REGULAR COMBINATION OF ACOUSTIC ELEMENTS 

 and Vi as 

 p,= .-'[A[coshS + [|j^,]si„h5] 



Fif 2Z^,Z„, 1 . ] 



283 



V. 



Fi 



cosh 5 + 



^R. + ^^n. J 



sinh 5 



(48) 



2 sinh 5 



If now the elements of our structure are non-dissipative straight 

 tubes, instead of a general filter structure, and the length of these 

 tubes between changes of area is made very small, it is evident that 

 the structure reduces to an exponential horn. We now let the ratio 

 S\lSi ■■= e^-' be expressed as 



So 



— e-^Tl 



where / is the distance between changes in area and T a new taper 

 constant. Then r, for a straight tube, neglecting dissipation, becomes 

 r — ioiljc and hence 



cosh b = cosh — ^ cosh Tl 



CO 



2/2 



CO 



,2/2 



11 - ■ ^^2 I 4 I (^4 "T 



p I r2 - '"' 



= 1 + 



a 



2! 



+ 1' 



. . {Tif (Tiy 

 1 ^^T'^^V"^ 



T^-6^,r^+% 



o 



o 



4! 



This reduces for small values of / to 



Hence 



cosh 5 = cosh ( ^ \/ ^^ ~ 7^ ) • 



Pn 

 Pl 



-nTln-n5 



= ,-n.(^+>-g)=,-^(^ + V^^-g) 



since nl = L, the total length of the horn. 



As long as T- > (coVC^), an attenuation band exists, while if or JO 

 > T^, the expression becomes 



pi 



cos I 



L ^,l-^, — T- ) — i s'm { L 



r~ 



- r- 



and a pass band occurs. 



19 



