﻿108 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  attachments 
  (n=l), 
  be 
  three 
  times 
  as 
  effective 
  as 
  the 
  simple 
  

   resonator 
  (n 
  = 
  0), 
  for 
  which 
  the 
  theory 
  is 
  given 
  by 
  my 
  book 
  

   on 
  Sound, 
  § 
  319 
  ? 
  

  

  The 
  answer 
  may 
  be 
  found 
  in 
  a 
  slightly 
  different 
  presenta- 
  

   tion 
  of 
  the 
  matter. 
  In 
  the 
  above 
  example 
  the 
  rigid 
  sphere 
  is 
  

   supposed 
  to 
  be 
  symmetrically 
  moored 
  to 
  a 
  fixed 
  point, 
  and 
  the 
  

   vibration 
  actually 
  assumed 
  is 
  in 
  a 
  direction 
  parallel 
  to 
  that 
  

   of 
  propagation 
  of 
  the 
  incident 
  waves. 
  Three 
  degrees 
  of 
  

   freedom 
  are 
  really 
  involved 
  here, 
  while 
  the 
  more 
  typical 
  

   case 
  will 
  be 
  that 
  in 
  which 
  the 
  motion 
  is 
  limited 
  to 
  one 
  direction. 
  

   The 
  efficiency 
  of 
  the 
  resonator 
  will 
  then 
  be 
  proportional 
  to 
  

   the 
  square 
  of 
  the 
  cosine 
  {/a) 
  of 
  the 
  angle 
  between 
  the 
  

   direction 
  of 
  vibration 
  and 
  that 
  of 
  the 
  incident 
  waves 
  ; 
  and 
  

   the 
  mean 
  efficiency 
  will 
  bear 
  to 
  the 
  maximum 
  efficiency 
  (/i<= 
  1) 
  

   a 
  ratio 
  equal 
  to 
  that 
  of 
  

  

  Jo 
  Jo 
  

  

  ifl, 
  

  

  that 
  is 
  of 
  I. 
  Thus, 
  if 
  the 
  vibration 
  in 
  the 
  case 
  of 
  n=l 
  be 
  

   limited 
  to 
  one 
  direction, 
  the 
  mean 
  efficiency 
  of 
  the 
  resonator 
  

   is 
  the 
  same 
  as 
  when 
  ?i 
  = 
  0; 
  and 
  a 
  similar 
  conclusion 
  will 
  hold 
  

   good 
  in 
  all 
  cases. 
  In 
  this 
  way 
  the 
  factor 
  2n 
  + 
  l 
  is 
  eliminated, 
  

   and 
  the 
  statement 
  assumes 
  a 
  form 
  more 
  nearly 
  capable 
  of 
  

   generalization 
  to 
  all 
  vibrating 
  systems. 
  

  

  Now 
  that 
  a 
  general 
  theorem 
  (52) 
  has 
  been 
  demonstrated, 
  

   it 
  will 
  be 
  of 
  interest 
  to 
  trace 
  its 
  application 
  to 
  some 
  case 
  of 
  

   a 
  uniform 
  medium, 
  for 
  which 
  purpose 
  we 
  may 
  take 
  the 
  

   simple 
  acoustical 
  resonator. 
  But 
  this 
  deduction 
  is 
  not 
  quite 
  

   a 
  simple 
  matter, 
  partly 
  on 
  account 
  of 
  the 
  extension 
  to 
  infinity, 
  

   and 
  also, 
  I 
  think, 
  for 
  want 
  of 
  a 
  more 
  general 
  theory 
  of 
  waves 
  

   in 
  a 
  uniform 
  medium 
  than 
  any 
  hitherto 
  formulated. 
  If 
  the 
  

   object 
  be 
  merely 
  to 
  obtain 
  a 
  result, 
  it 
  is 
  far 
  more 
  easily 
  

   attained 
  by 
  a 
  special 
  investigation 
  from 
  the 
  formulae 
  of 
  the 
  

   Theory 
  of 
  Sound, 
  on 
  the 
  lines 
  indicated 
  by 
  Prof. 
  Lamb. 
  

   It 
  may 
  perhaps 
  be 
  well 
  to 
  sketch 
  the 
  outline 
  of 
  such 
  an 
  

   investigation. 
  

  

  The 
  time 
  factor 
  e 
  ilcat 
  being 
  suppressed, 
  the 
  velocity-potential 
  

   <p 
  of 
  the 
  primary 
  waves 
  is 
  (§ 
  334) 
  e 
  ikx 
  , 
  or 
  e 
  ikr 
  ^ 
  y 
  and 
  the 
  

   harmonic 
  component 
  of 
  the 
  nth 
  order 
  has 
  the 
  expression 
  

  

  <*„=(2,-M)P„ 
  W 
  .P„(^-,)(^), 
  . 
  . 
  (53bis) 
  

  

  while 
  (§ 
  329) 
  the 
  corresponding 
  expression 
  for 
  the 
  divergent 
  

   secondary 
  waves 
  is 
  

  

  tn=(-l)»fc„P„ 
  W 
  . 
  P,,^,) 
  ^^-:"' 
  1 
  ^ 
  ). 
  (54) 
  

  

  