EFFICIENCIES 



15 



the asterisk denoting the complex conjugate and S 

 being any closed surface containing the transducer. 

 The electric power input is given by |/| 2 R c ,i where |/| 

 is the absolute value of the (complex) current and R 0l 

 is the real part of the effective electric impedance of 

 the transducer Z A . 



If S is chosen as a large sphere of radius d centered 

 at the transducer, we have 7',, = p/pc on S. 747 "' Then 

 the expression for the acoustic power output be- 

 comes: 



1 



P c 



fjP\ 2d% =i Id ~ 



rfS being now an element of area on the sphere and / 

 the sound intensity. Introducing the directivity fac- 

 tor 8, defined as 1 



8 = 





(30) 



with p axi8 the pressure at distance d on the axis of 

 symmetry of the transducer, or, generally, on any 

 fixed axis, one obtains for the projector efficiency 



£,= 



pc 



I'pRel 



4,rd 2 8|S| 

 pc Rei 



A 2 4R el 



(31) 



where S and M are the transmitting and receiving 

 sensitivities; see equations (16), (25), and (28). 



Another possible choice for 2 is the surface S of 

 the transducer. Then, v„ is the normal velocity of the 

 transducer surface, assumed constant over the dia- 

 phragm. Thus from equations (6) and (29) we have 

 the projector efficiency expressed as 



£„ = 



/| 2 R„ 



(32) 



where r r is the real part of the radiation impedance z r . 

 Finally, using equations (7) and (8) to find v n /I and 

 equation (9) from R el and substituting the results into 

 equation (32), we obtain 



E„ = 



\kr- 



+ Z r \ 



(33) 



(kk' \ 

 Z b J is the real part of the 



1 For further discussion of the directivity factor 5 and of the 

 directivity index = 10 log S, see Chapter 4. 



effective electric impedance, Z,,,. It may be shown 39 

 that a sufficient condition for E p to be 100 per cent 

 is R 6 = ) „ = 0, that is, the blocked electric and open- 

 circuit mechanical impedances have no real parts. 



It is worth pointing out in connection with equa- 

 tion (33) that the projector efficiency (at resonance) 

 of a resonant transducer may be determined by 

 purely electrical methods. (See reference 39.) In a 

 resonant transducer, the response as a function of 

 frequency has a sharp maximum when the impressed 

 frequency coincides with a natural frequency of the 

 transducer itself. If one measures the electric im- 

 pedance of the transducer at a frequency well above 

 and below its resonant frequency, the result will be 

 essentially Z b , the blocked impedance.™ Suppose now 

 one measures the electric impedance of the trans- 

 ducer at resonance in air. Then z r is approximately 

 zero and the motional impedance (the difference be- 

 tween the measured electric impedance and Z b ) will 

 be —kk'/z . Next one measures the impedance at re- 

 sonance in water. Then the motional impedance in 

 water — kk'/(z + z r ) is known. These three measure- 

 ments suffice for the determination of E p , if the trans- 

 ducer obeys the reciprocity condition \k\ = \k'\, so 

 that |/{| 2 = \kk'\. 



This may be seen as follows: At resonance the 

 imaginary part of z„ + z r vanishes," and |z + z,.j 2 = 

 (r + r r ) 2 . Then equation (33) may be written as 



E. 



\k\- 



RelO'o + >V) 



Rel 0"0 + >,)- 



\kk' 



\kk' 



1 - 



r n + r r 



\kk' 



(34) 



where the second form uses the reciprocity condition 



\ k \ 2 = \ kk '\- 

 It is now seen that R e , '**'' — A ^ kk " 



and 



are the 



r + >;. r 



only quantities needed for a knowledge of E p at re- 

 sonance. All of these can be found by the method 

 described, assuming that r is a slowly varying func- 

 tion of frequency so that r , at the resonance fre- 

 quency in air, is close in value to ?\, at the resonance 

 frequency in water. 



m Z h is in general a function of frequency hut does not show 

 resonance properties. Hence its value at the resonant frequency 

 of the transducer may he found hy joining the portions of the 

 curve found above and below resonance by a smooth curve. 



n When the imaginary part of : (1 -4- Z vanishes, the responses 

 S and M have their maximum values; see equations (16) and (25). 



