﻿Waves 
  generated 
  by 
  Impact, 
  107 
  

  

  3. 
  Two 
  spheres 
  of 
  the 
  same 
  diameter 
  but 
  of 
  

   different 
  materials. 
  

  

  We 
  have 
  seen 
  in 
  the 
  preceding 
  section 
  that 
  in 
  the 
  expres- 
  

   sions 
  for 
  F 
  and 
  Gr 
  for 
  two 
  spheres 
  of 
  the 
  same 
  material 
  but 
  

   of 
  unequal 
  diameters, 
  the 
  terms 
  containing 
  the 
  zonal 
  harmonic 
  

   of 
  the 
  second 
  order 
  P 
  2 
  (cos 
  6) 
  usually 
  preponderate, 
  and 
  that 
  

   the 
  intensity 
  diagram 
  is, 
  accordingly, 
  a 
  curve 
  which 
  consists 
  

   of 
  four 
  loops. 
  A 
  different 
  result 
  is 
  obtained 
  in 
  the 
  case 
  of 
  

   two 
  spheres 
  of 
  the 
  same 
  diameter 
  but 
  of 
  markedly 
  unequal 
  

   densities. 
  The 
  zonal 
  harmonic 
  of 
  the 
  first 
  order 
  preponde- 
  

   rates 
  in 
  this 
  case, 
  and 
  the 
  intensity 
  diagram 
  is 
  a 
  curve 
  

   consisting 
  of 
  only 
  two 
  loops. 
  To 
  obtain 
  this 
  result 
  theore- 
  

   tically, 
  we 
  have 
  to 
  proceed 
  on 
  exactly 
  the 
  same 
  lines 
  as 
  in 
  

   the 
  preceding 
  pages. 
  

  

  Taking 
  a 
  = 
  1 
  inch 
  and 
  b 
  = 
  l 
  inch, 
  we 
  easily 
  find 
  from 
  the 
  

   expressions 
  (7) 
  and 
  (8) 
  that 
  

  

  2<f>- 
  |_1- 
  g-3 
  + 
  35-43 
  + 
  53 
  --J 
  -2 
  

  

  .oTl-i^i 
  U 
  1 
  -\l2iooB0) 
  

  

  + 
  -[_ 
  2 
  1 
  + 
  3* 
  - 
  4 
  1 
  + 
  5 
  1- 
  -"J 
  ^ 
  

  

  , 
  of! 
  11 
  11 
  i 
  Ps(co^) 
  

  

  + 
  Ji_I.I 
  A 
  + 
  I 
  -| 
  P,(ooBfl) 
  

  

  + 
  L 
  2 
  6 
  + 
  3 
  6- 
  4 
  s 
  + 
  5 
  8- 
  -"J 
  r 
  5 
  

  

  + 
  &c, 
  (14) 
  

  

  „,, 
  r, 
  1 
  1 
  1 
  1 
  -lP^cosfl) 
  

  

  2^^-Ll- 
  F 
  + 
  33- 
  i5 
  + 
  53-...j- 
  i 
  V- 
  

  

  + 
  ri-Io.L 
  L 
  X 
  1 
  P2 
  (cos 
  6) 
  

  

  + 
  L 
  2 
  1 
  + 
  3 
  i_ 
  4 
  1 
  + 
  5 
  4 
  ~-J 
  r 
  3 
  

  

  L 
  2' 
  + 
  3 
  5 
  ~"4 
  5 
  + 
  5 
  5- 
  -J 
  r 
  4 
  

  

  + 
  ri_I.I 
  I 
  v 
  1 
  P4 
  (cos 
  6) 
  

   + 
  L 
  2 
  s+ 
  3 
  6_ 
  4 
  s 
  + 
  5" 
  6_ 
  -J 
  r> 
  

  

  -&c 
  (15) 
  

  

  and 
  

  

  