PLANE WAVES OF SOUND 171 



two reversals of s and none of f in the interval in question. 

 But if one end be closed and the other open, the signs of s and 

 f at P have each undergone one reversal only in the interval 

 2l/c, and a further interval of like duration must elapse before 

 the original state of things at P is restored. 



The foregoing theory explains one or two important points 

 in the theory of organ-pipes. Thus the frequency, in the 

 gravest mode, is inversely proportional to the length, and is 

 lower by an octave for a "stopped" pipe, i.e. a pipe closed 

 at one end, than for an " open " pipe, i.e. one open at both ends, 

 of the same length. It is, again, directly proportional to the 

 velocity of sound, and so increases with rise of temperature. 



In the analytical method for determining the normal modes 

 we assume as usual that f varies as cos (nt + e). The equation 

 59 (7) then becomes 



the solution of which is 



/ . nx D . nx\ /ON 



t=\A cos h B sin cos (nt + e), (2) 



\ c c J 



as in 25. The corresponding wave-length of progressive waves 

 in free air is X = 2?rc/n. Hence in any system of standing waves 

 there is a series of nodes ( = 0) at intervals of X, and a series 

 of loops, or places of zero condensation, (df/dx = 0), half-way 

 between these. 



For a tube closed at both ends (x = 0, x = I) we have 



-4=0, sin(y/c) = 0, (3) 



and therefore 



~ . rmrx frmrct \ 

 I f = w sm j- cos!-^ + m ), (4) 



\ 6 / 



where m 1, 2, 3, ..., the normal modes forming a harmonic 

 series. 



For a pipe open at both ends, the condition that s=d^/dxQ 

 for x = and x = I gives 



= 0, sin(^/c) = 0, (5) 



and the typical solution is 



n mirx frmrct \ 

 = G m cos -j cos j + e m , (6) 



t \ 6 / 



