﻿Plane 
  Waves 
  of 
  Sound. 
  451 
  

  

  The 
  motion 
  becomes 
  discontinuous 
  after 
  time 
  t 
  u 
  where 
  

   at, 
  = 
  1/2Y 
  2 
  " 
  = 
  i(fl-T), 
  

   ^ 
  and 
  t 
  being 
  connected 
  by 
  the 
  relation 
  

  

  * 
  = 
  i(*+T)-iY,'(*-r) 
  = 
  *(0+t)-Y,72Y,". 
  

   To 
  take 
  a 
  simple 
  numerical 
  example, 
  suppose 
  

  

  Y(.e 
  — 
  art) 
  = 
  A 
  sin 
  [_ir.(as—at)/a'] 
  m 
  

   Here 
  

  

  a* 
  = 
  c 
  2 
  /tt 
  3 
  A, 
  a^ 
  = 
  -c 
  2 
  /(2tt 
  2 
  A 
  sin 
  ttt/c), 
  

   where 
  

  

  u 
  — 
  r 
  = 
  — 
  ., 
  -cosec 
  — 
  , 
  

  

  ?HA 
  e 
  

  

  so 
  that 
  

  

  a 
  . 
  O 
  C 
  , 
  7TT 
  

  

  + 
  t 
  = 
  2.1'— 
  cot 
  — 
  ; 
  

  

  7T 
  

  

  <T 
  7TT 
  C 
  . 
  ITT 
  

  

  t= 
  ci' 
  + 
  -— 
  - 
  . 
  cosec 
  —cot 
  — 
  

  

  27T 
  2 
  A 
  6' 
  2<7T 
  C 
  

  

  For 
  a 
  low 
  note, 
  just 
  audible, 
  we 
  may 
  take 
  c 
  = 
  200 
  cm., 
  

   i.e. 
  a 
  frequency 
  of 
  80, 
  and 
  A 
  = 
  10" 
  6 
  cm.*; 
  so 
  that 
  the 
  

   equation 
  for 
  t 
  is 
  

  

  T 
  = 
  *+2.10»cpsec^ 
  & 
  — 
  j 
  cot 
  W() 
  -, 
  

  

  whence, 
  for 
  moderate 
  distances, 
  t= 
  — 
  (2 
  . 
  10 
  9 
  -j- 
  100) 
  is 
  the 
  

   smallest 
  negative 
  root, 
  and 
  

  

  at 
  1 
  = 
  i(0-r) 
  _ 
  *— 
  t 
  = 
  2.10 
  9 
  , 
  

  

  fx 
  = 
  10 
  5 
  sec. 
  nearly 
  ; 
  

  

  while 
  t 
  is 
  of 
  the 
  same 
  order 
  of 
  magnitude. 
  The 
  motion 
  

   of 
  the 
  air 
  due 
  to 
  a 
  low, 
  barely 
  audible 
  mte 
  is 
  therefore 
  

   such 
  that 
  viscosity 
  and 
  other 
  influences 
  will 
  cause 
  the 
  

   motion 
  to 
  cease 
  long 
  before 
  discontinuity 
  sets 
  in. 
  

  

  But 
  a 
  high, 
  loud 
  note, 
  on 
  the 
  other 
  hand, 
  gives 
  rise 
  to 
  a 
  

   motion 
  which 
  instantly 
  becomes 
  discontinuous. 
  Let 
  us 
  

   take, 
  for 
  instance, 
  c 
  = 
  2 
  cm., 
  i. 
  e. 
  a 
  frequency 
  of 
  8500, 
  

   and 
  A 
  = 
  10~ 
  2 
  cm.f 
  In 
  this 
  case 
  the 
  equation 
  for 
  t 
  is 
  

  

  nr 
  . 
  7TT 
  1 
  7TT 
  

  

  t 
  = 
  A'+zOcosec 
  -r 
  cot 
  -r-. 
  

  

  2 
  7T 
  2 
  

  

  * 
  This 
  is 
  well 
  within 
  the 
  range 
  of 
  audibility. 
  See 
  Rayleigh's 
  

   1 
  Sound,' 
  vol. 
  ii. 
  § 
  384, 
  p. 
  439. 
  

  

  t 
  This 
  is 
  the 
  amplitude 
  of 
  the 
  sound-wave, 
  at 
  a 
  distance 
  of 
  1 
  cm. 
  

   from 
  the 
  source, 
  in 
  the 
  experiment 
  described 
  on 
  pp. 
  434 
  & 
  435 
  of 
  

   Lord 
  Rayleigh's 
  ' 
  Sound' 
  (vol. 
  ii.). 
  

  

  