D. Schofield 7 
attention is restricted to a frequency region around a mechanical resonance, 
only this force need be considered, and the force law reduces to F = yV where 
y is a constant. It is easy to show that the displacement law is given by q= yé 
where & is the displacement. 
In all other force effects on matter due to the application of electric fields, the 
stress is proportional to the square ofthe electric field. There are three separate 
force effects: 
a. Dielectric stress, the attractive force between condenser plates 
b. Ve stress, where « is the dielectric constant 
c. Electrostrictive stresses 
In magnetism there are analogs ofthe Ve and the electrostriction effects, both 
proportional to the square of the magnetic induction, but analogs of the dielectric 
stress or piezoelectricity do not exist since there are no true free magnetic 
charges. There is also one force effect which has no analog in electricity. This 
is the force exerted ona conductor oflength / carrying a current i in a magnetic 
field of induction B given by F=Bli and is usually called the electrodynamic 
effect. It is a linear function of both magnetic induction and current. 
In many underwater acoustic experiments, it is important that the transducer 
not introduce any changes in the waveform and this dictates the use of a linear 
conversion effect. Therefore, only the piezoelectric and electrodynamic effects 
are immediately suitable for transducers. Compensation for any variation in 
response with frequency can usually be made in the electronics. 
It is easy to show that, ifthe mechanical stress is dependent on the square of 
the electrical or magnetic field, the frequency of the developed stress is twice 
that of an impressed sinusoidal variable. However, if a dc electrical quantity is 
added to the small alternating electrical quantity, itis possible, within limits, to 
obtain energy conversion without frequency change. This is the so-called polar- 
ized method of operation. Insome electrostrictive and magnetostrictive materials 
the remanent electric displacement or magnetic field of induction provides almost 
optimum polarization and an external dc field is not required. It is also important 
to note that with quadratic force effects the conversion of mechanical to electri- 
cal energy is associated with the existence of an electric or magnetic field. Dur- 
ing the remainder of the paper, electrostrictive and magnetostrictive materials 
will always be considered polarized, e.g., ferroelectric materials such as barium 
titanate and lead zirconate titanate will be considered as pseudopiezoelectric. 
1.3. EQUIVALENT ELECTRICAL CIRCUITS 
Every electroacoustic transducer is an electric storage element, e.g., capa- 
citor, or inductance, combined with an acoustic oscillator. The latter affects the 
input impedance of the transducer, and the analysis of a transducer as a com- 
ponent of an electrical network is greatly facilitated if the unit is replaced by an 
equivalent electrical circuit. It can be shown by relatively simple analysis [2] 
that, neglecting losses, a piezoelectric bar excitedina longitudinal mode of vibra - 
tion can be represented by the equivalent electrical circuit of Fig. 1.1. The 
parameters for the equivalent circuit are: 
