12 



GENERALIZED THEORY OF ELECTROACOUSTIC TRANSDUCERS 



The difference between Z el and Z b is called the mo- 

 tional impedance Z m . Thus 



Z e \ ~ Z b — 



kk' 



z + z,.' 



(10) 



Z,„ being the contribution to /„, which results from 



III O rl 



the motion of the diaphragm. 



Consider now the effective acoustic impedance z ac , 

 defined as the ratio of force to normal velocity, F/v n . 

 The term z. u . depends on the electric coupling condi- 

 tions. Suppose a load impedance Z L is connected to 

 the electric terminals so that, from equation (3), 

 E = — Z L I. Substituting (3) in the basic equations 

 (1) and (2), we obtain 



and 



Z L I 



;„7<„ + kl 



k' v„ + Z b I. 



Solution of these equations shows that 



- = L = z - hk ' 



" C v n Z L + Z b 



(11) 



(12) 



(13) 



3.4 SENSITIVITIES 



The various impedances associated with an electro- 

 acoustic transducer have been expressed in terms of 

 the fundamental transducer constants and the condi- 

 tions of coupling. The sensitivities of a transducer are 

 now considered. Here, two types of sensitivity are of 

 interest: the transmitting or electroacoustic sensitiv- 

 ity and the receiving or acoustoelectric sensitivity. Ex- 

 pressions for these are obtained and a proof of the 

 reciprocity theorem, which is basic in much of under- 

 water sound calibration work, is given. 



The transmitting sensitivity .S(R) of a transducer is 

 defined as the ratio of the pressure developed by the 

 transducer at the point R, when driven electrically, to 

 the input current to the transducer.' 1 In practice, the 

 point R is taken as a point at unit distance (1 meter) 

 on the axis of symmetry of the transducer. The value 



p The transmitting and receiving sensitivities defined here 

 bear a close relationship to the transmitting and receiving re- 

 sponses defined below. (See Chapter 4.) 



'i For simplicity we shall denote a point in the medium by its 

 position vector R from an arbitrary origin rather than by its 

 coordinates. 



of the sensitivity S(R) depends upon the properties of 

 both the transducer and the medium in which it is 

 operating. 



It is now necessary to express the pressure p(R) at 

 any point R in the medium in terms of the normal 

 velocity v„ of the diaphragm. This relationship can be 

 shown to be 



/>(R) = -??<„/ G(R,r)dr 



;<,jp_ 



'»£(*)• 



(14) 



Here the integral is taken over the acoustically ac- 

 tive portion of the transducer surface, the diaphragm 

 S,,; p is the density of the medium; a> is the angular 

 frequency of the sound wave (27r times the frequency); 

 and G(R, r) is the so-called Green's function. e Physi- 

 cally, it may be defined as the pressure which would 

 be produced at the point R as a result of a point 

 source of unit strength placed at the point R', if the 

 diaphragm of the transducer did not move. G(R, r) is 

 simply G(R, R') for R' taken at the point r on the 

 closed transducer surface S. It can be shown that such 

 a function can in principle be calculated for all ordi- 

 nary surfaces S, and that it is symmetric in its argu- 

 ments, that is, G(R, r) = G(r, R). The function g(R) 

 is introduced simply as an abbreviation for the 

 integral 



g(R) = / G(R, r) dr. 



(15) 



Since we are considering the transmitting sensitiv- 



e Green's function, G(R.R'), is defined mathematically as a 

 solution of the wave equation, which has a pole of residue unity 

 at the point R = R'. which satisfies the boundary condition 

 3G(R,R')/3» = on the closed transducer surface S, and 

 which, as |R — R'| _» », behaves like 



/ R - R ' V 



VIR-R'I/ 



R-R'j 



|R-R 



(X = wave length: /( — lis a function whose nature is de- 



lermined by the surface S), that is, it resembles an outward 

 travelling wave. Green's function can be shown to exist mathe- 

 matically for all ordinary surfaces S. It can also be shown that 

 in the particular case of a piston-like diaphragm in an infinite 



-.,-^|R_R'| 



rigid baffle, G(R.R') = : 

 surface of the piston or baffle. 



IR-R'I 



-.when R' = r lies on the 



