835 
PIEZOELECTRIC GAUGES 35 
0 
REDUCED FREQUENCY, Ff a/e 
Fic. 3. Frequency response of a circular gauge of radius a 
oriented edge-on to an advancing pressure wave. (f is frequency, 
and ¢ is velocity of wave propagation.) 
pressure of value Po, traveling with velocity c parallel to 
the faces of the disk. If the pressure discontinuity is at 
x=0 at time ‘=0, and the gauge of radius a has its 
center at x=0, the response of the gauge, R(t), is readily 
shown from geometrical considerations to be given by: 
0, when t<—a/c 
R(t)= (o= sing cosd)/7, when —a/c<t<a/c, (2) 
ile ~ when t>a/c 
where 
¢=cos!(—ct/a). (3) 
Equation (2) is plotted in Fig. 5 as a function of ct/a. 
From the symmetry, it is evident that the area under 
the response curve is the same as the area under the 
ideal curve if the integration is carried. from —a/c to 
times greater than +a/c. The interval 2a/c is obviously 
the time required for the pressure wave to cross the 
gauge. For a 12.6-mm diameter gauge in water, this 
crossing time is 8.4 usec. It is apparent that a gauge of 
diameter 2a cannot be expected to give very detailed 
information about pressure changes occurring in time 
intervals of 2a/c or less. 
(3) Response to saw-tooth wave. The response of a 
circular gauge to other arbitrary forms of transient can 
in principle be obtained either from Eq. (2) for the step 
response by use of the superposition theorem or from 
Eq. (1) for frequency response by use of the Fourier 
integral theorem. In many cases the practical difficulties 
of the calculations are considerable, and answers in 
closed form involving evaluated functions are not ob- 
tainable. In the special case of the type of wave en- 
countered in shock-wave measurements, however, the 
nature of the response can be adequately illustrated by 
assuming a saw-tooth shape. If the incident wave is 
given by, 
= when t<x/c (4) 
P(x, d= {Pr (1/0)(t—x/c)], when > 2/e 
the response of a circular gauge is shown to be, 
0, when t<—a/c 
$/ a— (sing cosp)/m— (a/37c8) 
(3 sin6—3¢ cos¢—sin*p), (5) 
when —a/c<t<a/c 
when t>a/c 
S()= 
1—1/0, 
where, as before, ¢=cos—!(—ct/a). 
The saw-tooth response of Eq. (5) is plotted in Fig. 6 
as a function of ¢/@ for various values of the ratio a/cé. 
If this ratio is less than about 2, the maximum response 
lies very nearly on the true response curve and is lower 
than the true maximum by the fraction a/c. The time 
required for the gauge to give the maximum response is 
very nearly the crossing time 2a/c. As a numerical ex- 
ample, consider a gauge 12.6 mm in diameter used to 
record a shock wave in water having a decay constant 
6=400 usec. The time to cross the gauge is 8.4 usec. and 
the indicated peak pressure will be approximately one 
percent low. If a gauge is to record a shock wave having 
6=SO usec. with less than one percent error in peak 
pressure, its diameter should not exceed 1.5 mm. 
The behavior of the saw-tooth response suggests that 
a measured value of peak pressure for a wave similar to 
a saw-tooth in form could be corrected for the effect of 
gauge size in either of two ways: (1) by adding a fraction 
(a/c0)/(1— a/c) of the maximum response to this value ; 
(2) by extrapolating the initial decay of the response 
back an interval a/c along the time scale, determined 
either from the measured radius a or from the measured 
Expenmentol for gouge 890. 
Diometer 3/47, including coating. Edge on. 
Response Rif) 
Theoretical response 
R (f= 24 (2TT fase) 
2TTfore 
f is frequency in cycles/sec 
Gis gouge radius 
¢ is velocity of sound 
400 is Bessel’s function of 
the first kind, index one 
YT TTT, 
Nell 
PT ert TT a bit 
TTT TTT J= Experimenta resuts for gouge 
Diometer V4", including coating. Edge on. 
L-- Theoretico! results 
=n a it 
NORG report:"Colibration of Hl 
‘Stonolind Ol 6 G: Co. Tourmolin na AL 
Mleseecinen tetas ene sou WA ST 
on 
CT 
Css Ep aly cores Ht 
0.2}—4Under Water Sound Reference 
Loborotories , Section "61-sr l130-1971 
1000 
frequency { (kc) 
Fic. 4. Comparison of experimental and theoretical frequency 
response curves for two gauges. (Experimental calibrations per- 
formed in sinusoidal acoustic field.) 
