Halbert Frederick Gates
candidate for the degree of
Doctor of Philosophy
Final examination, May 24, 1954, 1:00 P.M.
room, Physics-Mathematics Building

Conference

Dissertation:
Determination of the Ratio p/q of the
Photoelastic Constants of Optical Glasses by Means
of Two Different Ultrasonic Methods
Outline of Studies
Major subject:
Minor subject:

Physics
Mathematics

Biographical Items
Born, October 30, 1919, Milwaukee, Wisconsin
Undergraduate Studies, Milwaukee State Teachers College,
1936-1940
Graduate Studies, University of Wisconsin, 1940-1944,
continued 1947-1948, Michigan State College,
1950-1954
Experience:
Graduate Assistant, University of Wisconsin,
1941-1944; Officer in Charge, Principles of Elec­
tricity Section, Theater Signal Corps School
(Germany), 1946; Assistant Professor of Physics,
Berea College, 1948-1950; Special Graduate Research
Assistant, Michigan State College, 1951-1954
Member of American Physical Society, American Association
of Physics Teachers, Society of Sigma X i , Sigma Pi
Sigma, Pi Mu Epsilon, Kappa Delta Pi

DETERMINATION OP THE RATIO p/q OP THE PHOTOELASTIC
CONSTANTS OP OPTICAL GLASSES BY MEANS OF
TWO DIFFERENT ULTRASONIC METHODS

by
Halbert Frederick (Gates

AN ABSTRACT
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OP PHILOSOPHY

Department of Physics and Astronomy
1954

Approved

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Halbert F. Gates

ABSTRACT
Mueller

(l) has described, in theory, two dynamic

methods for the determination of the ratio p/q of the strainoptical constants of glass.

The strains are produced by

ultrasonic waves in the glass, while the optical effects are
observed in the diffraction patterns produced by light which
has passed through the sound field.

The theory of Mueller is

an extension of that of Raman and Nath (2) which explains the
diffraction of light by a sound wave on the basis of varia­
tions of the index of refraction of the medium traversed by
the sound wave.
Measurements were made on a series of thirteen American
optical glasses and fused silica, following Mueller's methods
,IB" and "C11,

The ultrasonic frequency was in the neighborhood

of ten megacycles per second.
Method "B" involves the methods of Hiedemann (3 ).
Measurements are made of the polarizations in the diffraction
pattern at several amplitudes.

An extrapolation to zero

sound amplitude leads to a value for p/q.
The experimental results are In excellent agreement with
the theory.

Values of p/q were determined for all of the

samples, the ratios ranging from 1.11 to 2.34.

Halbert F. Gates
Method "Cu involves the methods of Bergmann and Fues

(4)•

Measurements are made of intensity ratios in the diffraction
patterns at several sound amplitudes*
The experimental values at various sound amplitudes,
while of the correct order of magnitude, do not agree with
the theory.

Extrapolation cannot be performed as indicated

in the theory, and no value of the ratio p/q can be deter­
mined.

The behavior of the data is, however, consistent and

systematic and it is hoped that some significance may be
attached to it.
The values of the ratio p/q of the strain-optical con­
stants, as obtained by method MB", are given, together with
other data on the samples.

The behavior of the data as ob­

tained by method "C11 is described.
Literature Cited
1.

H. Mueller, Z. Kristallogr. A, 99, 122

(1938)

2.

C. V. Raman and N. S. N. Nath, Proc. Indian
Acad. Sci. A, 2, 406 (1935); 3, 75 (1936);
3, 459 (1936)
«

3.

E. Hiedemann, Z. P h y s ., 108, 592

(1938)

4.

L. Bergmann and E. Fues, Naturwissenschaften,
24, 492 (1936)

DETERMINATION OF THE RATIO p/q OF THE PHOTOELASTIC
CONSTANTS OF OPTICAL GLASSES BY MEANS OF
TWO DIFFERENT ULTRASONIC METHODS

by
Halbert Frederick Gates

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
1954

ACKNOWLEDGMENTS
The author wishes to express his sincere thanks to
Dr. E. A. Hiedemann who suggested this investigation and
has given valuable and helpful guidance throughout the
course of its development.

Greatful acknowledgment is

also due to Dr. C. D. Hause for his help with the photo­
metric measurements.
This work was supported by a grant from the OwensIllinois Glass Company.

The author greatly appreciates

the resulting Special Graduate Research Assistantship
which made it financially possible for him to pursue the
problem.

table

of

contents

Page
INTRODUCTION

1

Definitions of Photoelastic Constants

1

Static Determination of Photoelastic Constants

2

Dynamic Determination of Photoelastic Proper­
ties of Glass by means of Ultrasonic Waves

2

Purposes of this Investigation

5

Theory of Photoelasticity in Glass

5

THEORY

7

Basis of the Mueller Theory

7

Special Conditions for this Investigation

7

Conclusions of the Mueller Theory for Standing
Longitudinal Sound Waves

9

B and 0 as Functions of v and of Sound Ampli­
tude

12

Justification for Consideration of Only r = 0
Sub-components of First Diffraction Orders

15

Indication of Sound Amplitude

16

APPARATUS AND METHODOLOGY

18

Ultrasonic Generator

18

Transducer

26

Optical Arrangements and Measurements

26

Temperature

31

Glass and Fused Silica Samples

34

Page
presentation of data

40

Method nB"

40

Method 11C"

42

Values of R from Other Investigations

46

SUMMARY AND CONCLUSIONS

49

APPENDIXES
I

Theoretical Intensity Ratios in the
Diffraction Orders

52

II

Sample Data and Calculation for

Method "B"

56

III

Sample Data and Calculation for

Method "C"

59

BIBLIOGRAPHY

64

LIST OF FIGURES
Page
Figure
1.

Theoretical variation of tan 0 and
v2

-5b " with
13

2.

Oscillator and power amplifier

19

3.

Voltage-regulated power supplies

24

4.

High voltage power supply

25

5.

Transducer matching circuit

25

6.

Optical system for method "B"

27

7.

Optical system for method "C"

27

8.

Experimental variation of tan 6 with I2

29

9.
10.
11.
12.

Exoerimental variation of
in BSC-1

v—

with I

2
32

Experimental variation of
in DBF-1

-5B" with

Experimental variation of
in fused silica

-JB* with

Experimental variation of
in CF-1

-iRB with

I2
32
12
33
I2
33

list

of

Ta b l e s
Page

TABLE
I

ELECTRICAL COMPONENTS

20

II

DESCRIPTION OF GLASS SAMPLES

35

III

COMPOSITION OF GLASS SAMPLES

37

NUMERICAL RESULTS

43

VALUES OF p/q GIVEN BY OTHER INVESTIGATORS

47

IV
V
VI
VII
VIII
IX

TOTAL INTENSITY RATIOS:
ORDER, METHOD "B"

THIRD ORDER TO ZERO
53

SUB-COMPONENT INTENSITY RATIOS: FIRST TO ZERO
SUB-COMPONENTS, FIRST ORDER, METHOD "B"

53

TOTAL INTENSITY R a TIOS:
ORDER, METHOD f,C"

54

SECOND ORDER TO ZERO

SUB-COMPONENT INTENSITY R a TIOS: FIRST TO ZERO
SUB-COMPONENTS, FIRST ORDER, METHOD MC"

55

INTRODUCTION
Definitions of Photoelastic Constants
P o c ke ls 1 strain-optical constants.

Pockels

(l) has

given a set of thirty-six phenomenological constants which
relate the general strain tensor to the optical effects
which accompany it.

In the case of an isotropic material

these constants, p^j

(the subscripts run from one through

six), are reduced by symmetry considerations to two, p ^
and P 12 *
Neumann's strain-optical constants.

Neumann

(2) de­

scribes the photoelastic behavior of an isotropic medium
by means of two strain-optical constants, p and q.

These

constants relate a strain to the changes which it produces
in the index of refraction n for light vibrating parallel
and normal respectively to the direction of the strain.
Equation

(1) defines the Neumann constants in terms of a

typical strain zz .

n 3- n = 1

z.2
*

d n x = n x- n = — n p z *
Equation

U)

(2) shows the relation between the Neumann constants

and the Pockels constants for an isotropic material.

2

It is assumed that the fine-annealed optical glass used in
this investigation is an isotropic material,

and the dis­

cussion is in terms of the Neumann strain-optical constants,
p and q.
Stress-optical constants*

The optical effects may also

be described in terms of the stress which causes them.
deed,

In­

in measurements beyond the elastic limit, Filon and

Jessop

(3) find evidence that stress is the fundamental

quantity related to the optical effects.

It is assumed in

this investigation that the deformations are small and that
H o o k e 1s Law applies and that stress and strain constants are
equally significant.
Static Determination of Photoelastic Constants
Determinations of the photoelastic constants of glass
by means of the application of steady stress have been made
by Mach (4), Pockels

(5), Filon (6), Twyman and Perry

and Schaefer and Nassenstein (8).

(7),

These static methods often

involve difficult optical measurements requiring large or
specifically shaped samples, two "Identical11 samples, and/or
precise temperature control.

The samples are subject to re­

laxation or plastic flow as described by Coker and Filon
Dynamic Determination of Photoelastic
Properties of Glass oy means
of Ultrasonic Waves
The dynamic strains are produced in the glass by

(9).

3

ultrasonic waves.

The optical effects are observed in the

diffraction of a beam of light which has passed through the
sound field.
Diffraction of light by ultrasonic w a ve s.

Early obser­

vations of the scattering or diffraction effect of an ultra­
sonic field on a beam of light were made by Debye and Sears
and by Lucas and Biquard (ll).

The use of a point source of

light to show the diffraction of light by many sound waves
simultaneously was developed by Schaefer and Bergmann (12)
(13)

(14).

The use of a slit source of light to show the

diffraction by a single wave was developed by Hiedemann and
Hoesch (15)
vestigation.

(16)

(17).

This is the method used in this in­

The slit is made parallel to the sound wave

front and the light beam is normal to the sound beam.
line diffraction pattern is then formed.

A

This investigation

also makes use of the polarization techniques of Hiedemann
and Hoesch (18) for the separation of the effects of longi­
tudinal and transverse sound waves.
Raman and Nath (19) have given a theory of the dif­
fraction of light by a sound field.

The theory is based on

the phase changes in the light wave as it passes through
the periodic structure of variations of index of refraction
caused by the sound field.

The effect on the light is the

production of a "corrugated wave front11 which leads to a
diffraction pattern.

The theory gives the directions,

in­

tensities, and frequencies of the light in the diffraction

(10)

4

orders.

The theory has been verified on the basis of experi-

u
ments by Bar

/ \
(20) and Sanders

/
(21).

Determination of the ratio p/q of the strain-optical
constants.
theory.

Mueller

(23) has extended the Raman and Nath

He evaluates the index variations in terms of the

strain-optical constants of the medium.

This leads to ex­

pressions for the directions, polarizations, frequencies,

and

intensities of the elements of the diffraction pattern in
terms of the strain-optical constants and the sound amplitude.
Many of the conclusions have been experimentally verified
by Hiedemann (24)
Mueller

(25).

(23) gives three methods for the experimental

evaluation of the ratio p/q.
Method "B11 involves measurements of the polarizations
of the diffraction orders in the Hiedemann spectrum.
of this method have been used on glasses by Vedam (26)

Forms
(27)

and by Schaefer and Dransfeld (28).
Method "C" involves measurements of the intensity ratios
in the diffraction orders.

A partially similar method has

been used on glasses by Bergmann and Fues

(29).

Method "A” applies to certain crystals

(but not also

to glasses, as do "B" and ,?C ,f), yielding ratios of strainoptical constants.
Measurements have been made on crystals by Burstein,
Smith, and Henvis

(30) and by Galt

(31).

Galt reports agree­

ment with the theory of photoelasticity in cubic crystals

5

given by Mueller

(32), while Burstein and Smith

(33) in­

dicate the need of some modifications.
These dynamic determinations offer- some characteristic
advantages.

The results are obtained directly in terms of

the strain-optical constants, which are, according to
Mueller

(23), of greater theoretical importance than the

stress-optical constants.

Small samples are sufficient.

There are no effects due to relaxations with time constants
which are long with respect to the period of the sound.
Purposes of this Investigation
This investigation had two purposes.

The first was to

determine dynamic values of the ratio p/q for a series of
American optical glasses and fused silica.
to compare methods

The second was

MB ,r and "C" as a check on the values and

on the theory.
These results are made more interesting by other work
completed or projected in this laboratory.
vestigation was in progress Barnes

While this in­

(34) determined dynamic

values of the elastic constants of the same set of samples.
A projected problem involves static measurements leading to
values of p and q for the same samples.
Theory of Photoelasticity in Glass
A theory of photoelasticity in amorphous solids has
been given by Mueller

(35).

ascribed to two effects.

The photoelastic behavior is

The first involves elastic

6

alterations in the Lorentz-Lorenz interactions between di­
poles.

The second Involves the production of artificial

optical anisotropy of the atoms.

Under the action of p r es ­

sure the first produces positive birefringence while the
second produces negative birefringence.

The second is

usually the larger, but when the index of refraction reaches
large values the first may surpass it.
An effect due to the alignment

(by deformation)

of

optically anisotropic molecules is discussed by Treloar

(36)

and Braybon (37), but is probably of significance only for
long high-polymers•

7

THEORY
Basis of the Mueller Theory
The basic ideas in Mueller*s extension of the Raman
and Nath theory are outlined in the Introduction.

M ue ller’s

methods f,B" and "C,r for the determination of the ratio p/q
are identified.
Special Conditions for this Investigation
Homogeneous and isotropic m at e r i a l .

It is assumed that

fine-annealed optical glass is homogeneous and isotropic.
Each of the samples was good enough to permit almost complete
extinction when placed between crossed polarizers.
Longitudinal sound w a v e s .

The observed effects were

entirely due to longitudinal sound waves.
used was an x-cut quartz crystal

The transducer

(a thickness vibrator) so

that the primary wave in the glass was a longitudinal wave.
Frequencies were chosen at which the blocks exhibited strong
resonances for the longitudinal modes and weak resonances for
transverse modes.

In method ,fB" the polarizer and analyzer

are set at forty-five degrees to the sound wave front.
Hiedemann and Hoesch (18) have shown that this arrangement
eliminates the effect of transverse waves.

In method "C"

the absence of significant transverse waves can be checked
visually since their velocity differs from that of longi­
tudinal waves and the corresponding diffraction patterns

8

have different spacing.
Standing sound w a v e s .

In a one inch cube of a low-

absorption material like glass it would be very difficult
to produce anything other than standing waves.

The appear­

ance of intense diffraction patterns due to resonances as
the frequency was changed gave evidence of the standing
wave condition.

The theory of Raman and Nath notes that

each line in the diffraction pattern due to standing waves
is composed of sub-components having different frequencies
and polarizations.
by the Index m.

The diffraction orders are Identified

The value zero corresponds to the central

order while the index one corresponds to the first order and
so on.

The sub-components are identified by the index r,

which may be zero or have positive integral values.
Diffraction order and sub-component.
component

(r=0)

Only the zero sub­

of the first diffraction order

considered in any of the measurements.

(m = l) was

Justification for

this will appear below.
Applicability of Raman and Nath Theory.

The sound

frequencies used were in the neighborhood of ten megacycles
per second.

The optical path length in the sound field was

about two centimeters.

These values place this investigation

In the region in which the Raman and Nath theory applies.
Experimental verifications of the theory
been mentioned above.
involved in this work.

(in liquids) have

The wave lengths correspond to those
A theoretical condition for

9

applicability has been given by riytov (38).

(X*)

It is
(3)

z

where L is the optical path length in the sound field,

is the optical wave length,
Willard

X

XT is the acoustic wave length.

(39) gives for a criterion
(4)

including the Baman and Neth argument v, which,
work,

is seldom greater than unity.

in this

Both of these criteria

place these measurements in the province of the theory.
Conclusions of the Mueller Theory for
Standing Longitudinal Sound Waves
Arrangement for method MB ,f.

The arrangement for

method trB" requires the light beam to be perpendicular to
the sound propagation direction.
the sound wave fronts.
degrees to the slit.

The slit is parallel to

The polarizer is set at forty-five
This optical arrangement is diagramed

in Figure 6.
In this case the polarization of the sub-component r
of the diffraction order m of the Hiedemann spectrum is
given by

(5)

ixn ('«w,v+ 4s'°) if bo = X S

10

where s Is zero or any positive integer.

^ iY is the angle

of rotation of the analyzer from the crossed position which
causes extinction of the sub-component
order

m .

x

of the diffraction

J is the Bessel function of the first kind and

of order indicated by the subscript.
the strain-optical constants.

R is the ratio p/q of

v is the argument of the

Bessel functions in the Raman and Nath theory, given by

_ 2.TT./U-L

A

_

4-TT*l_n*ci A

~

A A'

. .

'

where j*. is the amplitude of variation of the index of re­
fraction associated with the strain-optical constant q, L
is the optical path length in the sound field,
optical wave length in air,

A

is the

A * is the acoustical wave

length, n is the index of refraction of the medium, and A
is the acoustical amplitude.

It is to be noted that the

quantity v is proportional to the amplitude A of the sound
wave •
In the case (used in this investigation) where r = 0 and
m = l, equation 5 becomes
J [Rv') T (Rv'j

to.* © = t o ^ ( T 0+4-S6) = —
where

9 =
0

(7)

,

is the experimental quantity measured in method "B11.

Arrangement for method irC n .

The arrangement for method

"C" requires the light beam to be perpendicular to the direc­
tion of propagation of the sound.

The slit is parallel to

11

the sound wave fronts.
gramed in Figure 7.

This optical arrangement is dia­

Natural light passes through a Wollaston

double image prism which splits it into two beams of equal
intensity*

the polarizations of which are normal to each

other and parallel and normal respectively to the slit.
two beams diverge slightly

(forty-two minutes in this work)

and give rise to two separate diffraction patterns.
ratio

The

The

of the intensities of corresponding lines in the

two patterns is given by

t)

—

(8)
xv\ = 2* S
T»-* I P

(p

where the symbols have the same meaning as in equation 5.
In the case
m =1,

(used in this investigation) where r - 0

and

equation 8 becomes
,____

To
<9>

B is the experimental quantity measured in method MC M .
Analytical similarity of methods

irB" and "C11,

A com­

parison of equations 7 and 9 indicates that
"tan 6 - -0 B

—

----------— “

Jo
where

0

ip

j

is the extinction angle

(10)

,t p
(from crossed position) plus

forty-five degrees for the sub-component r = 0 of the first
diffraction order of method "B" and B is the intensity ratio

12

of the r = 0

sub-components of the first diffraction orders

of method "C".

Thus two experimental quantities which may

be independently measured are given by the same analytical
expres s ion,
B and 0 as Functions of v
and of Sound Amplitude
B
solved

and

9

as functions of v .

Bquation 10 cannot be

to give values of R since v contains quantities which

cannot be accurately measured.

Neither sound amplitude nor

sound field width, for example,

can be precisely known.

However, as v approaches zero the right hand member of
equation 10 approaches the value R.

Thus, letting subscript

zeros indicate limiting values, equation 10 becomes

"tcun 0 o = i V B e —

R

(n)

Figure 1, drawn from equation 10 shows the mode of ap­
proach of these quantities to their limit as v approaches
zero.

Tan 9

and

iTB

are plotted against the square of v

merely in order to give a relation which is very nearly linear.
The straight lines are drawn to show the departure of the
points from the linear relation as v increases.
when v

For example,

is less than unity and

R is less than two, a straight

line is a good approximation.

Thus the "best" straight line

(criterion of least squares)

(40) through the four points

shown in Figure 1 for R equal to two, intersects the vertical
axis at 1.989, giving an error of 0.55 percent.

None of the

13

APPEARANCE OF
SECOND ORDER

9 = EXTINCTION ANGLE
B = INTENSITY RATIO
R * RATIO OF PHOTOELASTIC CONSTANTS

I I --------------------------- 1------------ J--------------1--------------------------- 1--------------------------- 1--------------------------0.2
0.4
_
0.6
0.8
1.0

V2 — ►

Pig.

1.

Theoretical variation of tan 0 and

-d B with

14

glasses examined exhibited values of R greater than 1.83.
B and

8

as functions of sound amplitude.

Since v is

proportional to sound amplitude, a similar linear relation­
ship exists between tan 9

or

and the square of the

sound amplitude, provided that the latter corresponds,
general, to values of v less than unity.
perimental points on a plot of tan 9

or

in

In this case ex­
-

5b

against the

square of the sound amplitude can be extrapolated linearly
to give a value for tan d0

or

t5B0 .

Such an extrapolation

was used in this investigation.
It is necessary to relate the experimental sound am­
plitudes to values of v in order to justify the linear
extrapolation.
An approximate value of v corresponding to any given
sound amplitude can be obtained from a comparison of the
light intensities in the diffraction orders due to that
sound.

It is thus possible to consider only values of

sound amplitude which correspond to acceptable values of
v.

The relationship between values of v and Intensities in

the diffraction orders is given,

In the case of liquids, by

the theory of Raman and Nath (19).

Good agreement between

experiment and theory has been obtained by Sanders

(21).

For solids the theory has been extended by Mueller

(23) to

include the effect of the photoelastic constants.

Since

the optics differ, the relationship is different for the
arrangements for methods

,fB M and "Cfl.

Appendix I gives

15

M u e l l e r !s theoretical conclusions and lists numerical values
in the range of Figure 1 and of this investigation.
For the special case of method 11B 11 and a glass having
R equal to 1.5, values of v corresponding to the appearance
of second and third diffraction orders are indicated in
Mueller's Figure 3 (23)

(due to Hiedemann).

These approxi­

mate values are indicated by the dotted vertical lines of
Figure 1.
In general, the values of v at which the diffraction
orders appear, decrease as R increases.
all conclusions may be drawn.

The following over­

For method "B", when the

third order diffraction line does not appear, the sound ampli­
tude corresponds to a value of v which lies on the essentially
straight portion of the appropriate curve of Figure 1.
extrapolation is then in order.

Linear

For method "Cu , when the

second diffraction order does not appear, the sound amplitude
corresponds to a value of v on the linear portion of the cor­
responding curve and extrapolation is justified.
An arbitrary criterion for the "appearance" of a dif­
fraction order is an intensity in that order of approximately
one percent of that in the central order.
gested by Sanders

One criterion sug­

(21) is 0.86 percent.

Justification for Consideration of Only r = 0
Sub-components of First Diffraction Orders
The numerical values given in Appendix I also indicate
the conditions under which it is permissable to consider only

16

the r - 0

sub-components or the first diffraction orders.

In method "B", when the third diffraction order does not
appear, the ratio of the intensities in the r = 0

and first

sub-components of the first diffraction order is not more
than a few percent.

Thus the extinction of the r = 0 sub­

component causes a distinct minimum in the intensity of the
line*

In method "C", when the second order does not appear,

the ratio of the intensities in the r = 0 and r = l sub-compo­
nents of the first diffraction order is not more than about
0*1 percent.

Thus an intensity measurement of the line Is

equivalent to an Intensity measurement of the r = 0

sub-com­

ponent •
In this work, third order lines never appeared during
measurements by method "B" and second order lines never ap­
peared during measurements by method "CTt.
Indication of Sound Amplitude
Since the curve of tan 0

or

v Tb

plotted against the

square of sound amplitude is extrapolated to zero sound, it
is, fortunately, not necessary to determine absolute values
of sound amplitude.

Values of some quantity which is pro­

portional to sound amplitude are sufficient.
is quartz

(transducer)

Such a quantity

current or the current at any point

In the series transducer circuit, as long as the geometry
and frequency remain constant.

A relationship between acoustic

intensity J and piezoelectric transducer current I is given

17

by Cady

(41).
T = r R s/ z

R g is the

(12)

(constant) series resistance of the transducer

equivalent network and involves frequency, piezoelectric
constants, dimensions,

acoustic impedances, and wave velocity.

Thus for a given transducer,

coupled to a constant load and

driven at a constant frequency,
tional to transducer current*

sound amplitude is propor­
The independent variable in

this investigation has been transducer current.
extrapolations to

and tan 90

The linear

were made on that basis.

18

APPARATUS AiJD MhThODOLOGY
Ultrasonic Generator
The ultrasonic generator designed and constructed as
a part of this investigation consists of, six main parts,
connected to operate as a unit and mounted in a r e l a y rack.
Oscillator and power amplifier.

Figure 2 shows the cir­

cuits of the oscillator and power amplifier which are con­
structed on the same chassis*

Table I identifies the com­

ponents •
The oscillator is an .electron-coupled Hartley circuit,
modified to give maximum stability.

Its frequency range is

continuous from about 5.2 megacycles per second to about 13.6
megacycles per second.
The radio-frequency power amplifier can deliver from
one hundred and fifty to two hundred electrical watts.

Plug­

in output coils cover the ranges from 5.7 to 12 megacycles
per second and from 12 to 24 megacycles per second, with the
output stage doubling in the upper range.
When the circuit is well warmed up it Is able to main­
tain a frequency In the neighborhood of ten megacycles per
second constant within one kilocycle per second for several
hours.
Stability was checked by means of a General Radio Type
620-A Heterodyne Frequency Meter while the approximate

19

x .9
U-O Q

u_

0
—

ouig
T"

rin

0 -i

UJ
U_

I•

li­
ar
C\l
U_

a:
a:
CT~

oc

Oscillator and power amplifi

■OX

(SJ
ci
•I— 1

20

TABLE I
ELECTRICAL COMPONENTS

Components for Oscillator and Amplifier

(Figure 2)

Ll,2,3

3.5 microhenrys

C14

130 mmfd mica 600 v

l4 a )

18 microhenrys

c17

0.001 mfd mica 3000 v

2*6 microhenrys

B1

50,000 ohm zero temp

%

swinging link

R2 .

68,000 ohm 1 w

C1 ,2> 3

13-300 mmfd

R3

10,000 ohm 1 w

c4

11-150 mmfd dual

r

4

25,000 ohm 1 w

C5

10 mmfd variable

R5

500 ohm 5 w

c6

0*5 mmfd variable

RFC1j2

2.5 mh 100 ma

C7

75 mmfd zero temp

r f c 3j4

2.5 mh 100 ma

ce

0*01 mfd

rfc5

1 mh 200 ma

Cg,10,13

0.01 mfd 500 v

r f c 6j7

14 turns -J”

c15,16

0.01 mfd 500 v

R FC 8

8 turns -J-ff

CI1

50 mmfd mica 600 v

b)

Components i for Voltage-regulated Power Supplies

(Figure 3)

R3

15,000 ohm 5 w

r4

90 ohm

600 v CT 120 ma;
5a
5 v 3a; 6.3v

R 5,6

1000 ohm lw

7.5 v

r

T-j_

5 v

T2

550 v

T,
T4

3 a
CT 250 ma

3 a

7

470,000 ohm 2w

21

TABLE I CONTINUED

Components for Voltage-regul ated Power Supplies
6.3 v

3a

(Figure 3)

R 8,9

100 ohm h w

6.3 v i a

R 10

12,000 ohm 25 w

L1

24 h

200 ma

R 1 1 ,12

1 meg 1 w

L2

25 h

75 ma

R 13

5,000 ohm 10 w

4 mfd 1000 v oil

R 14

250,000 ohm 1 w

C3, 4

16 mfd 600 v

R 15

100,000 ohm -J- w

C5, 5

0.1 mfd 400 v

r 16

390,000 ohm 2 w

c7

0.2 mfd 600 v

r 17

50,000 ohm 2 w

C8

4 mfd 600 v oil

r 18

120,000 ohm 1 w

R1

72 ohm 5 w

r 19

500,000 ohm 1 w

«2

5,000 ohm 50 w

T5
t6

Cl»

2

VH tube firing relay

Components for High Voltage Power Supply (Figure 4)
2.5 v 10 a
10,000 v insulation

L1

5/25 h 225 ma
swinging choke

1500 v (dc) 200 ma CT

L 2,3

8 h 225 ma filter
choke

T3

0.4 kva autotrans­
former

Cl,3

4 mfd 2,000 v

oil

R1

50,000 ohm 50 w

c2

2 mfd 2,000 v

oil

*1

S1

thermal delay relay

22

TABLE I COLTIBLBD

Components for Matching Circuit
2'1 diameter

L1

22 turns

Cl

20-200 mmfd

2 M length

(Figure 5)

23

frequency given for each run depends upon measurements made
with a General Radio Type 566-A Wavemeter.
Voltage-Regulated Power Supplies.

Figure 3 shows the

circuits of the voltage-regulated power supplies which are
constructed on the same chassis.

Table I identifies the

components•
The minus seventy-five volt bias supply is regulated
by two 0A3 gas discharge tubes operating in parallel.

The

load through these tubes must be carefully balanced in
order to avoid excessive current in either.

A delay relay

applies the firing voltage to this pair simultaneously.
the firing process is photosensitive,

As

it is sometimes neces­

sary to illuminate one or both tubes in order to fire both.
The four hundred volt plate supply is adapted from one
described by Elmore and Sands

(42).

It uses the series re­

actance tube principle and maintains a regulation of about
0.1 percent with a ripple of about five millivolts.
High voltage power supply.

Figure 4 shows the circuit

of the high voltage power supply.
components.

Table I identifies the

The filtering Is good and the circuit provides

voltages from zero to fifteen hundred volts at currents from
zero to two hundred mllliamperes•
Impedance matching circuit.

Figure 5 shows the circuit

of the impedance matching unit which is connected to the trans­
ducer.

Table I Identifies the components.

This circuit would

probably require modifications for operation above fifteen

-AAAHI-

11-o

Voltage-regulated power supplies

■vw

Fig. 3.

24

vw
AA/V

25

P 9
2 0Q >"na

866A

Pig* 4.

High voltage power supply

GENERATOR
OUTPUT

o--TO TRANSDUCER

Pig. 5.

Transducer matching circuit

26

megacycles per second.

The radio-frequency ammeter is the

one used to indicate sound amplitudes by means of their
proportionality to transducer current.
Transducer
The transducer used for all measurements was an x-cut
quartz crystal

(thickness vibrator), driven near its funda­

mental frequency of ten megacycles per second.

The crystal

was one inch square.

about three-

Aluminum foil electrodes,

quarters of an inch square provided the electrical connections.
Silicone grease
and the glass.

provided acoustic coupling between the crystal
The sandwich was held in olace against the

glass block by means of light spring pressure.
Optical Arrangements and Measurements
Method "B".
method "B".

Figure 6 shows the optical system for

The light source is a General Electric Type

A-H4 high intensity mercury lamp.

Condenser lens

luminates a slit of aperture about 0.05 millimeters.

il­
The

Gaertner Type L 541 E monochromatic filter combination iso­
lates the Hg5461 line.

Lens Lg collimates the light.

The

polarizing prism is set at forty-five degrees to the slit.
The light beam passes through the glass block normal to the
direction of propagation of the sound waves.
fronts are parallel to the slit.

The analyzer is initially

set in the "crossed" position for extinction.
field is present,

The sound wave

When the sound

a diffraction pattern is formed in the focal

27

filter

polarizer

analyzer

Hg arc
L|

quartz
glass block

slit L 2

Fig. 6*

filter

L3

eye piece

Optical system for method "B11

Wollaston

prism

Hg arc
slit L

Fig. 7.

quartz V
glass block L

Optical system for method “C"

camera

28

plane of lens

(focal length, twenty Inches).

The dif­

fraction pattern is viewed by means of an eyepiece.
analyzer is then rotated through an angle
position)

(from the crossed

to produce extinction (or a minimum)

diffraction order.

The

in the first

Ten minimum settings were made and

averaged to determine the angle.
This is the condition of extinction of the r = 0 sub­
component of the first diffraction order.
be computed as in equation 7.

Tan 8

can then

This measurement was repeated

for at least four different sound amplitudes corresponding
to four values of transducer current.

The maximum amplitude

was made at least twice the minimum, but not great enough
to produce third order diffraction lines.
Typical plots of tan 8

versus the square of the trans­

ducer current I are shown in Figure 8.
A linear extrapolation can then be made, as shown, to
give values of tan 0 O
by equation 11.

, which are values of R as indicated

It is more convenient and accurate, however

to perform the extrapolation analytically (40), using the
least squares criterion of best fit.

Appendix II gives a

sample calculation for the borosilicate crown glass of
Figure 8.

The data of this investigation was obtained

by analytical extrapolation.
Method »C».
method "C11.

Figure 7 shows the optical system for

The production of the collimated beam is the

same as for method "B” .

In this case a Wollaston double

29

2.5

2.3

2.1

BOROS ILICATE CROWN

TAN 9

1.9

1.7
20
[CURRENT IN AMP]

1.5

DENSE BARIUM FLINT
1.3

1.1

2

3

4

5

6

7

8

9

[CURRENT IN AMP. ] 2

Fig. 8.

Experimental variation of tan 9 with I2

1.0

30

Image prism is inserted before the glass block.

Two beams

are thus formed, having polarizations normal and parallel
respectively to the slit and to the sound wave fronts.
beams diverge at an angle of forty-two minutes.

The

When a

sound field is present In the glass, two diffraction patterns
are formed In the focal plane of lens L^.

These patterns

are recorded photographically, as are also, for a comparison
of Intensities, the images of the slit when no sound is present.
Exposures were made for four or more sound amplitudes, cor­
responding to as many values of transducer current.

The

maximum amplitude was made at least twice the minimum, but
not great enough to produce second order diffraction lines.
The ratios of the Intensities in the first order lines
of the two patterns were determined photographically.

Be­

fore development, an emulsion calibrating exposure was made
with Hg5461 light and a rotating step-wedge to give a pat­
tern of densities corresponding to ten or twelve exposures.
Each step represents an exposure of one and one half times
the previous value.

These densities were compared to the

densities of the diffraction pattern lines by means of a
Jarrel-Ash Type JA-200 Microphotometer.

Assuming the re­

ciprocity relation and no intermittency effect, the ratio
of intensities B of the diffraction pattern lines was cal­
culated from the ratio of exposures in the step-wedge pattern.
Two or three exposures were made for each sound ampli­
tude and two or three photometer readings were made on each
first order line on each exposure.

A correction was applied

31

for the

(small) difference in intensities of the no-sound

beams due to any slight misalignment of the Wollaston prism.
Plots of

i5"b

versus the square of the transducer current

I are shown in Figures 9, 10, 11, and 12.
tained by this method.

The data was ob­

No correspondence with the theoreti­

cal curves of Figure 1 is evident.

No value of

or

of R is indicated.
Appendix III gives a sample calculation by this method
for the crown flint glass of Figure 12 ,
Temperature
Importance of temperature.

Since a change of temperature

varies both the dimensions of the glass block and the velocity
of sound,

it may disturb the standing wave resonance condition

and so alter the proportionality between transducer current
and sound amplitude.

Temperature gradients in the block

might produce inhomogeneities or anisotropies which would
disturb the optical measurements.

Harris

(43) and Filon (44)

report small changes in photoelastic constants with changes
in temperature.
Room temperature.

Since the glass block was cooled by

a current of air, the temperature of this air

(’’room tempera­

t u r e ” ) was the nominal temperature of the block.

Room

temperature was measured by means of an ordinary mercury
thermometer placed in the neighborhood of the block.
general,

In

a run was not satisfactory when the room temperature

varied much more than one half of a degree centigrade.

32

1.95 r

|

J .
1.80

1.65
1.50
1.35

15

±
.30

Fig. 9.

_L

_L

.45
.60
.75
.9 0
i2
[CURRENT IN AMP]

1.05

Experimental variation of
12 in BSC-1.

1.20

1.35

-J"B with

1.55
1.40
V

q

1.25
.10

2

Fig. 10.

3
4
5
7
6
[CURRENT IN AM P]2 x I 0 2-

8

9

Experimental variation of xiB with
Id in DBF-1.

33

2.50
2 .3 5 -

t

1 2.20

-

2 .0 5 1.90
.75

2

3

4

[CURRENT IN A M P ]2 x I0 2-----^
Pig. 11.

Experimental variation of *\Jb " with
l2 in fused silica.

2.00

V

|

1.85

b

_
1.70

1.55
1.40

2

3

4

5

6

7

8

9

[CURRENT IN A M P ]2 x I 0 2 ------ ►
Fig. 12.

Experimental variation of t)B with
l2 in C F-1.

34

Block temperature.

The temper&ture difference between

the surface of the block and the surrounding air was measured
by means of a thermocouple in contact with the surface and
insulated from the air.

This measurement gives some indica­

tion of the internal temperature of the block.

Data was

taken only when this temperature difference was less than
one quarter of a degree centigrade.
Glass and Fused Silica Samples
Measurements were carried out on thirteen samples of
fine-annealed optical glass and one sample of fused silica.
The ordinary glasses were supplied by the Bausch and Lomb
Optical Company, the rare earth glasses by the Eastman Kodak
Company, and the fused silica block was fabricated in Germany
and secured by the Owens-Illinois Glass Company through the
Hanovia Chemical Company.

The Bausch and Lomb samples

and the fused silica block were one inch cubes.

The Eastman

samples were three quarter Inch cubes or near cubes.
Table II gives descriptions of the samples.

B&L re­

presents Bausch and Lomb, E represents Eastman, H represents
Hanovia.

The densities are those determined in this labora­

tory by Barnes

(34) with the exception of the value for

block number nine which is the manufacturer’s nominal density,
n^ Is the nominal index of refraction for Na5893 light.
V is the nominal dispersion, being given by the formula
V =

(n^p - l)/(np - nc) In which np refers to H4861

35

TABLE II
DESCRIPTION OF GLASS SAMPLES

Y o u n g !s
Modulus
dynes
cmfA
xlO11

Bulk
Modulus
dynes
-U
- f. 1
cm^
xlO1!

Jlock
No.

Mf g r .

1

B&L

BSC-1

2.48

1.51100

63.5

7.64

4.39

2

B&L

C-l

2.53

1.52300

58.6

7.57

4.30

3

B&L

CF-1

2.69

1.52860

51.6

6.20

3.54

4

B&L

LF-1

3.18

1.57250

42.5

5.91

3.50

5

B&L

LBC-2

3.14

1.57250

57.4

7.55

4.85

6

B&L

DBF-1

3.60

1.61700

38.5

5.98

3.64

7

B&L

DF-2

3.64

1.61700

36.6

7.81

5.58

8

B&L

DBC-2

3.78

1.61700

54.9

5.64

3.41

9

B&L

EDF-1

3.91

1.64900

33.8

7.98

4.40

10

B&L

EDF-4

4.72

1.75060

27.7

5.45

3.57

11

E

EK-110

4.13

1.69680

56.15

10.24

8.04

12

E

EK-330

4.57

1.75510

47.19

10.88

8.58

13

E

EK-450

4.63

1.80370

41.80

11.80

9.52

14

H

Fused
silica

2.20

7.30

3.67

Type

Dens ity
gra/cm*5

nD

V

56

and H q to H6565.

The elastic constants are preliminary

values of dynamic determinations made in this laboratory
by Barnes

(34) using ultrasonic waves of the same order

of frequency as those used in this work*
Table III gives the compositions of the samples as
supplied by the manufacturers.
The letters in the Bausch and Lomb designations for
the glass types have the following significance:
borosilicate crown;

BSC,

C, crown; CF, crown flint; LF, light

flint, LBC, light barium crown; DF, dense flint; DBF, dense
barium flint; DBG, dense barium crown; E D F , extra dense
fli nt •

37

TABLE III
COMPOSITION^ OF GLASB SAMPLES IK PERCENT BY WEIGHT

Block No.
Type
Melt No.

1
BBC-1
0-9560

2
C-l
0-7763

3
CF-1
0-9841

4
LF-1
0-9447

5
LBC-2
0-8765

SlOg

70.8

71.7

67.8

54.3

49.1

K 2°

12.1

2.0

11.2

8.0

7.8

7.5

13.7

2.0

3.0

0.5

9.0

32.5

Na20
PbO

2.1

BaO

31.0

SrO
CaO

1*3

9.0

b 2°3

8.0

1.1

ZnO

3.4

1*5

4.0

7.5

1.0

6.0

0.5

ZrOg
TiOg
Alg03
BbgO^
As 2 03

0.3

L a 203
Th02
Ta205
w o3
Al+Si+Zr+Ti

(oxides)

0.1

0.2

38

Block No.
Type
Melt No.

iii

6
DBF-1
0-9271

7
DF-2
0-9001

45*0

46.6

to
O

table

6.3

6.4

NagO

2.0

Si02

PbO

38.2

BaO

7.6

46.3

continued

8
DBC-2
0-8472

9
EDF-1
0-4432

10
EDF-4
0-8030

39.0

42.1

31.5

7.1

1.6

48.8

64.5

0.2

0.1

1.6
42.7

SrO
CaO
4.7

b 2°3

ZnO

0.8

0.4

5.4
0.5

ZrOg

1.5

TiOg
5.1

Alg03
0.3

Sbg03
^■s 2^3

0.1

L a 203
ThOg
Ta ^05
W03
A1+Si+Zr+Ti

(oxides)

2.0

0.8
0.2

0.3

0.3

39

TABLE III CONTINUED

Block No.
Type

11
EK-110

12
EK-330

13
EK-450

Si02

100

K 20
N a 20
PbO
BaO

14

SrO

6

12
}•

CaO
B 2°3

40

30

22

20

28

40

20

12

6

18

11

ZnO
Zr02
Ti02
a i 2°3

Sb203

AS2O3

ThO,
2
Ta2°5

4

W03
Al+Si+Zr+Ti

14
Fused silica

(oxides)

11

40

PRESENTATION OF D a TA
Method "B"
Only method "B" led to data which agreed with the theory
and permitted extrapolation.

Therefore values of R were ob­

tained only by this method.
Accuracy of d ata. Figure 1 shows the departure of the
theoretical points from a straight line.

As is mentioned

above, an analytical straight line extrapolation for the
case in which R equals two and v is not greater than unity
leads to an error of 0.55 percent in the value of R.

Since

all glasses tested exhibited values of R less than 1.83* the
"theoretical" error introduced by linear extrapolation is
certainly less than 0.5 percent.
The linearity of typical experimental data is shown in
Figure 8.
Since the result is obtained by extrapolation to zero
transducer current, absolute accuracy of current determina­
tion is unnecessary.
readings.

In one case

Linearity is sufficient for the meter
(block number’eight) successive de­

terminations were made with different ammeters.

The re­

sulting values for R differed by 0.31 percent, less than the
average percent of difference between runs using the same
meter.

Although the analytical extrapolation method

(40)

assumes deviations only in the values of the ordinates, errors

41

in abscissa

(current) values must also reflect in the

"probable error" of the y-intercept.

Although only four

or five points were involved in each determination, a formal
value of the probable error pR in the value of R due to the
deviation of the data points from the straight line was cal­
culated

(as in Appendix II) for each determination.

In every

determination of R for a glass sample p_ was less than 0.3

R

percent.

This gives some indication of the effect of any

non-linearity in the ammeter and of errors in the scale
rea d i n g s •
Each valu£ of extinction angle used in the calculations
was an average of ten settings and readings of the analyzer
circle.

Each value of the "crossed" position of the analyzer

was an average of ten settings and readings.

Angle determina­

tions are most difficult for the faint lines corresponding
to low sound amplitudes and for materials of large R which
require a greater analyzer rotation and consequently permit
more background illumination.

Since p^ was less than 0.3

percent for all glass determinations, it appears that the
contributions of errors in angle measurements to non-linearity
must be smaller than that.
Two Independent determinations of R were made for each
glass sample.

They agree, in every case, to within about

one percent.
Fused silica requires some special consideration.

Be­

cause of the large value of R it is especially necessary to

42

use small values of current
in this case.

(low sound amplitude, small v)

Thus the extrapolation was made from only three

points, not including a doubling of sound amplitude.

Be­

cause of this narrow base for extrapolation the uncertainty
in R is greater.
for fused silica.

Therefore four determinations were made
Each differs from the average by less than

0.8 percent.

It seems safe to conclude that the

average values

for

R are correct to within two percent.
Numerical r e s u l t s .

Table IV lists the individual de­

terminations with corresponding temperatures and approximate
frequencies, together with the final average values for R,
the ratio p/q of the Neumann strain-optical constants.
Method "C"
Accuracy of d a t a .

The remarks about the measurement of

transducer current which were made in regard to method '^B11
apply also in this case.'
Some idea of the dependability of the photographic method
can be gained from an examination of the spread of experi­
mentally determined values.
this experimental spread.

Figures

9,

10, 11, and 12 show

This check is especially signifi­

cant for the case of block number one

(borosilicate crown)

for which the experimental spread at each current includes
four determinations.

The spreads for block number six in­

clude two values while those for block number three include
two or three v a l u e s .

43

TABLE IV
NUMERICAL RESULTS

Freq.
me/sec

Room
Temp.
°C

Points for
Extrapolat ion

R

p/q

PR

1

BSC -1

9.8

24.0

5

1.827

.0025

1

BSC-1

9.8

25.0

4

1.816

.0011

2

C-l

10.2

25.0

4

1.823

.0004

2

C-l

10.0

25.0

4

1.819

.0004

3

CP -1

9.4

to
o
•

Block
No.
Type

5

1.672

.0014

3

CP -1

10.0

26.0

5

1.689

.0010

4

LF-1

9.9

26.5

4

1.440

.0038

4

LF-1

10.0

26.0

4

1.452

.0004

5

LBC-2

10.1

25.5

4

1.625

.0003

1.618

.0009

Aver.
R

1.82

CJ1

1.82

1.68

1.45

1.62
5

LBC-2

9.8

25.0

4

6

DBF-1

10.5

26.0

5

1.366

.0009
.0004

1.36
6

DBF-1

10.2

24.5

4

1.361

7

DF -2

10.2

28.0

4

1.456

.0010

7

DF -2

9.9

27.0

4

1.459

.0003

8

DBG-2

9.8

26.0

5

1.281

.0010

8

DBC-2

10.2

22.5

4

1.285

.0006

9

EDF-1

H

.
o

H

25.0

4

1.459

.0010

9.4

25.0

5

1.464

.0007

1.46

9

EDF-1

1.28

1.46

44

TABLE IV CONTINUED

Block
No.
Type

Freq*
mc/sec

Room
Temp.
°C

Points for
Extrapolation

Aver.
R

p/q

Pr

10

EDF-4

9.6

25.0

5

1.109

.0045

10

EDF-4

10.5

26.0

4

1.113

.0013

11

EK-110

9.9

24.0

4

1.530

.0017

11

EK-110

10.2

26.5

4

1.536

.0017

12

EK-450

10.2

27.5

4

1.548

.0016

12

EK-450

o
•
o
i
—1

27.0

4

1.554

.0016

13

EK-330

10.1

30.0

4

1.595

.0007

13

EK-330

9.8

27.5

4

1.596

.0007

14

Fused
silica

10.1

27.0

3

2.343

Fused
s ilica

10.4

27.0

3

2.349

k

1.11

1.53

1.55

1.60

14

2.34
14
14

Fused
silica

10.1

26.5

3

2.318

Fused
silica

10.0

25.5

3

2.334

45

Multiple diffraction cannot have caused trouble* for
only the first order lines appeared.
Transverse waves cannot have caused trouble since the
frequencies used were non-resonant for them* and* since their
diffraction pattern has a different spacing than that due to
the longitudinal waves* their presence would have been obvious.
Due to the action of the Wollaston prism* the beams of
light are not quite normal to the sound beam but have an
obliquity of twenty-one minutes,

Parthasarathy (45)

(46)

has investigated the asymmetrical intensity distribution in
the diffraction pattern due to oblique incidence of light
on progressive waves in liquids.

He reports an effect only

at wave lengths shorter than those involved in this work.
In addition* since the two beams from the Wollaston prism
are symmetrical In incidence and since the sound field con­
sists of standing waves* any effect should disappear when
the two first order intensities in each pattern are averaged
in the calculation.

Further*

an examination of the data in­

dicates no significant intensity asymmetry.
There is thus no apparent reason to doubt the experi­
mental values.
Numerical r e s u l t s .

Since the experimental points do

not follow the linear pattern indicated by the theory they
do not lead to values of R.

The values of the intensity

ratios for various currents are the only numerical results
of method 11C 11.

These values are shown on the curves of

46

Figures 9, 10 , 11 , and 12.
Values of R from Other Investigations
It is well known that glasses of similar composition
or even ’’identical1’ samples may differ significantly in their
properties.
(35).

See, for example, Coker and Filon (9 ) and Mueller

In addition, effects of aging on photoelastic properties

and temperature dependence are reported by Harris
Filon (44).

(43 ) and

So it is with grave reservations as to its signi­

ficance that the data in Table V is given.

The glass types

are merely similar, identified on the basis of incomplete
data on density,
composition.

index of refraction, elastic constants, or

The parenthesis In the Table indicate that the

similarity of types is quite doubtful.
to Pockels

(5), to Twyman and Perry

(7), and to Schaefer and

Nassenstein (8 ) are by static methods.
(26)

The measurements due

Those due to Vedam

(27) are by Mueller!s method "B” with sound intensities

inferred from input voltage to the power amplifier.

Those

due to Schaefer and Dransfeld (28) are by Mueller!s method
”B ” with sound amplitudes inferred from intensities in the
diffraction pattern and extrapolation from two points giving
an announced accuracy of five percent.

Bergmann and Fues (29)

used the intensity ratio method without extrapolation, being
content with sound amplitudes low enough to avoid second
order lines.
ficance.

Their results are, therefore,

of doubtful signi­

Bergmann has since remarked on the necessity of

extrapolation

(47).

It is to be noted that Schaefer and

47

TABLE V
VALUES OP p/q GIVEN BY OTHER INVESTIGATORS
(Glass types compared in this table are
merely sirrilar, with parenthesis
indicating especially doubtful
similarity)

This Investigation
Block No.
Type
p/ q
1

BSC -1

1.82

3

CP-1

1.68

4

LF-1

1.45

5

LBC -2

1.62

6

DBF-1-

1.36

7

DF-2

1.46

9

EDF-1

1.46

10

EDF-4

1.11

14

Fused
Silica

2.34

Other Investigations
Type
Reference
p/q
BK-1
BK -1
BK-1 2
BK-1 2
KF-7
KF-7
(LF-1 )
(LF-1)
LF-4
LF-4
0,8154
(x)
(10 )
BaK-4
BaK-4
(9)
BaSF -1
BaSF -1
(12 )
(0.1571)
(14)
(SF-2 )
(SF-2 )
(SF-9)
(SF-9)
(SF-4)
(3F-4)
0.500
(16)
(17)
Fused
Silica

(8 )
(28)
(8 )
(28)
(8 )
(28)
(8 )
(28)
(8 )
(28)
(5)
(7)
(27)
(8 )
(28)
(27)
(e>
(28)
(27)
(5)
(27)
(8 )
(28)
(8 )
(28)
(8 )
(28)
(5)
(27)
(27)

1.76
1.73
1.82
1.89
1.64
1.64
(1.45)
(1.50)
1.32
1.39
1.44
(1.51)
(1.70)
1.73
1.75
(1.84)
1.29
1.29
(1.50)
(1.27)
(1.41)
(1.18)
(1.26)
(1 .20 )
(1 .22 )
(1.07)
(1.07)
1.11
1.05
(1.18)

(26)

2.85

48

Nassenstein used the same set of glasses as Schaefer and
Dransfeld.

49

SUMMARY AND CONCLUSIONS
Mueller has described,

In theory, two methods for the

determination of the ratio p/q of the strain-optical constants
of glass.

Both methods Involve the diffraction of light by

means of ultrasonic waves.

Method "B" involves measurements

of the polarization of the diffraction orders.
involves measurements

Method HC"

of ratios of intensities in the dif­

fraction orders.
The appropriate experimental arrangement was provided
for each method.

Data was taken, by both methods,

on three

glass samples and on a fused silica sample.
The data given by method "B” was in excellent agreement
with the theory.

This method was then applied to a set of

thirteen American optical glasses,
glasses, and to fused silica.

including three rare earth

The ratio of the strain-optical

constants was calculated for each sample for dynamic harmonic
strains having a frequency in the neighborhood of ten mega­
cycles per second.
The data given by method "Cn did not agree with the theory
and did not lead to values for the ratio of the strain-optical
constants.

The data followed a definite pattern, however,

and the pattern was consistent among the four sarples tested.
It may be concluded that method "B" Is applicable as de­
scribed and leads to dynamic values of the ratio of the strain-

50

optical constants.

It may also be concluded that method

"C" does not apply as described.

However, on the basis

of the pattern and consistency of the data obtained by
method "C 11 it is hoped that it may be shown to have
significance.

51

APPENDIXES

52

APPENDIX I
Theoretical Intensity Ratios in the
Diffraction Orders
Arrangement of MuellerT3 Method "B11.

The optical system

for M u e l l e r ^ method nB ,! (23) is shown in Figure 6 ,

For the

case of standing longitudinal waves in glass, incident light
polarized at forty-five degrees to the slit, crossed analyzer,
and unit incident intensity, the intensity I
1
7
0 m,r in the r subcomponent of the m diffraction order Is given by the Mueller
theory as

I
where the quantities are the same as those in equation 5.
The total intensity I in the order of index m Is
J m
given by

(1-2 )

1
Numerical substitutions for R and v give the following
ratios of the total intensity in the third order to the total
intensity In the zero or central order.

53

TABLE VI
TOTAL INTENSITY RATIOS:
THIRD ORDER
TO ZERO ORDER, METHOD "B"

v

R 1.5

R 2.0

R 2.5

0.8

.0136

.0206

.0292

1.0

.0221

.0327

.0464

Numerical substitutions for R and v give the following
ratios of the intensity in the first sub-component to the
intensity in the zero sub-component for the case of the
first diffraction order.

TABLE VII
SUB-COMPONENT INTENSITY RATIOS:
FIRST TO ZERO
SUB-COMPONENTS, FIRST ORDER, M e T s OD "B"

v

R 1.5

R 2.0

R 2.5

0.8

.0223

.0591

.156

1.0

.0460

.155

.610

Arrangement of MuellerTs Method uC n .

The optical system

for M u e ller1s method UC ” (23) is shown in Figure 7 .

For the

case of standing longitudinal waves in glass, incident light
polarized parallel to the slit, and unit incident intensity,
the intensity I
in the r sub-component of the m difm, r
fraction order is given by the Mueller theory as

54

I ’v < ’

[7^)j^mx

if

(i-3

where the quantities are the same as those in equation 5 .
The total intensity in the m order is given by equation 1-2.
Numerical substitutions for R and v give the following
ratios of the total intensity in the second order to the
total intensity in the zero or central order.
TABLE VIII
TOTAL INTENSITY RATIOS:
SECOND ORDER
TO ZERO ORDER, METHOD "C,!

V

R 1.5

R 2.0

R 2.5

0.4

.000788

.00256

.00650

0.6

.00420

.0141

.0368

0.8

.0155

.0487

.127

1.0

.0368

.127

.299

Numerical substitutions for R and v give the following
ratios of the intensity in the first to that in the zero
sub-component of the first diffraction order.

55

TABLE IX
BUB-COMPONENT INTENSITY RATIOS: FIRST TO ZERO
SUB-COMPONENTS, FIRST ORDER, METHOD 11C"

0 •4

.000122

.000429

.00106

0.6

.000697

.00229

.00602

.00229

.00805

.0226

o
•
H

o
•
03

R 1.5

CO
•
o

R 2.5

V

.00602

.0226

.0702

56

APPENDIX II
Sample Data and Calculation for Method "B"
Glass;

Block number 1 , type BSC -1

Frequency:

9.8 megacycles

Room temperature:

24°

(oscillator dial 20.36)

centigrade

Thermocouple thermometer scale reading maintained less than
2.5

(temperature difference:

less than one quarter

degree centigrade)
Maximum permissable transducer current:

no third order at

1.5 amperes
Analyzer angle readings in degrees:

(for extinction or a

minimum)
No sound
("crossed")

.50 amp

.73- amp

.92 amp

1.10 amp

1.50 amp

44.75
44.80
44.85
44 .85
44.80
44.75
44.80
44.80
44.85
44.80

60.8
61.6
61.3
60.5
61.3
61.2
61.0
60.8
61.0
60.0

60.2
60.8
60.0
61.0
60.3
(59.4)
60.9
60.9
61.0
60.6

60.2
60.6
60.5
60.3
60.3
59.6
60.3
60.1
59.5
60.5

60.3
59.1
59.8
60.2
60.7
60.1
59.5
59.5
59.7
60.1

60.1
59.0
60.0
59.5
59.1
59.0
59.4
59.2
59.2
59.4

60.63

60.19

59.90

59.39

Analyzer angle averages:
44.80

60.90

57

Values of ©

i

(9 = minimum setting minus

plus forty-five degrees.

Tan 0

"crossed" setting

See equation 7 )

.50 amp

.71 amp

.92 amp

61.10

60.83

60.39

60.10

59.59

1.811

1,791

1.759

1.739

1.704

:

1.10 amp

1.50 amp

For linear extrapolation assume a curve of form y - a + b x ,
2
where x = I and y = 10(tan9 -1 .6 ), which gives best fit
in sense of least squares.
The y-intercept y Q is given by the value of the coefficient
a which is, in turn given (40) by

y

nZx--(Z*r

Evaluating a:
X

y

.250
.504
.846
.21
.69
4.50

2.11
1.91
1.59
1.39
1.04
8.04

xy

X2

.5275
.9625
1.345
1.6825
1.755
6.2725

.0625
.2540
.7140
1.464
2 •856
5.3505

Substituting In equation II-l,
yQ —

a —

2.2729

Evalutating R:
R =

tan 0O

R =

1.827

=

1.6 +

y 0/lO

Evaluating ppj;
If the data points are assumed to have a random dis­
tribution about a theoretical straight line, the probable
error ppj in the y-intercept is given formally by (40)

56

(II-2)
where
(II-3)
In this work the data points do not follow a relation
which is, theoretically, perfectly linear.

(See Figure 1 ).

Nevertheless, this calculation is carried through formally to
give some indication of the reliability of the data.
For this sample,
b

= -0.7394

pa =

0.0254

PR ~

0.0025

59

APPENDIX III
Sample Data and Calculation for Method "C"
Glass :

Block number 3, type CP -1

Frequency:

9.4 megacycles

Room temperature:

(oscillator dial 20.06)

23° centigrade

Thermocouple thermometer scale reading maintained less than 2 >
(temperature difference:

less than one quarter degree

centigrade)
Maximum permissable transducer current:

no second order at

300 milliamperes
Emulsion sensitivity exposure:

rotating step-wedge with

step to step exposure ratio of 1.5
Diffraction pattern exposure:
Exposure number
1
2
3
4
5
6
7
8
9
10
11
12
13

Exposure time
____ (s e c )____
1/2
1/2
1
1/2
1/2
1/2
1/2
1/2
1/5
1/5
1/5
1/2
1/5

Transducer current
(ma_)________
no sound
100
100
150
150
200
200
250
250
300
300
100
250

60

Photometer readings on successive sector steps:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10 )
11 )
12 )
13)

2.0
4.5
10.3
19.9
35.3
50.2
64.5
75.5
80.7
86.7
90.9
93.8
95.6

1.5
4.7
10.3
20.3
35.5
48.8
63.0
75.4
80.3
86.5
91.0
93.8
95.8

1.5
4.8
10.5
19.5
36.0
50.0
62.8
75.6
81.2
86.6
90.8
93.4
95.4

Average:

1.67
4.67
10.37
19.90
35.60
50.00
63.43
75.50
80.73
86.60
90.90
93.67
95.60

The average photometer readings are plotted against the
logarithm (to the base 1.5) of the rotating sector ex­
posures.

A standard s-shaped emulsion sensitivity curve

results *
Pairs of photometer readings on the two lines of exposure
1 (no s o u n d ) :

63.4
64.7

65.0
66.0

65.8 67.0
64.6 65.6

Average difference:

66.9

1.22

Difference in log exposure
curve):

65.8

(read from sensitivity

0.084

This diffrence, due to the inequality of the two beams
from the Wollaston prism, will be applied below as a
correction.
Photometer readings
L

on diffraction lines and calculations:

and L ?

;

photometer readings on the diffraction
lines of one beam.
(Light polarized
normal to slit).

L 3 and L a

;

photometer readings on the
lines of the other beam.

diffraction

61

A

12

average of L^ and L^.

A34

average of Lg and L 4 .

Bkg

photometer reading on background near
lines.

Bkg Cor

Correction to A due to presence of back­
ground.
Bkg Cor Bkg (100 - A)/ 100.

Ac

Corrected average photometer readings on
lines.

G_

Logarithm of average exposure of dif­
fraction lines of one beam.
G is de­
termined from A c by means of sensitivity
curve.

G2

Logarithm of average exposure of diffrac­
tion lines of the other beam.

D^Gg-Gi

Logarithm of ratio of exposures of dif­
fraction lines of the two beams.
Since
the exposures are made simultaneously, D
is the logarithm of the ratio of the in­
tensities of the diffraction lines of the
two b e a m s •

Da

Average value of three determinations
of D.

Dac = Da -*.08

Da corrected for beam inequality.

B = X.5^D a c ^

Intensity ratio of the diffraction lines.

62

Exp

L1

L2

A12

2
2
2
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
10
10
10
11
11
11
12
12
12
13
13
13

15 *4
13.9
13.1
28.8
25.3
21.3
28.1
25.5
22.1
47.2
42.8
36.0
47.0
42.0
34.9
61.2
56.4
51.0
6.1
4.7
5.8
13.7
11.6
14.5
11.6
9.9
9.7
13.4
12.9
12.3
6.1
4.7
6.3

15.9
14.0
12.5
28.9
25.2
22.0
29.2
25.2
22.2
49.0
43.4
37.5
48.0
43.3
35.8
61.6
56.8
51.7
6.4
4.3
6.0
14.6
12.2
15.2
12.5
11.2
9.4
13.1
13.0
12.3
6.2
5.4
7.6

15.65
13.95
12.80
28.85
25.25
21.65
28.65
25.35
22.15
48.10
43.10
36.74
47.50
42.65
35.35
61.40
56.60
51.35
6.25
4.50
5.90
14.15
11.90
14.85
12.05
10.55
9.55
13.25
12.95
12.30
6.15
5.05
6.95

Bkg Bkg
Cor
2
2
2
1
2
3
2
2
3
2
2
2
1
1
2
3
2
2
0
0
0
0
0
0
0
0
0
1
2
2
0
0
0

1.7
1.7
1.7
0.7
1.5
2.3
1.4
1.5
2.3
1.0
1.1
1.3
0.5
0.6
1.3
1.3
0.9
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.9
1.7
1.8
0.0
0.0
0.0

Ac

G1

L3

L4

13.95
12.25
11.10
28.15
23.75
19.35
27.25
23.85
19.85
47.10
42.00
35.45
47.00
42.05
34.05
60.10
55.70
50.35
6.25
4.50
5.90
14.15
11.90
14.85
12.05
10.55
9.55
12.35
11.25
10.50
6.15
5.05
6.95

3.43
3.24
3.11
4.53
4.23
4.00
4.47
4.25
3.93
5.84
5.49
5.04
5.83
5.50
4.94
6.75
6.44
6.07
2.34
1.95
2.28
3.47
3.20
3.52
3.21
3.02
2.88
3.25
3.13
3.02
2.31
2.08
2.45

41.2
36.4
32.2
67.6
60.2
54.0
65.1
59.4
54.4
81.6
77.0
72.2
81.0
76.4
70.7
87.0
84.4
81.4
39.7
35.0
38.0
59.2
56.3
59.7
58.0
52.3
50.0
37.3
35.7
33.1
41.5
35 .6
43.8

40.2
35.4
31.2
66.4
59.4
54.3
65.0
58.8
53.5
81.9
77,4
73.0
81.4
77.4
71.2
87.0
84.4
81.4
39.8
34.7
39.0
59.9
56.0
60.3
57.6
52.9
50.6
36.8
35.6
31.3
42.0
36.7
43.3

63

2
2
2
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
10

10
10
11
11
11
12
12
12
13
13
13

A 34
40 •70
35 •90
31 •70
67 .00
59.80
54.15
65.05
59.10
53.95
81.75
77.20
72.60
81.20
76.90
70.95
87.00
84.40
81.40
39.75
34.85
38.50
59.55
56.15
60.00
57.80
52.60
50.30
37.05
35.65
32.20
41.75
36.15
43.55

6kg Bkg
Cor
1
2
2
1
2
2
1
1
1
1
1
1
1
1
2
2
2
2
0
0
0
0
0
0
0
0
0
1
1
2
0
0
0

The values of

0.6
1.3
1.4
0.6
0.8
0.9
0.3
0.4
0.5
0.2
0.2
0.3
0.2
0.2
0.6
0.3
0.3
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.6
0.6
1.4
0.0
0.0
0.0

Ac

G2

40.10
5.36
34.60
4.98
30.30
4.68
66.40
7.18
6.67
59.00
6.27
53.25
7.06
64.75
58.70
6.65
53.45
6.29
81.55
9.00
77.00 8.21
72.30
7.70
81.00
8.92
76.70 8.24
7.51
70.35
86.70 10.00
9.47
84.10
8.91
81.00
5.34
39.75
4.99
34.85
5.25
38.50
6.71
59.55
56.15 6.48
6.74
60.00
57.80 6.59
6.22
52.60
50.30
6.06
5.10
36.45
5.00
35.05
30.80 4.71
5.48
41.75
5.08
36 •15
5.60
43.55

D
1.93
1.74
1.57
2.65
2.44
2.27
2.59
2.41
2.36
3.16
2.72
2.66
3.09
2.74
2.57
3.25
3.03
2.84
3.00
3.04
2.97
3.24
3.28
3.22
3.38
3.20
3.18
1.85
1.87
1.69
3.17
3.00
3.15

Gac

-5b ~

1.75

1.67

1.40

2.45

2.37

1.62

2.45

2.37

1.62

2.85

2.77

1.75

O
CO
•
c\i

Exp

2.72

1.74

3.04

2.96

1.82

3.00

2.92

1.81

3.25

3.17

1.90

3.28

3.20

1.91

1.80

1.72

1.42

3.11

3.03

1.85

Da

from the last column are plotted aga:

the square of trie transducer current in Eigure 12 .

64

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