﻿440 
  Mr. 
  J. 
  R. 
  Wilton 
  on 
  

  

  Hence, 
  at 
  a 
  considerable 
  distance 
  from 
  the 
  centre 
  amplitude 
  

  

  V 
  2 
  AK* 
  . 
  1 
  

  

  is 
  , 
  this 
  being 
  where 
  r—~ 
  is 
  small. 
  

  

  Now 
  returning 
  to 
  equation 
  (25), 
  if 
  plate 
  extends 
  to 
  infinity 
  

   V 
  must 
  be 
  zero 
  when 
  x 
  is 
  infinite, 
  

  

  L 
  e. 
  V= 
  A{( 
  r 
  - 
  log 
  2) 
  J 
  (px/KLx) 
  -h 
  Y 
  (j»/Khx)}< 
  

   He 
  nee 
  , 
  when 
  x 
  is 
  large, 
  but 
  not 
  infinite, 
  

  

  V 
  '*-/.{ 
  V(KLW 
  sin 
  U 
  -V(KL)^j. 
  . 
  (31) 
  

  

  Hence, 
  comparing 
  (30) 
  and 
  (31), 
  we 
  see 
  that 
  at 
  large 
  

   distances 
  from 
  the 
  origin 
  on 
  the 
  elementary 
  theory, 
  the 
  

   amplitude 
  of 
  the 
  oscillation 
  is 
  inversely 
  proportional 
  to 
  the 
  

   square 
  root 
  of 
  the 
  distance, 
  and 
  when 
  radiation 
  is 
  taken 
  into 
  

   account, 
  inversely 
  proportional 
  to 
  the 
  distance. 
  

  

  In 
  conclusion 
  I 
  must 
  express 
  my 
  thanks 
  to 
  Prof. 
  Howe, 
  

   of 
  the 
  City 
  and 
  Guilds 
  (J3ng.) 
  College, 
  to 
  whom 
  I 
  am 
  

   indebted, 
  both 
  for 
  the 
  suggestion 
  of 
  the 
  original 
  problem 
  

   which 
  led 
  to 
  the 
  deductions 
  in 
  the 
  paper, 
  and 
  also 
  for 
  many 
  

   helpful 
  criticisms 
  in 
  the 
  course 
  of 
  the 
  work. 
  

  

  XXXIV. 
  On 
  Plane 
  Waves 
  of 
  Sound. 
  By 
  J. 
  R. 
  Wilton, 
  

   M.A., 
  B.Sc, 
  Assistant 
  Lecturer 
  in 
  Mathematics 
  in 
  the 
  

   University 
  of 
  Sheffield*. 
  

  

  1. 
  TT 
  is 
  well 
  known 
  that 
  the 
  exact 
  equation 
  for 
  plane 
  waves 
  

   X 
  of 
  sound 
  leads 
  to 
  a 
  result 
  which 
  cannot 
  hold 
  beyond 
  a 
  

   certain 
  time, 
  owing 
  to 
  the 
  fact 
  that 
  the 
  motion 
  becomes 
  dis- 
  

   continuous 
  f 
  . 
  The 
  proof 
  given 
  in 
  the 
  text-books 
  depends 
  on 
  

   Earnshaw's 
  solution 
  of 
  the 
  equation 
  of 
  motion, 
  which 
  assumes 
  

   a 
  relation 
  between 
  the 
  velocity 
  and 
  the 
  rate 
  of 
  variation 
  of 
  the 
  

   displacement 
  with 
  the 
  position. 
  There 
  does 
  not 
  seem 
  to 
  be 
  

   any 
  reason 
  for 
  this 
  relation, 
  and 
  the 
  fact 
  of 
  discontinuity 
  does 
  

   not 
  depend 
  upon 
  it. 
  The 
  following 
  paper 
  was 
  undertaken 
  

   with 
  a 
  view 
  to 
  discovering 
  when 
  discontinuity 
  sets 
  in 
  in 
  

   any 
  given 
  case, 
  the 
  initial 
  displacement 
  and 
  velocity 
  being 
  

   arbitrarily 
  assigned. 
  Incidentally 
  it 
  will 
  be 
  shown 
  that 
  the 
  

   ordinary 
  approximate 
  solution, 
  in 
  which 
  the 
  displacement 
  is 
  

   regarded 
  as 
  a 
  small 
  quantity 
  whose 
  square 
  may 
  be 
  neglected, 
  

   does 
  not 
  begin 
  to 
  depart 
  widely 
  from 
  the 
  truth 
  until 
  the 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  See, 
  for 
  example, 
  Lord 
  Rayleigh's 
  l 
  Sound,' 
  2nd 
  edit. 
  vol. 
  ii. 
  p. 
  36. 
  

  

  