86 ACOUSTICAL ELEMENTS 



increasing the radiation from a sound source. The principal virtue of a 

 horn resides in the possibility of presenting practically any value of acoustic 

 impedance to the sound generator. This feature is extremely valuable for 

 obtaining maximum overall efficiency in the design of an acoustic system. 

 As, an example, in a horn loud speaker high efficiency is obtained by de- 

 signing the system so that the driving force works against resistance in- 

 stead of inertia of the diaphragm. Employing suitable combination of 

 horns, directional characteristics which are independent of frequency, as 

 well as practically any type of directional pattern, may be obtained. The 

 combination of high efficiency and the possibility of any directional pattern 

 makes the horn loud speaker particularly suitable for larger scale sound 

 reproduction. It is the purpose of this section to consider some of the 

 factors which influence the characteristics of a horn. 



5.14. Fundamental Horn Equation 13,14,15,16,17,18,19,20, _ Consider a tube 

 with a certain rate of flare and with the diameter small compared to 

 the wavelength of the sound passing through it. Let the axis of the tube 

 coincide with the x axis. Take an element of volume of the tube de- 

 fined as 



^-A;^ 5.22 



where S = cross-sectional area of the tube at x, and 

 A;c = length of the element of volume. 

 The growth of matter in this volume is the difference between the influx 

 and efflux of fluid through the faces and may be expressed as 



diSp'u) . ^^ 



A;^ 5.23 



dx 



where u — component of the particle velocity along the axis, and 

 p' = density of the medium. 



The principle of continuity was expressed in Sec. 1.3. Applying the 

 principle, the difference between the influx and efflux of the fluid into the 



13 Webster, A. G., Jour. Nat. Acad. Sci., Vol. 5, p. 275, 1919. 

 " Stewart, G. W., Phys. Rev., Vol. 16, p. 313, 1920. 

 15 Goldsmith and Minton, Proc. hist. Rad. Eng., Vol. 12, p. 423, 1924. 

 I*' Slepian and Hanna, Jour. Amer. Inst. Elec. Eng., Vol. 43, p. 393, 1924. 

 " Ballantine, G., Jour. Frank. Inst., Vol. 203, p. 85, 1927. 



18 Crandall, " Vibrating Systems and Sound," D. Van Nostrand Co., New York. 

 " Stewart and Lindsay, " Acoustics," D. Van Nostrand Co., New York. 

 "^^ Olson and Massa, " Applied Acoustics," P. Blakiston's Son and Co., Phila- 

 delphia. 



