﻿Waves 
  generated 
  by 
  Impact, 
  99 
  

  

  directions 
  in 
  which 
  the 
  intensity 
  is 
  a 
  minimum 
  are 
  also 
  

   asymmetrically 
  situated. 
  

  

  For 
  the 
  explanation 
  of 
  these 
  and 
  other 
  results, 
  we 
  have 
  

   naturally 
  to 
  turn 
  to 
  the 
  mathematical 
  theory 
  which 
  rests 
  

   upon 
  the 
  fact 
  that 
  the 
  sound 
  is 
  due 
  to 
  the 
  wave-motion 
  set 
  

   up 
  in 
  tie 
  fluid 
  by 
  the 
  sudden 
  reversal 
  of 
  the 
  motion 
  of 
  the 
  

   spheres. 
  Let 
  a 
  and 
  b 
  be 
  the 
  radii 
  of 
  the 
  two 
  spheres 
  and 
  

   p 
  a 
  and 
  p 
  b 
  be 
  their 
  densities. 
  Then 
  the 
  masses 
  of 
  the 
  spheres 
  

  

  are 
  ^Jrp 
  a 
  a 
  z 
  and 
  ^irp 
  b 
  b* 
  respectively. 
  Denoting 
  the 
  changes 
  

  

  in 
  velocity 
  which 
  the 
  balls 
  undergo 
  as 
  a 
  result 
  of 
  the 
  impact 
  

   by 
  U 
  and 
  U 
  6 
  respectively, 
  by 
  the 
  principle 
  of 
  constant 
  

  

  momentum 
  we 
  have 
  ^ 
  a 
  l^ 
  b 
  = 
  p 
  b 
  b 
  d 
  lp 
  a 
  a 
  3 
  . 
  The 
  ratio 
  ^r 
  thus 
  

  

  depends 
  only 
  on 
  the 
  diameters 
  and 
  the 
  densities 
  of 
  the 
  

   spheres, 
  while, 
  of 
  course, 
  the 
  actual 
  values 
  of 
  U 
  a 
  and 
  U 
  6 
  

   would 
  depend 
  on 
  the 
  relative 
  velocity 
  before 
  impact 
  and 
  the 
  

   coefficient 
  of 
  restitution. 
  It 
  is 
  obvious 
  that 
  if 
  we 
  leave 
  out 
  

   of 
  account 
  the 
  duration 
  of 
  impact, 
  that 
  is, 
  regard 
  the 
  changes 
  

   in 
  velocity 
  of 
  the 
  spheres 
  as 
  taking 
  place 
  practically 
  instan- 
  

   taneously, 
  the 
  character 
  and 
  the 
  ratio 
  of 
  the 
  intensities 
  of 
  

   the 
  sound 
  produced 
  in 
  different 
  directions 
  would 
  be 
  com- 
  

   pletely 
  determined 
  by 
  the 
  sizes 
  of 
  the 
  spheres 
  and 
  the 
  ratio 
  

   of 
  their 
  changes 
  of 
  velocity, 
  that 
  is, 
  by 
  their 
  diameters 
  and 
  

   their 
  masses 
  ; 
  when 
  the 
  spheres 
  are 
  of 
  the 
  same 
  material, 
  

   the 
  nature 
  of 
  the 
  motion 
  in 
  the 
  fluid 
  set 
  up 
  by 
  the 
  impact 
  

   would 
  depend 
  only 
  on 
  the 
  radii 
  of 
  the 
  spheres. 
  

  

  The 
  complete 
  mathematical 
  problem 
  of 
  finding 
  the 
  nature 
  

   of 
  the 
  fluid 
  motion 
  set 
  up 
  by 
  the 
  reversal 
  of 
  the 
  motion 
  of 
  

   the 
  spheres, 
  taking 
  the 
  finite 
  duration 
  of 
  impact 
  into 
  account, 
  

   would 
  appear 
  to 
  be 
  of 
  great 
  difficulty. 
  In 
  my 
  first 
  paper, 
  I 
  

   have 
  shown 
  that 
  when 
  a 
  single 
  sphere 
  of 
  radius 
  a 
  undergoes 
  

   an 
  instantaneous 
  change 
  of 
  velocity 
  U, 
  the 
  wave-motion 
  

   produced 
  is 
  given 
  by 
  the 
  expression 
  

  

  'Ot+a—r 
  

  

  + 
  4— 
  SL? 
  C0S 
  l^ 
  jirjjcostf, 
  (1) 
  

  

  which 
  indicates 
  that 
  it 
  is 
  of 
  the 
  damped 
  harmonic 
  type, 
  

   confined 
  to 
  a 
  small 
  region 
  near 
  the 
  front 
  of 
  the 
  advancing 
  

   wave. 
  The 
  wave-motion 
  set 
  up 
  in 
  the 
  case 
  of 
  two 
  spheres 
  

   in 
  contact 
  assumed 
  to 
  undergo 
  instantaneous 
  changes 
  of 
  

   velocity 
  would 
  be 
  of 
  a 
  more 
  complicated 
  type. 
  In 
  order 
  to 
  

   obtain 
  a 
  general 
  idea 
  of 
  the 
  results 
  to 
  be 
  expected, 
  particu- 
  

   larly 
  as 
  to 
  the 
  intensity 
  and 
  character 
  of 
  the 
  sound 
  in 
  

  

  H2 
  

  

  