OPEN AND CLOSED PIPES 



47 



3,8. Open and Closed Pipes. — The vibrations of a column of gas or 

 fluid in a cylindrical tube are analogous to the longitudinal vibrations in 

 a solid bar. For the open pipe there must be a loop of displacement at 

 the open ends. 



The fundamental resonant frequency of a pipe, open at both ends, 

 Fig. 3.7, is 



f-h 



3.9 



where / = length of the pipe, in centimeters, and 



c = velocity of sound, in centimeters per second. See Table 1.1. 

 The overtones of an open pipe are harmonics of the fundamental. That 

 is /a = 2/i, /a = 3/i, fi = 4/i, etc. 



FUNDAMENTAL FIRST HARMONIC 



SECOND OVERTONE 



THIRD HARMONIC 



FIRST OVERTONE THIRD HARMONIC 



Fig. 3.7. Modes of vibration of the air column in a pipe open at both ends and in a pipe 

 closed at one end and open at the other end. The velocity nodes and loops are indicated 

 by A'" and L. 



The fundamental resonant frequency of a pipe closed at one end and 

 open at the other end. Fig. 3.7, is 



/ = 



4/ 



3.10 



The overtones of the pipe closed at one end are the odd harmonics. 

 That is /2 = 3/i, fs = Sfu etc. 



In the above examples the end connection has been omitted. Rayleigh ^^ 

 shows the added length at the open end to be .S2R where R is the radius 

 of the pipe. If the pipe is terminated in a large flange the end connection 

 will be that given in Sec. 5.7. 



Organ pipes and whistles have been built to cover the range from 16 



Rayleigh, " Theory of Sound," Macmillan and Co., London. 



