  . 
  B 
  2 
  ^ 
  ~d 
  2 
  v 
  ~d 
  2 
  v 
  

  

  on 
  the 
  curve, 
  where 
  r, 
  ,9, 
  t 
  denote 
  ~-p, 
  X£S~~ 
  ' 
  "j*"^' 
  sa 
  ^ 
  r 
  °' 
  °' 
  °' 
  

  

  "0, 
  A>, 
  To, 
  So- 
  For 
  

  

  °o 
  — 
  ^>— 
  , 
  — 
  i 
  

   ?o 
  + 
  ?7o 
  

  

  a 
  „ 
  ^0 
  ,9 
  Po 
  + 
  ?o 
  „ 
  ._ 
  o_A_ 
  + 
  9 
  JV^io_ 
  

  

  jV 
  = 
  r 
  of 
  0' 
  + 
  W, 
  £0' 
  = 
  *o£o' 
  + 
  W> 
  

  

  *o' 
  = 
  a 
  o£o' 
  + 
  Awo\ 
  *o' 
  = 
  7ofo' 
  + 
  ^oV« 
  

  

  To 
  determine 
  the 
  forms 
  of 
  <£ 
  and 
  ^, 
  we 
  have 
  then, 
  from 
  

   equations 
  (4) 
  and 
  (5), 
  

  

  ^ 
  ) 
  = 
  (?^f 
  ™~u&& 
  • 
  • 
  (7) 
  

  

  <f> 
  and 
  a|t 
  are 
  thus 
  to 
  be 
  determined 
  by 
  quadratures, 
  and 
  the 
  

   results 
  when 
  substituted 
  in 
  equation 
  (6) 
  give 
  the 
  solution 
  

   required. 
  

  

  4. 
  The 
  initial 
  conditions 
  are 
  given 
  in 
  terms 
  of 
  the 
  a:,y, 
  and 
  t 
  

   of 
  equation 
  (1), 
  say, 
  

  

  /=/(.*), 
  |f=F(«), 
  whent=0*, 
  

  

  These 
  conditions 
  necessitate 
  

  

  |^= 
  fix) 
  when 
  t 
  = 
  0. 
  

  

  When 
  the 
  initial 
  conditions 
  are 
  transformed 
  so 
  as 
  to 
  apply 
  

   to 
  equation 
  (3), 
  they 
  become 
  

  

  7 
  

  

  fo=-^ 
  I 
  {/W}"^+F(X) 
  = 
  5a{/^)}- 
  1/5 
  + 
  F(X). 
  

  

  ^ 
  = 
  ?« 
  = 
  i 
  -^)t/ 
  , 
  w} 
  J 
  ? 
  = 
  - 
  \m\fw** 
  

  

  where 
  X 
  is 
  the 
  x 
  of 
  equation 
  (1). 
  

  

  * 
  Earnshaw's 
  solution 
  (which 
  is 
  the 
  same 
  as 
  Poisson's 
  integral) 
  of 
  

  

  9n 
  -Tz} 
  

  

  1/5 
  

  

  equation 
  ( 
  1 
  ) 
  assumes 
  F 
  {x) 
  = 
  ^-p 
  { 
  /' 
  (at) 
  } 
  2 
  , 
  i. 
  e. 
  , 
  F(x) 
  =5a{ 
  f(x) 
  ] 
  

  

  in 
  this 
  case. 
  There 
  does 
  not 
  seem 
  to 
  be 
  any 
  reason 
  for 
  this 
  relation 
  

   between 
  the 
  initial 
  displacement 
  and 
  the 
  velocity. 
  

  

  