D. Schofield 13 
no insulation problems; it does not have tobe enclosed in a watertight container. 
The disadvantages are that the unit, if made of metal, must be laminated to keep 
the eddy-current loss low, and this affects the useful top frequency (about 50 
keps); for optimum operation magnetostrictive materials require a bias field, 
necessitating an auxiliary magnetic circuit. 
1.6. EXAMPLES OF TRANSDUCERS 
The remainder of the paper will be devoted to a consideration of two typical 
transducer designs: the first is a design of a high-power projector, the basic 
ideas of which can be used from a few kcps to over 100 kcps; the second is a 
hydrophone for operation deep in the ocean. 
1.6.1. Projectors 
Simple resonant bars do not have optimum characteristics for projectors. 
For example, the mechanical Q of a longitudinal resonant bar of barium titanate 
radiating into water is about 50; whereas for the maximum bandwidth condition 
(0.%0,) and with a coupling factor of 0.2, 0, should be 5. A more complex 
acoustic oscillator design has a larger number of variables with which to achieve 
the required characteristics. 
1.6.2. Resonant Frequency 
Consider an idealized compound-bar resonator [14,15], of the type shown in 
Fig. 1.4. The center of the ceramic sectionis a velocity node and each half of the 
element may be considered separately as a quarter-wave vibrator, the head 
being 3-ply and the tail 2-ply. By setting up the equations of the dynamic dis- 
placements in the various plys of the element and substituting the various bound- 
ary conditions at the interfaces, the conditions for resonance can be determined. 
For the 3-ply vibrator we have 
234s D LIES 5 5 AED oo ai (13) 
Z2A2 ZA, ZA, 
where Z=p,c, is the characteristic impedance; p; is the density of the ply 7; 
c, is the velocity of sound in plyi; t; =tank,l, ; k, is the wave number of ply i 
1 is the length of the ply7; and a is the cross-section area of plyi. 
If 13 KA3, tan k3l3 ~k3l3 and Z3A3t3=@M, where M is the mass of section 3. 
CERAMIC SANOWICH SS 
HEAD MASS 
“A 
WHY Fig. 1.4. Compound-bar piezoelectric 
element. 
|e — 
HEAD SHANK ar ans 5 —l TAIL 
