38 A. 
Be JA‘RLOON:S: VAIN D) &Ry. oH IG OLE 
Fic. 7. Amplifier error vs. 
non-dimensional amplifier ad- 
mittances 8 for incident waves 
of various a. (See text for ex- 
planation of notation.) 
WON- DIMENSIONAL AMPLIFIER ADMITTANCE, @ 
impedance can generally be made sufficiently high by 
use of cathode follower preamplifiers. 
The situation with respect to high frequency require- 
ments is slightly more complex in that a certain amount 
of high frequency distortion is unavoidably present be- 
cause of the geometry of the gauge. (This effect was 
computed for some simple wave forms in Section V.) 
In general, therefore, the demand upon amplifier re- 
sponse will be conditioned by the gauge response, and a 
useful quantitative calculation can be made for the 
special case of steep-fronted waves, where high frequency 
considerations are most important. 
We assume that a saw-tooth wave is incident at the 
gauge and that the shape of the signal reaching the 
amplifier can be represented to a first approximation by 
a linear rise to a peak value followed by a linear decay, 
i.e.: 
0 when ‘<0 
F(t)=4 t/y when 0</<y, (7) 
1—(t—-y)/@ when i>y 
where 6 is the decay constant of the wave and y=2a/c, 
the transit time of the shock wave across the gauge. 
The step response of the amplifier is assumed to have 
the form: 
R)=1-e"«, (8) 
Expressing the above relations in terms of dimension- 
less parameters with 6 as the scaling factor (a=7/8, 
8=x/@), and applying the superposition integral, it is 
shown that: 
A=8 Infi+(1—e-**)/a], (9) 
where A represents the fractional error introduced by 
the amplifier in reproducing the peak of the applied 
function, F(/). F(t), however, already contains an error 
a/2, with respect to the peak pressure of the incident 
shock wave. Thus the total error introduced by the 
combined effects of gauge size and amplifier high fre- 
quency response is: 
E=A+a/2 (16) 
with respect to the incident shock wave peak pressure. 
Figure 7 is a representation of Eq. (9) which should 
prove useful in estimating error due to a given frequency 
response for various gauge conditions or for design 
purposes where one seeks to find the frequency response 
necessary to keep the error below a certain selected limit. 
The problem of calibrating piezoelectric gauges is a 
vital one, involving rather highly specialized instru- 
ments and techniques. Owing to fairly recent develop- 
ments in the field, calibration will be discussed in a 
separate paper at some future date. 
Vil. ACKNOWLEDGMENTS 
The knowledge and experience upon which this report 
is based could not have been accumulated were it not for 
the cooperative effort of a large number of individuals 
who participated in the work of the Underwater Ex- 
plosives Research Laboratory. The authors are pleased 
to acknowledge in particular the significant contribu- 
tions of Professor E. B. Wilson, Jr., Dr. W. D. Kennedy, 
Dr. W. G. Schneider, Dr. A. M. Shanes, and Dr. W. E. 
Gordon, and regret that space does not permit a more 
adequate identification of the contributions of each 
individual to the fundamental understanding of the use 
and design of piezoelectric gauges. Much credit for de- 
velopment of techniques of construction of the present 
gauge design is due Professor C. Frondel of the Depart- 
ment of Mineralogy, Harvard University. Earlier con- 
struction techniques were developed by Dr. D. Silver- 
man and Mr. H. M. Lang of the Stanolind Oil and Gas 
Company. 
