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ABSTRACT

A REFINED OPTICAL TECHNIQUE FOR THE
MEASUREMENT OF ULTRASONIC ABSORPTION

by William J. Taczak

The optical method has been used quite extensively for
the measurement of ultrasonic absorption in liquids. However,
acoustic diffraction has limited the accuracy of such measurements
because of large local variations in the optical effect. It is
shown in this paper, both theoretically and experimentally, that
a light beam traversing an acoustic beam radiated from a square
transducer, parallel to a diagonal of the transducer, produces
an on-axis integrated optical effect that is nearly constant in
the near field of the sound beam. In the present experiment,
absolute and relative ultrasonic absorption coefficients are
determined for aqueous solutions of the electrolytic salt,
manganese sulfate. Absolute ultrasonic absorption measurements
are made in this region by assuming that the on-axis effect is
constant. For relative ultrasonic absorption measurements, the
following procedure is used: (1) the integrated optical effect
is measured in water, which has negligible acoustic absorption;
(2) the total optical effect is measured with manganese sulfate
added to the water; and (3), (1) is divided into (2) to correctfbr
the acoustic diffraction. With proper experimental care and with
a nearly constant region of sufficient range, relative measure-
ments will give reliable values for ultrasonic absorption coeffi-

cients to<x = .001 em-l.

A REFINED OPTICAL TECHNIQUE FOR THE

‘MEASUREMENT OF ULTRASONIC ABSORPTION

by

. “\

H
William J? Taczak

A THESIS

Submitted to
‘Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics

1968

ACKNOWLEDGMENT

The author wishes to express his gratitude to
Professor E. A. Hiedemann and Dr. B. D. Cook for their guidance
in this work. The discussions with Dr. F. Ingenito also have
been very helpful. The financial support of this research by

the Office of Naval Research is gratefully acknowledged.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENT
CHAPTER
I. INTRODUCTION
II. THEORY
III. EXPERIMENTAL STUDY
IV. RESULTS
V. CONCLUSION
BIBLIOGRAPHY

iii

Page
ii

11+
18 '
25

26

LIST OF FIGURES

Figure Page

1 Schlieren photograph of the sound field produced
by a square transducer with the light beam cros-
sing the sound field parallel to (a) one of the
sides of the transducer, and (b) a diagonal of

the transducer. . . . . . . . . . . . . . . . 2
2 Coordinate system of the transducer configura-

tion. 0 O O O O O O O O O I O O O O 3
3 Theoretical curve of the integrated optical effect

for a light beam crossing the sound field, radia-
ted by a square transducer, parallel to one of

the transducer diagonals. . . . . . . . . . 12
A Schematic diagram of experimental arrangement. . 15
5 Comparison of theoretical and experimental results;
normalized to flat region (f* = h.0 mHz.,
a = 005 me). o o o o o o o o o o o o o o o 19
6 Frequency range of various techniques used for
measuring the ultrasonic absorption in.MnSOh. . . 20
*
7 Parameter oz/f 2 vs.oacoustica1 frequency for
001 m MnSOLI- at 2105 C. o o o o o o o o o o e 21
*
8 Parameter<1'x 'vs. acoustical frequency for
0.1 m MnSOh at 2105 Co 0 e o o o o o o o o o 22
9 Relative absorption coefficignt orvs. solution
concentration of MnSOh at 25 C. . . . . . . . 2h

iv

CHAPTER I

INTRODUCTION

Optical methods have been used by many investigators for
the measurement of ultrasonic absorption in liquids. The methods
have been restricted to ultrasonic frequencies above 2 mHz., because,
at low frequencies, the ultrasonic absorption is very small, requir-
ing the measurement to be made over a substantial path length in
order to obtain measurable effects. However, over large path
lengths, diffraction of the acoustic beam.becomes significant. In
addition, the angular separation of the Optical diffraction orders
is prOportional to the ultrasonic frequency, and at low frequencies,
it becomes difficult, if not impossible, to isolate an individual
diffraction order.

The latter difficulty can be overcome by keeping the
optical configuration constant. The more serious problem, dif-
fraction of the acoustic beam, is illustrated in Figure lal, a
schlieren photograph of the sound field radiated by a square trans-
ducer, with the light beam traversing the sound field parallel to
one of the transducer sides. The darker areas in the photograph
correspond to regions of higher acoustic pressure. It can be seen
that the diffraction of the acoustic beam produces large variations
of the Optical effect over relatively small regions. In making an
absorption measurement, the sound field is sampled by a narrow light

beam and the optical effects at various distances from.the source

.uoosmmcouu onu mo Hmeowmav m Any use «noosmmemuu
osu mo macaw onu mo oao Adv ou HoHHmuwQ macaw venom onu wcfimmouo Boon uzmHH
onu nufiz Hooswwemuu sheave a he vooswoum macaw venom can mo amonOuosm souowanom .H ouswfim

3v

 

 

 

 

 

l INCIDENT LIGHT

f (x:Y:z)
J (xoy )
3

5%

 

 

T

Y

Figure 2. Coordinate System of Transducer Configuration

are compared. It is clear that the large variations due to the
diffraction of the acoustic beam will tend to obscure the effect
of absorption.

Recently, Cook and Ingenito1 have devised a transducer
configuration which produces a slowly varying optical effect in
the near field of the transducer. Figure lb1 is a photograph of
the schlieren pattern produced by a square transducer with the
light beam traversing the sound field parallel to a diagonal of
the square. In this case, the optical effect is nearly uniform
close to the axis of the sound beam. Cook and Ingenito have also
calculated the theoretical on-axis expressions corresponding to
Figures la and lb, and their results agree with the experiment.

In the next section, we derive an expression for the on-

axis optical effect of the transducer configuration shown in Figure 2,

and it is shown that the resulting expression is the product

of the non-absorptive result of Cook and Ingenito and an exponent-

ial absorption factor. In a later section we describe the measure-
ment of the ultrasonic absorption in aqueous solutions of manganese
sulfate using the transducer configuration of Figure 2. Our technique
consists of measuring the total optical effect and dividing out the
non-absorption result, which is due to the transducer configuration,
leaving the factor that is due to the absorption of the solution,

from which the absorption coefficient can be found. Aqueous solu-
tions of the electrolytic salt mnsoh were chosen because (1) manganese
sulfate has a convenient relaxation frequency of about 3 mHz.,

(2) reasonable solution concentrations produce measurable absorption
and (3) other experiments have been done with this salt and these

will be used for comparison.

CHAPTER II

THEORY

In the presence of a sound beam, the local optical re-
fractive index is altered through the changes in density of the
liquid induced by the sound waves. If the sound beam ideally con-
sists of plane waves localized in a beam of width L, the region
containing the sound beam would act as a EElEE.°Ptical phase grating.
The nature of the diffraction of the light depends not only upon the
change of refractive index and the width of the sound field, but
also upon certain combinations of the wavelengths, k and X*, of the
light and sound respectively, with the width L and the angle of
incidence of the light beamz. At low frequencies and for the light
beam at normal incidence to the sound beam, many of the multiple
interference effects vanish and the diffraction phenomena can be
described by assuming that the sound beam.acts as a simple phase
grating. That is, for a plane light wave impinging of the sound
beam, only the phase of the light beam is modified and there is
no amplitude modulation. For light traveling in the y direction
and the sound in the z direction, the modulation of the light can
be described as

ei(kywnt) eiV(x,z,t)

, (1)

E = E
o

where
2 +°°
V(x,z,t) = -£; U/‘

-oo

P(x:Y:z)t) dy ° (2)

In Eq. 2, x is the piezooptic coefficient of the liquid and
p(x,y,z,t) is the localized acoustic pressure in the liquid. For
the idealized situation of plane sound waves of acoustic pressure
p = po sin (k* z -(fi*t), the light is diffracted into discrete dif-

fraction orders with the angular separation
sin en=3§ , (3)
x

and normalized intensity

I. = J§<v) , (u)

where Jn is the ordinary Bessel function and v = 2&5 po .
In the practical situation, these results are modified

by diffraction of both the light and the sound, as beams of finite

width are necessary. Limiting the size of the optical beam so it is

comparable to the acoustic wavelength causes an angular spreading

of the diffraction orders, even to the point of overlapping. The

acoustical beam radiating from a transducer of finite size is not

confined to a collimated beam, but undergoes Fresnel diffraction

in the region near the transducer. This diffraction causes

p(x,y,z,t) to vary locally in space, both in magnitude and phase.

Integration over y of p(x,y,z,t) in Eq. 2 yields a V(x,z,t), the

magnitude of which varies with both x and 2 because of diffraction.
It is desirable to have the variations of V(x,z,t),

arising from the acoustical diffraction, to be as small as possible

when using the optical technique for absorption measurements. As

 

 

the diagonal transducer configuration, discussed earlier, provides

a means of reducing the variations, we shall theoretically calculate
the integrated optical parameter V(x,z,t) from a sound field pro-
duced by this configuration. Since the schlieren photograph in
Figure lb indicates that the integrated optical effect does not

vary much in a region near the transducer axis, only the axial values
will be calculated.

In order to obtain the integrated optical effect, an
expression for the sound pressure field of a square transducer
must first be found. Then the optical effect of a light beam
traveling through the sound field, parallel to a diagonal of the
transducer can be determined.

Consider a square transducer of side 2a, in an infinite
rigid mounting, radiating into a fluid of average density p0. The
transducer is assumed to act as a plane piston, so that every point
on its face has a velocity u = uoe-iwfitnormal to the piston face,
where uo is a constant and¢fi* is the angular frequency. The trans-
ducer is parallel to the x-y plane and a diagonal lies on the y-axis.
See Figure 2.

To find the acoustic pressure at some point (x,y,z),
the contribution from an infinitesimal element of the transducer
is first considered. According to Huygens' principle, the infinitesi-

mal element dS (POint source) will radiate a hemispherically symmetric

wave producing, at a distance R, a pressure dp of the form

-)(- * *
-1p u w i(K R -w t)
d — ——-° ° 6 ds (5)
p _ 2n R ’

 

where K* = Kr + 10! and a > O. The complex propagation constant K*
includes the absorption coefficient a'of the medium, in addition to
the real propagation constant Kr' If we integrate over the surface
S of the transducer, the acoustical pressure at (x,y,z) will be

-ipouocn eiK R
P(X:Y:Z) =7- LT— dxo dyo .9 (6)

where the time dependence e-Lw tis understood and

= [(x-xo)2 + (y-yo)2 + 22]1/2

(7)

is the distance between a point on the transducer face (xo,yo,0)
and a field point (x,y,z). Having found the sound pressure, we
can now calculate the integrated optical effect.

By substituting Eq. 6 into Eq. 2, we can write the

integrated optical effect as

 

21/2
-i u * m*[ - z 2]
V(x,z) = 90x0 (D K :/T dyg/dxo dyo (x X o) 2+(Y Yo )2 + . (8)
Mac-2:02) +(y-yo) +221”2

By changing the order of integration, and using the following
3

relation ,

T eiK*[ (x-xo )z+(y-yo )2+ 22]1/2 1211/
Je [(x-xo ) 2-+(y -oy )2 + z 211/2 dy = 1TH£){K*[(X x °)2 + 22} , (9)

 

-oo

the expression for Eq. 8 becomes

*

u
V(x,z) = TEXTS-T fdxodyo Hgl) K*[(x_x0)2 + 22]1/2} . (10)
S

Since we are interested in obtaining the on-axis effect, one can see
that with x = O, the symmetry of Eq. 10 is such that its result is
equivalent to four times the result of one quadrant of the trans-

ducer. Using this fact, and setting x = 0, Eq. 10 becomes

Mn u * {Ea {Pa-x0
V(O,z) = -p—°>\2w—E fdxof dyO Hg1){K*[x§ + 22]1/2} . (11)
O O

The Hankel function Hgl) is not a function of yo, and we are left

with a single integral. Using the standard result that
l l
f 5 Hi ) (aB) d8 = E H; ) (8.8) , (12)

the on-axis integrated optical effect becomes

An u
V(O,z) = fl {mi/T a 11(1) K* [xi +2211”) dxo

- *
K

2 2 1/2
[2 + 28 I H51) K*[22+ 232]1/2}+ :* H(1)[K* 2]} (13)

The expression can be further simplified if we consider the case

* 2 2
where <JE' K.a >> 1 and 2 >> a . For large values of the argument

9, the Hankel functions can be approximated asll

10

Hi” (9)9(fl%)1/2 e1(6 - Ir/h) ’ (1h)

and

H§1)(6)zé‘%)1/2 ei(e'3“/h) . (15)

Using these expressions in Eq. 9, we get

 

 

42 a
* . * 2 2 1/2
V(O z) _ Tflpou6w Kjlhai)1/2 u/T e1[K (xo+ z - n/h] dx
) “ * 1
x (xx 0 (x:+ 22) ’1' °

_ (22+ 2a2)1/2

 

2 1/2 e1[K*(22+ 282)1/2_ 3n/h] (16)
K* nK*(zz+ 2a2)1/2

*.
+ z_*(_2_*)1/2 ei(Kz-3n/)+)} .
K nK 2

In Eq. 16, the exponential factors in the numerators vary more rapidly
with 2 than the demoninators; hence, more care must be given to ap-
proximating the exponentials. Expanding the exponents by the binomial
series, and neglecting all terms in the denominator except 2, we get

from Eq. 16,

112

* 2 * 1/2 YE a * 2
i -
V(O,z) = ZPOwa*K e (K 2 n/h) (%E_E7) U/T e1K xo/Zz de

O

*2
+ 1(gz_*)1/2 (eiKa/z_1)} . (17)

nK

11

Finally, the on-axis integrated optical effect can be expressed as

1 52
V(O,z) = 2pouow*1c ei<Krz' “/10 c(5)+ 13(5)+ 1(n5)'1(e'jzr_ -1)}e'az,(18)

2 2K*2
where 6 =-—-JE- ‘,
nz
8
u 2
and C(5) = ‘/p cos E-t dt , (19)
o
5
3(6) = \/o sin-g t dt , (20)

are Fresnel integrals.

For liquids with low absorption coefgicients,
2 2K a

G r . Equation 18 is
12

 

{Krl >> I 0: I, so that we can write 8
now the product of a factor that is due to the transducer configura-
tion, and an exponential, e-Qz, that is due to the absorption of the
medium. By plotting V(O,z) as a function of 8, excluding the absorption
term, one can see in Fig. 3, a flat region for 1.h5 < 5 < 2.20 where

the effect remains nearly constant.

At this point, the question arises as to how well the
theoretical curve agrees with experimental results - if the theory
correctly predicts the position, range and "flatness" of the integrated
optical effect in the near field for the chosen transducer configuration.

We also are confronted with the problem of whether the nearly constant

12

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venom oeu weHmmouo anon uanH w you uoommo HnoHumo pounuwoueH onu mo o>h:o HoOHuouooea .m ousmHh
. . . . . . m. o.H .o
z m m N o m w H m H w : H H _ w—
_ H _ _ _ _ _
rlu
I
HUMhhm
.2qu

 

QB éoEzH

13
integrated optical effect can be useful in obtaining meaningful
and reproducible ultrasonic absorption measurements, and, if so,
what the limits are for reliable results. These questions form the
basis for the present experiment and will be explored in later

sections.

CHAPTER III

EXPERIMENTAL STUDY

A schematic diagram of the Debye-Sears Optical method used
for the absorption measurements is shown in Fig. h.

The transducer assembly consists of a holder that provides
air backing for a circular, x-cut quartz transducer. A square elec-
trode painted in the center of the transducer with metallic silver
paint gave the desired transducer configuration. The transducer
assembly is mounted on an overhead optical rail and can vary 0.5 to
30.0 cm. from the light beam. Transducers of different thickness
can be easily interchanged in the holder. The resonant frequency
of each transducer is Obtained by adjusting the radio frequency
transmitter to a frequency that produces the maximum number of
optical diffraction orders for a fixed r.f. voltage. The trans-
ducers vary in resonant frequencies from.300 kHz. to 7.5 mHz.

The sound beam.is adjusted to be normal to the light
beam by means of two screws on the transducer holder, which pivot
the transducer about two perpendicular axes in the plane of the
transducer face. The sound and the light are judged to be normal
when the maximum light diffraction occurs and the zero order light
intensity is a minimum, for small acoustic intensitiess.

In order to orient the square transducer so that a diagonal
is parallel to the direction of propagation of the light beam, the
transducer holder is rotated about an axis perpendicular to the

center of the transducer face, while observing the schlieren pattern

1h

15

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mmHHmHBADZUHomm

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16

of the sound beam. It is found that changes in the integrated
optical effect are symmetric about the desired position.

The optical system consists of the following: a He-Ne
gas laser; a beam.expander that spatially filters and recollimates
the light; lens L1, which focuses the optical diffraction pattern
on a plane at the entrance slit of the photomultiplier; and a
photometer, which records the relative intensity of light entering
the photomultiplier. The filter is inserted to reduce the intensity
of the laser beam so that the entrance slit of the photomultiplier
can be opened to a suitable width. If the entrance slit is too
narrow, instabilities occur due to mechanical vibrations Of the
optical system. An aperture A Of 2 mm. diameter allows the light
beam to pass only through the center of the symmetrical sound
field.

The photomultiplier is set on the zeroth order of the
Optical diffraction pattern to measure the light intensity as a
function of r.f. potential. The zeroth order was chosen for the
following reason. An ultrasonic wave of finite amplitude that is
initially sinusoidal becomes saw-toothed-shaped as it propagates
away from.the source. The saw-toothed waves produce an asymmetrical
optical diffraction pattern. NOmoto and Kegishi6 have theoretically
shown that while the higher orders Of the optical diffraction pattern
are greatly affected by finite amplitude waves, the zeroth order is
not. In fact, the intensity of the zeroth order is the same whether

sinusoidal or saw-toothed waves exist when the Raman and Nath phase

17

grating approximation is applicable.

From Raman and Nath theory, the intensity of the zeroth
order of the diffraction pattern is given by Eq. h. The relative
acoustical effect is determined from l/E, where E is the r.f. Poten-
tial needed for a constant Io. The largest gradient in IQ for an
incremental change in v can be shown to be Ia:50% of its maximum
value. Thus, the greatest accuracy for determining the applied
potential can be attained for this constant value of 10. The
quantity l/E is proportional to V and includes the effects of
both the transducer configuration and the acoustic absorption Of
the fluid.

For frequencies less than 1 mHz., the distance between
the optical diffraction orders is so small that the orders overlap.
The intensity recorded by the photomultiplier is still primarily
due to the zeroth order, but news small function of the light
intensity is due to higher orders. Fortunately, the finite amp-
litude effects of the higher orders are quite small at low frequen-
cies and do not affect the accuracy of the absorption measurements.
The problem of overlapping orders is solved by keeping the optical

configuration constant during the experiment.

CHAPTER IV

RESULTS

The first problem to be considered is to determine how
well the theoretical curve predicts the measured on-axis integrated
optical effect. Water is used to investigate the on-axis effect,
since the acoustic absorption is not significant over the desired
range of measurements. In Fig. 5, the experimental results are
compared to the theoretical curve, calculated from Eq. 18. As one
can see, the two curves are not perfect fits, with differences up
to 3% in the predicted flat region. This is interpreted to mean
that the theoretical on-axis integrated optical effect can be used
when the exponential decay due to the ultrasonic absorption is much
greater than the error between the two curves. For practical purposes,
the on-axis effect is assumed to be constant and an absolute absorp-
tion measurement is made. When the ultrasonic absorption is quite
small, the error of 3% becomes significant, and an absolute absorp-
tion coefficient cannot be determined. For the latter condition,
the on-axis effect must be determined experimentally for each in-
dividual case, and a relative change in absorption can be measured,
for example, when small amounts of electrolytic salts are added to
water.

Aqueous solutions of the polyvalent electrolytic salt,
manganese sulfate, were chosen for the present experiment, for
reasons stated earlier. Measurements were taken for frequencies in
the range 0.3 to 7.5 mHz. at solution concentrations of 0.01 to

9

0.5 m. Kurtz and Tamm7, Smithson and Litovitz8 and Kor and Verma

l8

19

1:
_

A.EO m.o u u ..nma 0.: u *mv GOHwou ume

Ou OONHHmEuos “muHsmou HuueoaHHomxo can HooHuouomsu mo :OmHuomaOU .m oustm

mmmHMZHBzmo

0.. H w

——d-

 

AdHZHZHmmmNm AU
HdUHHmmommH nll

 

 

 

Bowman

H553

20

performed similar experiments with‘MnSO1+ in the frequency range of
present interest, using various techniques, as shown in Fig. 6.
Absolute absorption measurements were taken over the range
of frequencies for a fixed solution concentration of 0.1 m and
constant temperature of 21.50C. Figure 7 shows the dependence of
OVf*20n frequency. The curve has a constant value at low frequencies
and decreases with increasing frequency. For frequencies greater
than 1 mHz, our results are somewhat lower than those of Kurtz et.al.
and Kor et. a1. With increasing frequency, the curves begin to ap-
proach each other due to the increasing importance of the absorption
of water. For frequencies less than 1 mHz., we found that the para-
meterOt/f*2 begins to level off at about 1.5 x lO-lh secZ/cm., more
than double the results of Kor et. al. The only explanation given
for the different results is that the error associated with the
determination of absolute absorption coefficients by the Optical
technique is large at low frequencies. The relaxational nature

of‘MnSOh is illustrated in Fig. 8, which is a plot of the parameter

 

 

 

 

 

 

 

REVERBERATION
‘QT : j'Kurtz
Tamm
OPTICAL 'L g. s 1th
l ‘m son
PULSE F " Litovitz
OPTICAL : : Kor and Verma
OPTICAL : ' : present
I I I I I f*(mHz) I I
0.1 0.2 0.5 1.0 2.0 5.0 10.0

Figure 6. Frequency Range of Various Techniques Used for
Measuring the Ultrasonic Absorption in‘MnSOh.

sec /cm.

a/f*2 x 10'17

21

 

2000 _
O G
1000 .—
O
0 O
c a
500 “ .Ahx
o 01
\U
o 'A
\ .
o
mx>'—‘ o n
G)
o
100 —‘
El Kurtz and Tamm
A Kor and Verma
0 Present
50 .—
g)__ vmmER -——3F

 

 

10
o!3 ols l g I 1b

f*(mHz.)

Figure 7. Parameter Ot/fdx2 vs. acoustical frequency for
0.1 M MnSOLL at 21.5 C.

22

 

 

20 --
A
1..
,. 4" °
A V
O O Q
o D
O
10 ._. ¢,°
O
:1
Q G
5 —- D
.d'
o
H
N
*.«_<
23
Kurtz and Tamm
¢ Kor and Verma
2 —'1 u 9 Smithson and Litovitz
0 Present
D
1 I I I I I I
0.1 0.2 0.5 1 2 5 10

f*(mHz.)

-)I-
Figure 8. Parameter 05'}. vs. acoustical frequency for 0.1 m

Mnsoll at 21-5°c.

23

*
a'x vs. frequency. (1' is the excess absorption due to the salt,

defined as<a' =<1 Our results show a relaxation

solution - O":Iplater'
peak near h mHz. with somewhat smaller value than the results of
others, and does not drop off as rapidly at lower frequencies as
the curve of Kurtz et. al.

Figure 9 shows the effect of various concentrations of
MnSOh on the relative measurements of ultrasonic absorption at
the fixed frequencies of 0.8 and 4.0 mHz. At h.0 mHz., the absorp-
tion increases very linearly for concentrations from 0.01 to 0.3 m,
and increases slightly less than linearly for higher concentrations.
There is little agreement between our results and the non-linear
results of Kor et. al. at h.0 mHz. The ultrasonic absorption at

0.8 mHz also appears to increase linearly for concentrations from

0.05 to 0.5 m.

21+

A ZmHz.

0.8 mHz

 

 

 

 

0.002—
A Kor and Verma
G G) Present
0.001
I I I I I I molar
0.01 0.02 0.05 0.1 0.2 0.5 1.0 concentration

Figure 9. Relative absorption coefficient 05 vs. solution con-

centrat ion of MnSOh.

CHAPTER V

CONCLUSION

It has been shown, both theoretically and experimentally,
that a light beam traversing an acoustic beam radiated by a square
transducer, parallel to a diagonal of the transducer, produces an
integrated optical effect that is nearly constant in the Fresnel
field of the sound beam. Figure 5 shows that the calculated and
experimental results, when normalized to the flat region, agree
quite well in the near field, but differ as to where the optical
effect begins to fall off. The theory is quite satisfactory for
our purpose, as we are concerned only with the flat region of the
integrated optical effect. Using the flat region for absorption
measurements has produced results comparable to those of other
investigators, though our results are generally lower. The
optical method for absorption measurements is not limited to plane
acoustic waves, as earlier, since the optical effect can now be
determined by integrating across the sound beam. With proper
experimental care and with a nearly constant optical effect of
sufficient range, relative measurements will give reliable values

for ultrasonic absorption coefficients to a'= .001 cm-1.

25

BIBLIOGRAPHY

B. Cook and F. Ingenito, to be published in Proc. IEEE.

W. Klein and B. Cook, IEEE Trans. on Sonics and Ultrasonics,
13, 126 (1967).

W. Magnus and F. Oberhettinger, Functions of Mathematical
Physics, Chelsa Publishing Co. 21959;, p. 27.

 

See Reference 3, p. 22.
0. Raman and N. Nath, Proc. Indian Acad. Sci. A2, h13 (1935).
O. Nomoto and K. Negishi, Acustica 15, 223 (1965).
G. Kurtz and K. Tamm, Acustica 3, 33 (1953).

Smithson and T. Litovitz, J. Acoust. Soc. Am. 28,
M62 (1956).

S. Kor and G. Verma, J. Chem. Physics 29, 9 (1958).

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