﻿344 
  Lord 
  Rayleigh 
  on 
  the 
  

  

  Thus, 
  i£ 
  L 
  be 
  the 
  force 
  corresponding 
  to 
  /, 
  we 
  get 
  by 
  (16) 
  

  

  Jf4J(T-¥)* 
  

  

  in 
  which 
  

  

  E 
  representing 
  as 
  before 
  the 
  constant 
  total 
  energy. 
  Accord- 
  

   ingly, 
  L 
  now 
  representing 
  the 
  mean 
  value, 
  

  

  L=<^ 
  (24) 
  

  

  In 
  the 
  case 
  of 
  a 
  medium, 
  like 
  a 
  stretched 
  string, 
  propagating 
  

   waves 
  of 
  all 
  lengths 
  with 
  the 
  same 
  velocity, 
  m=— 
  1, 
  n—\ 
  r 
  

   and 
  L= 
  — 
  E/Z, 
  as 
  was 
  found 
  before. 
  

  

  In 
  the 
  application 
  to 
  a 
  rod 
  vibrating 
  flexurally, 
  m= 
  — 
  3,. 
  

   n 
  = 
  l, 
  so 
  that 
  

  

  L=-2E/Z 
  (25> 
  

  

  If 
  m 
  = 
  n, 
  L 
  vanishes. 
  This 
  occurs 
  in 
  the 
  case 
  of 
  the 
  line 
  of 
  

   disconnected 
  pendulums 
  considered 
  by 
  Reynolds 
  in 
  illustra- 
  

   tion 
  of 
  the 
  theory 
  of 
  the 
  group 
  velocity*, 
  and 
  the 
  circumstance 
  

   suggests* 
  that 
  L 
  represents 
  the 
  tendency 
  of 
  a 
  group 
  of 
  waves 
  

   to 
  spread. 
  This 
  conjecture 
  is 
  easily 
  verified. 
  If 
  in 
  conformity 
  

   with 
  (13) 
  we 
  suppose 
  that 
  

  

  V=V,l»tf, 
  T=T 
  J»&, 
  

  

  and 
  also 
  that 
  

  

  , 
  . 
  Iirt 
  • 
  2ir 
  2irt 
  

  

  <p 
  1 
  = 
  sin 
  — 
  , 
  fa 
  = 
  —cos 
  — 
  , 
  

  

  T 
  T 
  T 
  

  

  t 
  being 
  the 
  period 
  of 
  the 
  vibration 
  represented 
  by 
  the 
  co- 
  

   ordinate 
  fa, 
  we 
  obtain, 
  remembering 
  that 
  the 
  sum 
  of 
  T 
  and 
  V 
  

   must 
  remain 
  constant, 
  

  

  V 
  ^ 
  = 
  T 
  Z'\47t/t 
  2 
  . 
  

  

  This 
  gives 
  the 
  relation 
  between 
  t 
  and 
  /. 
  Now 
  v, 
  the 
  wave- 
  

   velocity, 
  is 
  proportional 
  to 
  1\t 
  ; 
  so 
  that 
  

  

  vozl 
  l 
  -i 
  n 
  ^ 
  m 
  (26) 
  

  

  Thus, 
  if 
  u 
  denote 
  the 
  group-velocity, 
  we 
  have 
  by 
  the 
  general 
  

   theory 
  

  

  u/v=.\n— 
  \m 
  ; 
  (27) 
  

  

  and 
  in 
  terms 
  of 
  u 
  and 
  v 
  by 
  (24) 
  

  

  *~ 
  s 
  '■"<> 
  

  

  * 
  See 
  Proc. 
  Math. 
  Soc. 
  ix. 
  p. 
  21 
  (1877) 
  ; 
  Scientific 
  Papers, 
  i. 
  p. 
  322. 
  

   Also 
  Theory 
  of 
  Sound, 
  vol. 
  i. 
  Appendix. 
  

  

  