14 



GENERALIZED THEORY OF ELECTROACOUSTIC TRANSDUCERS 



electrically terminated in a load impedance Z L is con- 

 sidered, so that E = - Z L I, t'„ can be eliminated be- 

 tween the equations and the following is obtained for 

 the voltage output of the transducer: 



E = -Z L 1 



Z, 



k' 



z L + z b - 



kk' z„ + z r 



F - 



Z + z r 



(22) 



We are interested in the open-circuit voltage E oc . This 

 is obtained from equation (22) by letting Z, — > oo; 

 thus 



E„„ 



k'F 



z + z r ' 



(23) 



Since 



p inc (R„) = * ■ 



-'f "«•-*• 



|R — R,. 



(24) 



due to a point source of spherical sound waves 3 at R, 

 at a distance d from the acoustic center of the trans- 

 ducer to the transmitting sensitivity S, measured at 

 the point R, is a constant independent of all particu- 

 lar characteristics of the transducer. The value of the 

 constant is: \M/S\ = 2d\/pc, where A and c are the 

 wave length and velocity of sound, respectively, and 

 p is the density of the medium. The applications of 

 this theorem are discussed in Chapters 4 to 7. 



The theorem is readily proved by taking the ratio 

 of M as given by equation (25) with R,. = R, and S as 

 given by equation (16): 



4^/AVT 



]otp\k / 



(27) 



Taking the absolute value of this equation, and re- 

 membering that \k'\ = \k\, we obtain 



the receiving sensitivity, using equation (20) for F , is 



M 



*'g(R )|R -R c | 



fr„c(Ro) 



(z + z,.) e 



-;^|R -R r | 



(25) 



35 PROOF OF RECIPROCITY THEOREM 



The reciprocity theorem will now be proved. This 

 theorem applies to all transducers which obey the 

 condition 



I k I = I k' 



(26) 



that is, equality of the absolute values of the electro- 

 acoustic and acoustoelectric coupling constants. It is 

 simple to show that this condition is satisfied for the 

 various idealized transducers 1 considered in Chapter 

 2. The statement of the theorem is as follows: The ab- 

 solute value of the ratio of the receiving sensitivity M 



i Thus, for the case o£ the electrodynamic moving ribbon 

 pressure-gradient transducer (see Chapter 2) k = — Bl, k' = BI. 

 where B is the magnetic flux density in the region where the rib- 

 bon moves, and / is the effective length of the ribbon. Values 

 of /< and k' for other types of transducers are given in the 

 literature.80, 81 



i This restriction is not necessary. It may be shown that the 

 reciprocity theorem is valid for any source distribution for the 

 incident waves just so long as the distance between sources and 

 transducer is large compared to the dimensions of either of 

 them. 





2rfA 

 pc' 



(28) 



The reciprocity theorem, proved here for the case 

 of a rigidly vibrating diaphragm, can be shown to 

 hold for any general mode of diaphragm vibration 

 where v n (r) is a function of position on the dia- 

 phragm. k 



3.6 



EFFICIENCIES 



Having discussed the sensitivities of a transducer 

 and their relationship through the reciprocity theor- 

 em, a treatment of efficiencies follows, beginning 

 with the efficiency of the transducer on transmission, 

 the projector efficiency. This is defined as the ratio 

 of the total acoustic power output of the transducer 

 to the electric power input. Several expressions for 

 the projector efficiency E p will be derived which will 

 be useful for different purposes. 



The acoustic power output may be shown to be 74,75 



1 



/j(R) r>„*(R) d.% + i>(R)* y„(R) dl 



(29) 



k Another extension of the theorem is to the case of a series of 

 transducers individually obeying reciprocity and coupled by 

 electric and mechanical transformers. Then the condition |A'| = 

 |ft| will hold for the coupling between the input E, I and output 

 F, v , if it holds for the individual transducers, and the reci- 

 procity theorem wilfbe valid for the series considered as a unit. 



