204  On  Resonance }  and  Change  of  Phase  accompanying  Reflection, 
same  direction  in  which  x  is  measured  positively;  we  have  fffx 
and  /2  all  positive. 
But,  from  what  has  preceded,  it  results  that  the  disturbance 
in  the  plane  of  the  aperture  must  be  represented  by  systems  of 
equations  of  one  or  other  of  the  three  following  forms,  viz. : — 
velocity  .     .  =  <j>{at), 
(6) 
condensation 
velocity  .     .   =      ^jr{at)} 
condensation  =  —  ~ — -; 
(7) 
velocity.     .   =  <j>(at)  +  ^  {at) , 
■•I 
condensation   =  -^ — - — -^ — '-  •   f 
and  it  is  clear  that  no  values  which  can  be  assigned  to  fv  /2 
will  reduce  (5)  to  the  form  of  (6)  or  (7) .  It  must  therefore 
become  identical  in  form  with  (8),  in  order  to  which  we  must 
havey2=/i;  whence  it  is  clear  that  the  disturbance  within  the 
tube  during  the  period  in  which  the  original  wave  of  condensa- 
tion is  endeavouring  to  escape  from  the  aperture  is  resolvable 
into  two  disturbances,  one  of  which,  propagated  to  the  right,  is 
identical  with  the  original  condensation  represented  by  (4),  while 
the  other  will  be  propagated  to  the  left,  and  will  be  represented  by 
velocity  .     .  =      f\{at-\-x), 
condensation  =  —  ~ -, 
a 
and  therefore,  the  expression  for  the  condensation  being  negative, 
will  be  a  rarefaction. 
In  precisely  the  same  manner  we  might  prove  that  a  wave  of 
rarefaction,  after  traversing  a  tube  open  at  both  ends,  when 
finally  emerging  from  the  tube  will  send  back  a  condensation. 
In  like  manner  also  it  may  be  shown  that  when  two  gases 
touch  each  other  along  a  given  plane  without  intermingling,  and 
a  pulse  is  transmitted  through  the  one  to  the  other  in  a  direction 
normal  to  the  plane  of  contact,  the  wave  reflected  from  the  latter 
will  be  of  the  same  phase  as  the  incident  wave,  or  the  opposite 
phase,  according  as  the  density  of  the  second  gas  is  greater  or 
less  than  that  of  the  first. 
The  same  holds  when  a  disturbance  is  propagated  along  a 
stretched  cord  consisting  of  two  pieces  of  unequal  density. 
It  may  in  fact  be  stated  as  a  general  truth,  that  whenever  a 
