under  which  Change  of  Phase  accompanies  Reflection.     203 
presented  by  (2),  will  be  propagated  to  the  left* :  and  in  this 
case  the  combined  disturbance  will  be  represented  by  the  system 
velocity  .     .   =  cf)(at— x)-\-ijr(at  +  x),  -\ 
condensation  =  »(«<-*)-*("  +  «).  j     .     (3) 
If  the  tube  be  supposed  to  lie  on  the  right  of  the  bell,  it  is 
clear  that  the  wave  of  condensation  which  the  first  movement 
of  the  bell  tends  to  propagate  within  the  tube  may  be  repre- 
sented before  it  reaches  the  further  extremity  of  the  tube  by  the 
system 
velocity  .     .   =  f(at—x),  -> 
-  .  f{at-x)     I    ....     (4) 
condensation  =  — — » 
a         J 
As  this  wave  emerges  from  the  tube,  the  condensation  of 
any  thin  stratum  of  air  which  occupies  the  plane  of  the  aper- 
ture will  be  diminished ;  i.  e.,  by  reason  of  its  being  in  contact 
with  the  free  atmosphere  outside  the  tube,  its  condensation 
will  be  less  than  it  would  have  been  had  the  tube  been  pro- 
longed indefinitely  beyond  the  aperture.  Moreover  such  dimi- 
nution of  condensation  of  the  air  occupying  the  aperture  will  ne- 
cessarily be  accompanied  by  an  increase  of  its  velocity,  since,  the 
direction  of  particle-motion  in  a  wave  of  condensation  being 
always  the  same  as  that  of  propagation,  a  diminution  of  density 
on  the  side  towards  which  the  motion  takes  place  must  lead  to  an 
acceleration  of  the  latter. 
Hence,  measuring  x  from  the  aperture,  instead  of  the  velocity 
and  condensation  at  the  aperture  being  represented  by 
velocity  .         =  f(at) , 
condensation  = -> 
a 
as  they  would  be  if  the  tube  were  prolonged  indefinitely  beyond 
the  aperture,  they  must  be  of  the  form  indicated  by  the  system 
velocity  .     .   =  f(at)  +fx(at),  -> 
condensation  =  M~M^  .  \     ■     ■    •    ■     W 
a  J 
where,  assuming  the  velocity  to  be  measured  positively  in  the 
*  See  Encyc.  Met.  art  Sound,  No.  128.  It  may  be  remarked  that 
although  the  equations  in  the  text  do  not  apply  to  what  takes  place  outside 
the  tube  as  the  wave  emerges  from  it,  they  do  apply  to  what  takes  place; 
within  the  tube,  which  is  sufficient  for  our  purpose. 
