A study of the regular combination of acoustic elements, with applications to recurrent acoustic filters, tapered acoustic filters, and horns

The use of combinations of tubes to produce interference between 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 similar to electrical filters. The assumptions made by Stewart in the development 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 assumption 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 fitters between 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 transforming action in addition to its filtering properties. It is shown that if straight tubes arc 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.