P. M. Kendig 249 
R is perhaps the least significant; yet, as a sound projector, it is the most 
important because the energy delivered to this element represents the radiated 
acoustic energy. Therefore, the value of this element, relative to the other re- 
sistive elements, determines the efficiency and, hence, the equivalent noise 
pressure of the transducer. 
Examination of the equivalent circuit of Fig. 13.8 shows that the electrical 
input admittance is ; 
V 2G, tand + jaCs +e Ons jo€ 
(RoC)? toll 
and since RaC «<1 at these low frequencies, * 
Y =@Cp tand + R(@C)? + ja@(C + Co) 
Again, since w(C + Co) > [wCp tand + R(@C)*], 
2 
wCy tand + R(wC) (1) 
Rao Ban= 
Bo ice arC= Gyr 
Since the second term is that portion of the series resistance produced by 
acoustic radiation, the efficiency is simply this term divided by R7, or 
Rw?C? 
See Sa ESE RO 2 
@Cy tand + Rw*C (2) 
7 
Now, R; depends upon the specific acoustic impedance of the medium (py) and 
the dimensions of the transducer. At low frequencies, where the radius is con- 
siderably smaller than the wavelength, the expression for the real part of the 
mechanical radiation impedance is approximately 
R, = Ampaso (3) 
Vv 
where p is the density and vis the sound velocity of the medium. 
For a thin-shelled hollow sphere, the stiffness is approximately 
87bE’ (4) 
where E’ is Young's modulus of elasticity and o is Poisson's ratio. The modulus 
E'is the short-circuit or zero-field modulus, and it is related to the open-circuit 
or zero-electric-displacement modulus E by the relation 
E'= E(1 -k?) (5) 
where k, is the electromechanical coupling coefficient. This may be expressed 
approximately by 
2) Co 
eG Gh (6) 
For a thin-walled hollow sphere, the clamped capacity is approximately 
2 
Cy = cents (7) 
*The equality sign (=) will be used throughout, even though most expressions are not exact equalities. 
