248 Lecture 13 
a 
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ENON SID Cc 
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R 
R=— 
a2 
Fig. 13.8. Equivalent transducer circuit for low frequencies, 
The first factor on the left is the equivalent sound intensity of the ambient 
thermal noise for a 1-cps band and the second factor is the area of the radiating 
face. Therefore, this expression states that for a 100%-efficient transducer, 
the ambient thermal noise contained in a 1-cps band is equivalent to a rate of 
flow of acoustic energy equal to kT, which, as we know, is the energy of one de- 
gree of freedom and is less than the kinetic energy of a single gas molecule.* 
Note especially that this is true no matter how large the radiating face of the 
transducer. On the other hand, Eq. (9) of Section 13.3 shows that the sensitivity 
or free-field voltage response of a hydrophone varies as the square root of the 
directivity, and hence it does increase with the size of the radiating face. Thus, 
we see that, at least for ambient thermal noise and indeed for any isotropic 
noise, an increase in the directivity provides an equivalent increase in the 
signal to ambient-noise ratio when the main beam of the receiving hydrophone 
is on the target. 
13.3. FACTORS THAT DETERMINE THE EQUIVALENT NOISE PRESSURE, FREE-FIELD VOLT- 
AGE RESPONSE AND EFFICIENCY OF A TRANSDUCER AT LOW FREQUENCIES [13] 
13.3.1. The Equivalent Circuit and the Efficiency 
Consider a thin-walled hollow sphere of radius a and wall thickness »b that 
is vibrating in the radial mode at a frequency well below resonance. At these 
frequencies, the equivalent circuit may be represented by the circuit diagram 
shown in Fig. 13.8, where Co is the clamped capacitance, C is the motional ca- 
pacitance, R is the resistance resulting from the mechanical load R;, XK is the 
stiffness, a is the electromechanical transformation ratio, tan 6 is the loss tan- 
gent (ratio of clamped resistance to clamped reactance), and is the angular 
frequency. The purely mechanical losses will be omitted in this discussion. 
At the low frequencies under consideration, the impedance is almost en- 
tirely capacitive. Indeed, from impedance considerations alone, the resistance 
*This expression appears to have the units of energy, because the bandwidth which has dimensions of 
reciprocal time was taken to be unity. 
