A two transducer correction formula valid for both solid and liquid specimens is presented. Using computer simulations of velocity measurements, the accuracy and range of validity of the results are discussed and are compared with previous approximations

Miller, J. G.

Moerner, W. E.

Ringermacher, H. I.

English

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TWO TRANSDUCER FORMULA FOR MORE PRECISE DETERMINATION OF ULTRASONIC PHASE VELOCITY
FROM STANDING WAVE MEASUREMENTSt N75-18544'
)RIGINAL PAGE 1i- I. Ringermacher, W. E. Moerner, and J. G. MillerLaboratory for Ultrasonics
O'F POOR1 QUALITY Department of Physics 3 p HC Unclas
Washington University CSCL 20A G3/35 12487
St. Louis, Missouri 63130
ABSTRACT. Ultrasonic velocity measurements in solids and liquids using standing-wave techniques cannot
be accurately analyzed without the use of an appropriate transducer correction formula. We discuss the two-
transducer (transmission) case. An improved transducer correction is derived which is substantially more
accurate than all previous approximations over the range of parameters corresponding to velocity determinations
in both solids and liquids. Previous approximations are useful only over very limited ranges. We discuss the
relationship between the present result and a previously derived result for the one-transducer case. Computer
t-0 ',simulations of velocity measurements demonstrate the accuracy of our formula under a wide variety of conditions.
SD I. Introduction In the absence of transducers, the isolated
vn En W W specimen resonances would be equally spaced at intervals
'a tt Z Standing wave ultrasonic measurements can be Avs = vs/2e s. However, the observed resonance spacingsS1 performed in either the reflection mode (one transducer n+ n
a W r r the transmission mode (two transducers). For the Ac = v - v of the composite system, vary with :the
t r c- Ietermination of ultrasonic phase velocity it is es- particular mechanical resonance pair. This leads to
Z H -4 sential in either case to incorporate an appropriate certain inaccuracies encountered with the so-calledi
3 C transducer correction. "uncorrected formula"
-4 t 1 An improved velocity correction for the one v = 21s AV (3)
S N tJ transducer case was discussed previously.' The two c
n W transducer problem is discussed only briefly in the For resonances not too far from the transducer reso-
W 0 W literature.. , The approximations adequate under nance frequency vt, an approximation2 to Eq. (1) yields
S~ 13 Hnarrow constraints in solids, for example, are found
E to be totally inadequate in liquid studies. vs = 2 s  +2) (4)
o > 1 - In this paper we present a substantially more where 6 = ptt/psZ. Anticipating the results of4 't accurate two transducer correction formula valid for Section III, this '1+26 formula" is found to be
t 0 4 both solid and liquid specimens. Using computer adequate only for experiments with solids, where 6
a ( simulations of velocity measurements.the accuracy is typically less than -0.02.
P-" 0 tv and range of validity of the new results are discussed
S and are compared with previous approximations. Using Eq. (1), we develop an improved transducer cor-
- 0 cl b rection useful over a wider range of 6. Equation (1)
N4 to II. Theory is quite complex in comparison with the one transducer
W-H resonance equation (Eq. (4) of Ref. 1). Using trigono-
SThe system we treat is a one dimensional com- metric substitutions, however, Eq. (1) is found to be
w ~, posite resonator consisting of a specimen (with proper- factorable into two simpler equations,
0 t ties characterized by subscript s) and two identical r-
transducers (subscript t). We wish to find the sound tan ---s -) tan t = 0 (5)
H3 , velocity vs in the specimen having measured a set of
mechanical resonance frequencies vn of the composite e
I system. Using a transmission line analogy, Bolef and cot + (r) tan t = (6)
o Menes3 treat the three-element system consisting of a 2 r+
transducer, bond, and specimen. The transducer-
specimen-transducer problem can be formulated using (These two equations are identical to the one-transducer
a similar approach. 2  resonance equation of Ref. 1 provided that Zs is re-
placed by £s/2.) The solutions of (5) and (6) comprise
Provided that the ultrasonic attenuation is the full set of two transducer resonances, symmetric
not excessive, the mechanical resonance frequencies and antisymmetric, s respectively. Taken separately,
vn of the composite system are the solutions of Eqs. (5) and (6) describe alternate resonances.
tan 
_ 
2(_rrTtan 2t T n t e 0
tan -r - tan2 t tan s =0 (1) In order to develop a velocity approximation
dependent upon single spacings 
~  
= vn+l _vn, wewhere nc c c
vn n subtract Eq. (6) written for the nth resonance from
es = V and et = - Eq. (5) written for the (n+ l)th resonance. (The
s t approximate expression for the velocity which is
and where obtained is identical to that obtained if Eq. (5) for
and where issubtracted from Eq. (6) for vn+Z.) Following
r (pv - Pvt)/(pv + v) a procedure similar to that used for the one transducer
ss tt ss ptt) case and employing the- approximation.av n* - , we
arrive at the velocity correction formula for the two
is the reflection coefficient from specimen to trans- transducer case,
ducer. The notation employed is the same as in the n 2 1
one transducer case.' For computer evaluation an 2 +c v7 vt DT2 (7)
purposes Eq. (1) can be written in the equivalent form s 2 n +26 + DT + T , (7)
iin(2et + a ) - r in(2et- es) + 2r sin e= 0 .(2) C
https://ntrs.nasa.gov/search.jsp?R=19750010472 2020-03-22T22:41:48+00:00Z
where Tn = tan i vn/vt  . .0 , .
UNCORRECTED FORMULA
D = (Tn+ 1 - TO o Eq. (31) .-
Although this expression differs only slightly from Eq.
(6) of Ref. 1, the appropriate generalization is by no+28) FORMULA
means obvious a priori. In particular, the replacement 10 (+2) FORMULA
of 6 by 26, as might be suggested by Eq. (4), is not Eq.(4)
appropriate. F /
In Section III we compare the accuracy of this z 0, -
new result with that of the 1+26 formula, the uncor-
rected formula, and an empirical approximation tech-
nique sometimes used in solid studies. W 16 BOLEF-MENES FORMULA
Eq (8)
ILl. Discussion ...
In order to examine the behavior of the various
approximate formulas for vs , numerical iteration was
4used to find to an accuracy of 1 part in 1010 the R0W
,frequencies vn satisfying Eq. (2) for an assumed set VPRESENT WORK
of parameters ps, pt, Vs, ,s, t. In these simulations, Eq.(7)
all arbitrary parameters were given values typical of 04
those encountered in ultrasonic experiments. The re-
sulting set of simulated mechanical resonance frequen-
cies was used in conjunction with the expressions for
v to compute approximate values for the phase velocity. 10 '
The percent error for each approximation was computed 0.o0, 0002 0.003 oo5 0.0o07 01 0015 0.02
with respect to the value of vs assumed in the initial a (dimensionless units)
iteration. Figure 1. Absolute value of the percent error
In general, experiments involving solid samples in the velocity of sound vs versus
feature smaller values of 6 than do experiments on 6 = pt/Pss for 6<0.02 ("solids
liquids. Accordingly, we somewhat arbitrarily choose region).
to divide the range of 6 studied into two parts. We
call the region where 6 < 0.02 the "solids" region Also shown in Figure 1 are the results of the
and the region where 6 > 0.02 the "liquids" region. present work (Eq. (7)). The improved accuracy of Eq.
The approximations analyzed include the uncorrected (7) over the Bolef-Menes expression (Eq. (8)) may be
formula (Eq. (3)), the 1+26 formula (Eq. (4)), our of value only if measurements to an accuracy of 1 part
new approximation (Eq. (7)), and an additional expres- in 0I7 are available.
sion which we shall call the Bolef-Menes formula.
3
'
67
The Bolef-Menes formula involves rounding to the O2  ,
nearest integer m the following expression for m': 10
n (--
m' =- (l- 26) (Sa)
C / (1+28) FORMULA
/ Eq.(4)
The approximate expression for the velocity is t Eq. (4
2ps c t >V n"
v s  1 + 26 V-t- (8b) ---
C / UNCORRECTED FORMULA
o rEq.($)
In Figure 1 we display the percent error for loo (
each of the various approximations over the range of
6 typical of experiments on solid specimens (6 < 0.02). z
The pair of mechanical resonances used in the calcula- w
tions were the fourth and fifth on the high-frequency
side of vt . The 1+26 formula (Eq. (4)) is seen to w PRESENT WORK
result in errors of roughly 1 part in 104 over most Eq.(7)
of the range. In contrast, the uncorrected formula 10
(Eq. (3)) yields errors approximately two orders of
magnitude larger than those resulting from the 1+26
formula. (The cusp-like behavior near 6 = 0.01 for
Eqs. (4) and (8) and near 6 = 0.004 for Eq. (7) is
due to a change in sign of the error.)
2 I I I
The Bolef-Menes expression (Eq. (8)) is also L00 .05 0.1 02 0.3 0.4
presented in Figure 1. We note that Eq. (8) is 8 (dimensionless units)
superior to Eq. (4) over the range of 6 shown by
roughly two orders of magnitude. Equation (8) rapidly Figure 2. Absolute value of the percent error
deteriorates as 6 nears 0.02, rendering it unusable in in Vs v ersus 6 E p~pt/p n for
the "liquids" region. 6 > 0.02 ("liquids region).
ORIGINAL PAGE IS
OF POOR QUALITY
Figure 2 shows the error curves in the "liquids" Acknowledgements
region (6 >0.02) for Eqs. (3), (4), and (7). (As noted
above, the Bolef-Menes formula (Eq. (8)) diverges in The support and encouragement of Professor R.
this region.) As in Figure 1, the pair of resonances E. Norberg and the contributions of Or. D. E. Yuhas
used in the calculations were the fourth and fifth on are gratefully acknowledged.
the high-frequency side of vt. The 1+26 formula (Eq.
(4)) results in large errors for 6 greater than 0.02. References
We notq in particular that the uncorrected formula
becomes more accurate than the 1+26 formula as 6 Supported in part by the Air Force Office of
increases. The accuracy of the present result (Eq. Scientific Research, the National Science Foundation,
(7)) is superior to all previous results by roughly and the National Aeronautics and Space Administration.
a factor of 10. 1H. I. Ringermacher, W. E. Moerner, and J. G.
Miller, J. Appl. Phys. 45, 549 (1974).
The behavior of the various approximations . 2D. I. Bolef and J. G. Miller in Physical
depends upon the distance in frequency between the Acoustics Vol. 8, edited by W. P. Mason and R. N.
resonance pair (vc, v~+l) and vt in a fashion similar Thurston (Academic Press, New York, 1971).
to that for the one transducer case.' Briefly, the 3D. I. Bolef and M. Menes, J. Appl. Phys. 31,
1+26 formula is most accurate near vt, while the 1010 (1960).
errors for the uncorrected formula and the present 4W. P. Streett, H. I. Ringermacher, and J. L.
work decrease rapidly with distance from vt. Burch, J. Chem. Phys. 57, 3829 (1972).
sF. Eggers, Acoustica 19, 323 (1967/1968).
In situations where electromagnetic leakage 60. I. Bolef, N. T. Melamed, and M. Menes, J.
complicates velocity measurements,8 a formula in- Phys. Chem. Solids 17, 143 (1960).
volving the spacing between resonance n and n+2 may 7R. L. Melcherand D. I. Bolef, Phys. Rev. 8178,
be useful. Such a double-spacing formula is given 864 (1969).
by Eq. (6) of Ref. 1 with 6 replaced by 26 and ts  8J. G. Miller, J. Acoust. Soc. Am. 53, 710 (1973).
replaced by Zs/2. The error behavior for this double-
spacing formula is very similar to that for the
present result, Eq. (7).
ORIGINAL PAGE IS
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Two transducer formula for more precise determination of ultrasonic phase velocity from standing wave measurements

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