PIPES AND RESONATORS 279 



found by Helmholtz, by direct observation, to be of the simple- 

 harmonic type, but the fluctuations in the current of air are 

 necessarily of a more complex character. If the periodic 

 current be expressed by a Fourier series 



C + Cicos(n -{-!)+ C 2 cos(2nt + 2 ) + ..., ...(1) 



the coefficients C z , G 3 , ... are usually by no means insensible 

 as compared with C lt and accordingly if the sound is heard 

 directly it has a very harsh and nasal character. In practice, 

 the reed is fitted with a suitable resonator, or " sound-pipe," 

 which specially reinforces one or more of the lower elements in 

 the harmonic series (1). 



For the purposes of mathematical treatment we may 

 idealize the question somewhat, and imagine that at a given 

 point in the interior of the resonator we have a simple source 

 of the type corresponding to one of the terms in (1). It 

 appears from the elementary theory of 62 that in the case of 

 a cylindrical pipe, with the source at one end, the frequencies 

 of maximum resonance are very approximately those of the 

 free vibrations when that end is closed. Hence a reed fitted 

 with a cylindrical sound-pipe of suitable length will emit 

 a series of tones whose frequencies are proportional to the odd 

 integers 1, 3, 5, .... In a conical pipe, on the other hand, with 

 the source near the vertex, we have the complete series of 

 harmonics with frequencies proportional to 1, 2, 3, 4,... (see 

 84). But in either case the harmonics of high order are 

 discouraged by the increasing deviation of the frequencies of 

 maximum resonance from the harmonic relation which neces- 

 sarily holds in the expression for the essentially periodic current 

 of air. 



As the question is instructive in various ways it may be 

 worth while to examine more in detail the case of a cylindrical 

 sound -pipe (of any form of section), applying the correction 

 for the open end, and allowing for the dissipation due to the 

 escape of sound outwards. The plan of the investigation is 

 similar to that of 87, the difference being that we now have 

 a source Ce int (say) at the end x = I. For simplicity we will 



