202    Mr.  R.  Moon  on  Resonance,  and  on  the  Circumstances 
same  periodic  time  as  the  vibrating  body,  effects  similar  in  cha- 
racter but  still  more  prominent  and  striking  will  occur  (Tyndall 
'On  Sound/ p.  176). 
If  we  further  substitute  for  the  pipe  stopped  at  one  end  an 
open  tube  twice  the  length  of  the  former,  like  results  will  con- 
tinue to  be  perceptible. 
The  method  of  explanation  I  have  already  developed  applies 
in  principle  to  this  latter  case  also. 
For,  suppose  the  first  wave  propagated  by  the  bell  down  the 
tube  to  be  a  condensation.  At  the  moment  when  the  front  of 
the  wave  reaches  the  aperture,  a  reflection  will  begin  to  take  place 
from  the  latter;  the  reflected  wave,  as  I  shall  presently  show, 
being  a  wave  of  rarefaction,  and  not  a  wave  of  condensation,  as 
would  have  been  the  case  had  the  reflection  taken  place  from  a 
fixed  obstacle.  Such  being  the  case,  if  we  represent  by  2t  the 
period  of  vibration,  going  and  returning,  of  the  bell,  we  shall 
have  within  the  tube  at  the  end  of  2t  a  rarefied  wave  travelling 
from  the  aperture  and  completely  filling  the  tube.  Hence,  du- 
ring the  third  interval  t  (i.  e.  while  the  bell  is  moving  for  the 
second  time  towards  the  tube),  the  rarefied  wave  reflected  back 
from  the  aperture  in  manner  already  explained  will  be  under- 
going a  second  reflection  at  the  bell,  and  so  will  bring  to  bear 
upon  the  latter  a  rarefaction  which  must  have  the  effect  of  acce- 
lerating its  motion ;  and  so  for  each  successive  demi-vibration  of 
the  bell. 
The  fact  that  a  wave  of  condensation  after  traversing  a  tube 
will  send  back  a  wave  of  rarefaction  when  it  arrives  at  the  open 
extremity  of  the  tube,  may  be  proved  either  popularly  or  analyti- 
cally. As  the  point  is  of  considerable  importance  I  shall  here 
pursue  the  latter  method. 
Adopting  the  notation  of  the  Encyclopedia  Metropolitana, 
any  disturbance  of  the  air  within  a  cylindrical  tube  may  either 
be  represented  by  one  or  other  of  the  two  following  systems  of 
equations;  viz. 
velocity     .     .      =  <j>(at—x),-^ 
condensation      =  — -,  \ 
or 
velocity     .     .     =  ty(at  +  x)3^ 
condensation  =  —  — ->  \ 
a         J 
or  else  will  be  capable  of  resolution  into  two  disturbances — one 
of  which,  represented  by  (1),  will  be  propagated  to  the  right  if 
p  be  measured  positively  in  that  direction,  while  the  other,  re- 
