﻿35t> 
  Mr. 
  C. 
  Godfrey 
  on 
  Discontinuities 
  of 
  Wave-motion 
  

  

  tables 
  are 
  due 
  to 
  the 
  fact 
  that 
  the 
  substances 
  in 
  question 
  are 
  

   chemically 
  related 
  ; 
  and 
  perhaps 
  it 
  is 
  scarcely 
  justifiable 
  

   to 
  generalize 
  from 
  such 
  a 
  limited 
  number 
  ; 
  and 
  it 
  may 
  

   well 
  be 
  that 
  a 
  wide 
  range 
  of 
  elements 
  would 
  show 
  other 
  

   peculiarities. 
  

  

  I 
  desire 
  to 
  express 
  my 
  hearty 
  appreciation 
  of 
  the 
  efficient 
  

   service 
  rendered 
  in 
  this 
  work 
  by 
  Mr. 
  C. 
  R. 
  Mann, 
  and 
  

   especially 
  to 
  recognize 
  the 
  patience 
  and 
  skill 
  shown 
  in 
  the 
  

   tedious 
  and 
  delicate 
  process 
  of 
  preparation 
  of 
  the 
  vacuum- 
  

   tubes, 
  to 
  which 
  in 
  great 
  measure 
  the 
  success 
  of 
  the 
  investi- 
  

   gation 
  is 
  due. 
  

  

  XXXIX. 
  On 
  Discontinuities 
  connected 
  with 
  the 
  Propagation 
  

   of 
  Wave-motion 
  along 
  a 
  Periodically 
  Loaded 
  String. 
  By 
  

   Charles 
  Godfrey, 
  B.A., 
  Scholar 
  of 
  Trinity 
  College, 
  

   Isaac 
  Newton 
  Student 
  in 
  the 
  University 
  of 
  Cambridge*. 
  

  

  1. 
  rj 
  1HE 
  system 
  described 
  below 
  shows 
  rather 
  remarkable 
  

   JL 
  discontinuous 
  properties. 
  The 
  work 
  was 
  suggested 
  

   by 
  a 
  passage 
  in 
  Sir 
  George 
  Stokes' 
  Read 
  lecture, 
  and 
  formed 
  

   part 
  of 
  an 
  essay 
  written 
  in 
  December 
  1896. 
  

  

  A 
  heavy 
  string 
  of 
  density 
  p 
  under 
  tension 
  T 
  extends 
  from 
  

   — 
  go 
  to 
  +00. 
  From 
  -co 
  to 
  it 
  is 
  free 
  from 
  loads 
  ; 
  from 
  

   to 
  +oo 
  it 
  is 
  loaded 
  at 
  equal 
  intervals 
  I 
  with 
  equal 
  particles 
  

   of 
  mass 
  M. 
  To 
  avoid 
  ambiguity 
  we 
  will 
  suppose 
  that 
  the 
  motion 
  

   of 
  each 
  mass 
  is 
  retarded 
  by 
  a 
  small 
  viscous 
  force 
  ; 
  this 
  will 
  

   finally 
  be 
  neglected. 
  We 
  will 
  investigate 
  the 
  steady 
  vibration 
  

   of 
  the 
  system 
  when 
  simple 
  transverse 
  waves 
  are 
  travelling 
  

   along 
  the 
  string 
  from 
  -co 
  . 
  These 
  impinge 
  on 
  the 
  system 
  of 
  

   masses 
  ; 
  a 
  reflected 
  wave 
  is 
  generated 
  which 
  travels 
  back 
  

   along 
  the 
  string 
  ; 
  furthermore, 
  the 
  masses 
  are 
  agitated 
  in 
  a 
  

   certain 
  manner. 
  

  

  2. 
  We 
  will 
  denote 
  by 
  £ 
  the 
  lateral 
  displacement 
  of 
  a 
  point 
  

   on 
  the 
  string 
  between 
  #= 
  —go 
  and 
  # 
  = 
  0. 
  The 
  velocity 
  of 
  

  

  /T 
  

   propagation 
  along 
  the 
  string 
  is 
  a 
  / 
  — 
  =v. 
  For 
  a 
  motion 
  

  

  whose 
  frequency 
  is 
  given 
  by 
  e 
  mi 
  we 
  have 
  

  

  ^Ae 
  ini 
  '- 
  ± 
  h-Be 
  <n 
  (' 
  + 
  ^ 
  (i.) 
  

  

  Let. 
  the 
  displacement 
  of 
  the 
  mass 
  B 
  y 
  at 
  time 
  t 
  be 
  denoted 
  by 
  

   y 
  r 
  . 
  For 
  a 
  point 
  in 
  the 
  ?*th 
  string 
  B 
  r 
  _iB 
  r 
  let 
  the 
  displacement 
  

   be 
  £ 
  r 
  ; 
  and 
  let 
  the 
  distance 
  of 
  such 
  a 
  point 
  from 
  B 
  r 
  _, 
  be 
  <r 
  r 
  . 
  

  

  * 
  Cominmiicated 
  by 
  R. 
  T. 
  Glnzebrook, 
  F.R.S. 
  

  

  