290 BELL SYSTEM TECHNICAL JOURNAL 



connection with electrical networks are the image parameters which 

 include two image impedances and an image transfer constant. We 

 define these constants as follows for the acoustic case. 



If we have a network terminated in impedances Z/, and Zj^ (per 

 square centimeter of area) at the beginning and at the end of the net- 

 work, then these impedances are the image impedances of the structure 

 if they terminate the structure in such a way that at either termina- 

 tion junction, the impedance looking in either direction is the same. 



The image transfer constant 6 may be defined as one half the 

 natural logarithm of the vector ratio of the product of the pressure 

 by the volume velocity, at the input junction point, and this product 

 for the output junction point, when the network is terminated in its 

 image impedances. 



Hence 



2 PvVv 



To determine the image impedances, we have one set of equations 



Vr, = FlC - ^ D. 



(62) .1 



This gives the pressure and volume velocity propagated in one direc- 

 tion. We need also the equation of propagation in the opposite 

 direction. This can evidently be written 



Pr,'= PM' - Vr'^B', 



, " \ (63) 



where ^,' and V^l represent the pressure and volume velocity at 

 the beginning and F/ and pi at the end of the structure. A' can be 

 obtained from A by cyclically permuting the subscripts. By writing 

 the expansions for these quantities we can show that 



A' = C; C'^A; B' = ^^B- D'^^^D- (64) 



Eliminating the ratio VjVi from (62) and writing pi/Vi = ZjSi 

 and pnlVr, = ZjJSr,, we obtain 



Z,,Z,,D + Zo, ^Zj,(^^A - Z,,C - Zo/^ (^^b]= 0. (65) 



