A Study of the Regular Combination of Acoustic Ele- 

 ments, with Applications to Recurrent Acoustic 

 Filters, Tapered Acoustic Filters, and Horns 



By W. P. MASON 



Synopsis: The use of combinations of tubes to produce interference be- 

 tween sound waves and a suppression of certain frequencies originates with 

 Herschel (1833), and was applied by Quincke to stop tones of definite pitch 

 from reaching the ear. Following the development of electrical filters, G. 

 W. Stewart showed that combinations of tubes and resonators could be 

 devised which would give transmission characteristics at low frequencies sim- 

 ilar to electrical filters. The assumptions made by Stewart in the develop- 

 ment of his theory are that no wave motion need be considered in the 

 elements, and that the lengths of the elements employed are small compared 

 to the wave-length of sound. 



The present paper considers primarily regular combinations of acoustic 

 elements, such as straight tubes, and shows that the equations for recurrent 

 filters, tapered filters and horns can be obtained in this manner. The as- 

 sumption of no wave motion in the elements, made by Stewart, is removed 

 and also account is taken of the viscosity and heat conduction dissipation. 

 The principal difference between acoustic and electric filters is that the 

 former have an infinite number of bands. The effect of using filters be- 

 tween varying terminal impedances is also determined. 



Studying next the combination of filters having the same propagation 

 characteristics but in which the conducting tube areas increase in some 

 regular manner, it is shown that a tapered filter results which has a trans- 

 forming action in addition to its filtering properties. It is shown that if 

 straight tubes are employed and the distance between successive changes in 

 areas is made small we obtain the horn equations first developed by Webster. 

 The general combination of acoustic elements is then considered, and a 

 proof of several theorems has been given. 



STEWART, in a series of papers,' has studied the recurrent acoustic 

 filter as an analogue of the electric filter with lumped constants. 

 If due account is taken of the wave motion occurring in the individual 

 elements themselves, it appears that the nearest electrical analogue of 

 the acoustic filter is a combination of electric lines. 



In the present paper we study primarily regular combinations of 

 acoustic elements, such as straight tubes, and show that the equations 

 for recurrent filters, tapered filters, and horns can be obtained in this 

 manner. The efifect of viscosity and heat conduction dissipation has 

 been taken into account, and a consideration of the efifect of varying 

 terminal impedances has been included. 



I. Equations of Propagation of a Plane Wave in a Uniform 



Tube 



The propagation of plane waves of sound in uniform tubes has been 

 discussed in a number of places,^ but generally the results obtained are 



^Phys. Rev., 20, 528 (1922); 23, 520 (1924); 25, 90 (1925). 



^ Rayleigh's "Theory of Sound," Vol. II, p. 318. Lamb's "The Dynamical 

 Theory of Sound," p. 193. 



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