172 DYNAMICAL THEORY OF SOUND 



where ra = l, 2, 3, .... Here, again, the sequence of normal 

 modes is harmonic. The figure illustrates the cases m=l, 

 m = 2. The arrows shew the direction of motion at the loops, 

 whose position is indicated by the dotted lines, in two opposite 

 phases ; the nodes are indicated by the full transverse lines. 



Fig. 60. 



In the case of a pipe closed at # = and open at x=l, 



we have 



.4=0, cos(ttZ/c) = 0, .................. (7) 



whence nl/c ^mjr, the integers m being odd. We thus obtain 



~ mirx (rmrct 

 =---- 



where m= 1, 3, 5, .... The absence of the harmonics of even 



Fig. 61. 



order determines the characteristic " quality " of stopped pipes 

 ( 91). The figure shews the cases m = 1, m = 3. 



The formula (2) can be applied also to the case of forced 



