﻿along 
  a 
  Periodically 
  Loaded 
  String, 
  359 
  

  

  These 
  are 
  equivalent 
  to 
  

  

  A 
  

  

  e 
  

  

  (e 
  m 
  -2e 
  fi 
  cosa-^ 
  + 
  1) 
  

  

  2 
  sin 
  iir 
  

  

  B 
  y 
  ■ 
  (**■> 
  

  

  where 
  

  

  g£ 
  COS 
  a 
  — 
  COS 
  *Jr 
  , 
  ^ 
  COS 
  a 
  — 
  • 
  COS 
  \/r 
  • 
  . 
  v 
  

  

  tan 
  t= 
  -g— 
  : 
  : 
  — 
  1-, 
  tan 
  r' 
  = 
  -3-^ 
  : 
  — 
  f 
  . 
  (xvi.) 
  

  

  <? 
  p 
  sin 
  a 
  — 
  sin 
  y 
  7 
  <?p 
  sin 
  a 
  + 
  sin 
  yjr 
  K 
  ' 
  

  

  3. 
  We 
  are 
  now 
  provided 
  with 
  a 
  complete 
  solution 
  of 
  the 
  

   motion 
  ; 
  this 
  we 
  will 
  proceed 
  to 
  interpret. 
  

  

  The 
  quantities 
  a 
  and 
  ft 
  are 
  determined 
  by 
  equations 
  (ix.) 
  : 
  

  

  cos 
  a 
  cosh 
  )8 
  = 
  cos 
  *^r— 
  nty 
  sin 
  -\fr 
  = 
  z 
  say, 
  \ 
  

  

  . 
  , 
  Q 
  vk 
  . 
  I 
  . 
  (ix.) 
  

  

  sm 
  a 
  sinn 
  ft 
  = 
  — 
  7p 
  sin 
  y. 
  v 
  ' 
  

  

  We 
  shall 
  clearly 
  perceive 
  the 
  drift 
  of 
  the 
  matter 
  if 
  we 
  

   neglect 
  the 
  friction 
  and 
  put 
  k 
  = 
  0. 
  Then 
  either 
  sin 
  a 
  or 
  

   sinh 
  ft 
  vanishes 
  ; 
  the 
  former 
  or 
  the 
  latter 
  being 
  the 
  case 
  

  

  according 
  as 
  s^l. 
  

  

  If 
  z 
  2 
  < 
  1, 
  /3 
  = 
  0, 
  and 
  a 
  wave-like 
  motion 
  will 
  be 
  propagated 
  

   through 
  the 
  masses, 
  for 
  

  

  y 
  r 
  — 
  Q 
  e 
  Kr*+nt) 
  m 
  

  

  If 
  z* 
  > 
  1, 
  ft 
  is 
  finite, 
  while 
  a 
  is 
  a 
  multiple 
  of 
  tt. 
  The 
  equation 
  

  

  y 
  r 
  =C 
  e 
  Kr*+nt)-rfi 
  

  

  will 
  represent 
  an 
  exponential 
  falling 
  off 
  of 
  motion, 
  consecutive 
  

  

  masses 
  being 
  either 
  in 
  the 
  same 
  or 
  in 
  opposite 
  phases. 
  In 
  

  

  this 
  case 
  the 
  deeper 
  masses 
  will 
  be 
  practically 
  unaffected 
  by 
  

  

  the 
  incident 
  wave. 
  

  

  In 
  order 
  to 
  understand 
  how 
  these 
  phenomena 
  depend 
  upon 
  

  

  the 
  frequency 
  of 
  the 
  incident 
  wave, 
  we 
  must 
  trace 
  the 
  changes 
  

  

  n 
  I 
  

   of 
  z 
  for 
  different 
  values 
  of 
  yjr 
  , 
  or 
  — 
  . 
  

  

  T 
  v 
  

  

  4. 
  Graph 
  of 
  z 
  = 
  cos 
  -^r—fj,yjr 
  sin 
  yjr. 
  

   This 
  is 
  readily 
  constructed 
  on 
  finding 
  the 
  roots 
  of 
  

  

  z=0, 
  2=1, 
  2= 
  — 
  1. 
  

   For 
  2 
  = 
  we 
  have 
  cot 
  yjr=fiyfr. 
  By 
  the 
  usual 
  graphic 
  

   method, 
  we 
  find 
  that 
  the 
  roots 
  of 
  this 
  equation 
  lie 
  between 
  

  

  and 
  -x, 
  7r 
  and 
  ^-, 
  27T 
  and 
  -^-, 
  &c, 
  

  

  £l 
  u 
  u 
  

  

  approaching 
  closer 
  to 
  the 
  lower 
  limit 
  for 
  the 
  greater 
  values. 
  

  

  