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TH ESYS

  

ABSTRACT

SUBHARMONIC GENERATION IN AN
ACOUSTIC FABRY-PEROT INTERFEROMETER

by Jack A. Bamberg

Above a certain energy density threshold in an intense ultra-
sonic standing wave, subharmonics of the driving frequency are sdmetimes
present. To study this effect, an acoustic Fabry-Perot interferometer
was constructed, consisting of two air-backed quartz transducers submerged
in water. One of the transducers is driven at frequencies around its
3 MHz resonance. Electronic analysis of the waveforms present in the
cavity is accomplished with a spectrum analyser connected to the other
quartz transducer. Optical methods are used as a supplementary means of
analysis.

It was observed that subharmonics exist only at frequencies
for which the cavity is in resonance and always occur in pairs, such
that the sum of their frequencies is equal to the driving frequency.
These discrete frequencies appear only in regions near integral sub-
multiples of the driving frequency. The pressure in the cavity at the
onset of subharmonic oscillation may be as low as 1.5 atmosphere. In
this study, subharmonics were obtained only when the cavity reflector

had a frequency dependent reflection coefficient.

SUBHARMONIC GENERATION IN AN

ACOUSTIC FABRY-PEROT INTERFEROMETER

by

Jack A. Bamberg

A THESIS

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

MASTER OF SCIENCE

Department of Physics

1967

 

ACKNOWLEDGMENT

The author wishes to eXpress his gratitude to Professor
E. A. Hiedemann and Dr. B. D. Cook for the assistance received in the
formulation and solution of this problem. The discussions with Dr.
W. R. Klein, Dr. F. Ingenito, and the other members of the ultrasonics
group at Michigan State University also have been very helpful. The
financial support of the Office of Naval Research is gratefully

acknowledged.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENT . .
Chapter
I. INTRODUCTION .
II. APPARATUS . .
III. PROCEDURE . .
IV. RESULTS .
V. DISCUSSION . .

APPENDIX: Ultrasonic Light

BIBLIOGRAPHY

Diffraction

111

Theory

Page
ii

10
13
29
33
hl

 

LIST OF TABLES

Table Page
1.1 Terminology for various frequencies

Of the system C O O O O O O O O O O O O O 3

h.1 Typical subharmonic data sample . . . . . . 1h

iv

Figure

2.1

h.8
h.9

h.10

h.ll

LIST OF FIGURES

Experimental arrangement of electronic
equipment

Experimental arrangement of optical
equipment

Electronic pulse alignment arrangement
A typical subharmonic threshold curve
Graph of observed subharmonics as a
function of frequency, with a 2.0 MHz
reflector . . . . . . . . . . . .
Graph of subharmonic thresholds as a
function of frequency, with a 2.0 MHz

reflector . . . . . . . . . . . . .

Graph of observed subharmonics; 1.0 MHz
reflector . . . . . . . . .

Graph of subharmonic thresholds; 1.0 MHz
reflector . . . . . . . . . . . . . .

Spectrum analyser tracing of subharmonics
around f/2 for odd m . . . . . . . .

Bands of observed subharmonics in terms of
the driving frequency f

Brass reflector . . . . . .

Typical cavity resonance, showing regions
for which subharmonics occur

Response curve for cavity, showing relative
sound pressure . . . . . . . .

Diffraction pattern photographs obtained
during subharmonic generation . . . .

Page

11

15

16

18

19

20

21

21

22

23

25

27

Figure

A.l

Graph of average light intensities
in the first few diffraction orders,
for a stationary wave . . . . . . .

vi

Page

36

CHAPTER I

INTRODUCTION

It is sometimes possible to solve problems in acoustics by
the elimination of nonlinearities in the equations describing the system;
this approximation often leads to valuable information about the system.
However, there are also many systems for which linear treatments will
not suffice. For example, the deviation from a linear Hooke's Law
relationship becomes significant when large amplitude vibrations occur
in an elastic medium.

A more accurate description may be obtained in such instances,
by including a few nonlinear terms as perturbations on the linear situa-
tion. In other cases, new phenomena occur in a nonlinear system which
can not occur in a linear system. An example of this type is the occur-
rence of oscillations at a lower frequency than the resonant frequency
of the system, i.e. subharmonics. The above approximate techniques
are not useful here, since the effect depends upon the very terms we
need to neglect. It is difficult to treat such problems analytically.

The existence of subharmonics has long been known; the sub-
ject was discussed as early as 1877 by Lord Rayleigh [1]. Since then,
numerous examples have appeared in the literature. Most of these exhibit

a mechanical nonlinearity leading to an equation of motion of the form

m x + c x + f(x)

F Out) (1.1)

where x is the vibration amplitude, F is the external driving force,

m and c are constants of the system, and

Solutions of Eq. 1.1 predict subharmonic reSponse under certain
conditions. Stoker [2], Hayashi [3], and others have eXplored these
solutions in some detail.

Recently, nonlinear wave interactions have been shown to
produce subharmonic waves at ultrasonic frequencies in liquids. In
1965, Kuljis [h] observed subharmonic generation in water at around
300 kHz in a cavity, using two Spherical bowl transducers to focus the
sound. The cavitation threshold was very low, so that subharmonics
were present only during the nonlinear cavitation process.

Cavitation and focusing, which immensely complicate an
analytical analysis, are not required for subharmonic generation.
Korpel and Adler [5], using an acoustic Fabry-Perot interferometer
Operating around 5 MHz in water, observed subharmonics at intensities
well below the cavitation threshold. Analysis of the waveforms present
in the cavity was accomplished by observation of the optical diffraction
pattern produced by passing monochromatic light through the standing
wave sound field. From Eq. A.h in the Appendix, it can be seen that
the spacing of diffraction orders is directly proportional to the sound
frequency. Hence, the appearance of additional orders between the orders
due to the fundamental wave was attributed to the presence of subharmonic

waves in the cavity. If the cavity was driven at a multiple of the

TABLE 1.1

Terminology for various frequencies of the system.

SYMBOL TERM
f fundamental (or

pump) frequency

f fundamental cavity
resonant frequency

fS subharmonic
frequency

ftr transducer resonant
frequency

EXPLANATION

The frequency used to excite the
cavity.

The lowest frequency at which
a standing wave can exist in
the cavity, i.e. the effective
cavity length is equal to x /2,
where Kb is the wavelength 8f a
wave of frequency fo in the fluid.

Frequencies in the cavity lower
than f, due to nonlinear coupling
to the pump frequency.

Resonant frequency of transducer
which is driven at the pump
frequency.

fundamental cavity resonant frequency (see Table 1.1), they observed
subharmonics above a certain energy density threshold in the water.

The subharmonics occurred in pairs such that the sum of the two fre-
quencies was always equal to the pump frequency. Also, the subharmonics
were observed to be multiples of f0, and consequently are at resonance
in the cavity. The subharmonic frequencies were all in the vicinity of
f/Z. In 1966, McCluney [6] and Breazeale [7], utilizing the optical
diffraction pattern and electronic means to determine the subharmonic
frequencies, concluded that subharmonics occur at integral sub-multiples
of the pump frequency. Since a given subharmonic could be obtained

at several successive cavity resonances, the subharmonic is very

seldom at resonance in the cavity, a result incompatible with the
results of Korpel and Adler.

Several questions arise from these two studies. Are the
subharmonics at a cavity resonance or not? What are the acoustic
pressures of the fundamental and subharmonics at threshold? What
factors influence the thresholds for the various subharmonics? How
does the type of reflector and its thickness effect subharmonic genera-
tion? In this study, initiated late in 1965, subharmonic generation
in an acoustic Fabry-Perot cavity is systematically investigated, in

an attempt to answer these and other questions.

CHAPTER II

APPARATUS

The acoustic Fabry-Perot interferometer is composed of two
air-backed, x-cut quartz transducers, submerged in water, and accurately
aligned such that the front surfaces are parallel. Each transducer
assembly is mounted on an optical bench, which allows the separation
between them to be varied from 0.5 cm. to 30 cm. Each transducer I
may be easily replaced with other transducers of varying thickness. rs
TWO Spring loaded adjustment screws, pivot the reflecting surfaces
about two perpendicular axes in the plane of the transducer faces to
within 0.05 degree.

A stable, variable frequency transmitter with a maximum
output power of 100 watts excites one transducer, whose resonant
frequency in water is 2975 kHz. The other transducer serves as a
reflector and as.a receiver for electronic analysis of waveforms
present in the cavity.

Both optical and electronic techniques are used to study
the waveforms present in the cavity. Electronic analysis is accomp-
lished with a Singer Metrics SB 12 spectrum analyser connected to the
quartz reflector, as in Fig. 2.1. The analyser displays amplitude as
a function of frequency, causing a sinusoidal signal to appear as a
vertical "pip" of height proportional to its amplitude. In order to
compensate for the non-uniform response of the quartz to various fre-
quencies, the reaponse curves as a function of frequency of each quartz
transducer to be used as a reflector, was ascertained. A frequency

counter measures the transmitter frequency directly; the frequencies of

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the-subharmonics seen on the analyser screen may be measured by super-
imposing a "pip" from a low power oscillator, on top of the subharmonic
"pip" on the analyser screen. Then the counter measures the frequency
of the oscillator. If care is used when superimposing the two "pips",
it is possible to measure the subharmonic frequencies to better than
500 Hz.

The transmitter used to excite the quartz transducer which
drives the cavity has the facility for frequency adjustment to about
one Hz at 3 MHz, and has a short term stability of better than one Hz
at 3 MHz.

The Optical system is shown in Fig. 2.2. A helium-neon gas
laser with 1.0 milliwatt output power, in conjunction with a beam
expander, provides a 5 cm diameter beam of parallel light. The entire
interferometer assembly is oriented so that the standing wave sound
field in the cavity is perpendicular to a beam of parallel monochromatic

light. Stop S allows only that light which passes through a uniform

1

portion of the sound field to pass on to lens L A single lens reflex

1.
camera, with its ground glass screen removed, is positioned at the focal

point of lens L The image formed at the camera location is magnified

1.
and projected onto a large screen by the prism and lens L2.
The standing wave sound field in the cavity produces periodic
changes in the index of refraction in the liquid, which causes a periodic
retardation in the phase of the light as it passes through the liquid,
and results in a diffraction pattern at the camera location and on the

screen. The detailed structure of the diffraction pattern can be

observed on the screen and then recorded on film immediately by actuating

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the shutter mechanism. High magnification, necessary for visual
observation, and good resolution on the film are easily obtained.
The camera can be replaced with a photomultiplier, to measure relative

intensities of the orders in the diffraction pattern when needed.

 

CHAPTER III

PROCEDURE

To align the interferometer system, two alignment criteria
must be satisfied: 1) the two transducers must be parallel to each
other, and 2) the light beam must be perpendicular to the sound field
between the transducers. The first requirement may be accomplished
electronically by observing the rate of decay of a pulsed signal
introduced into the cavity. As is shown in Fig. 3.1, the impedance
of an Arenberg PG 650 pulsed oscillator is matched to the 2975 kHz
quartz transducer, and the transducer is driven at its resonant
frequency. An oscilloscope protrays the transmitted and reflected
pulses, and the alignment screws are adjusted for an exponential
decay envelope with a maximum in the number of visible reflected
pulses. This method produces 30 or more reflected pulses and is a
sensitive indication of correct alignment within the stated accuracy
of 0.05 degree.

The second requirement is obtained next, by removing the
reflector assembly to allow a progressive wave to interact with the
light beam. From Raman and Nath [8] theory, the diffraction pattern
from a progressive wave will be symmetric around the zeroth order,
only if the light and sound are perpendicular. This adjustment is
made visually. After replacing the reflector assembly, the first
procedure is repeated to guarantee parallelism. The pulse alignment
is repeated whenever either transducer assembly is moved along the
optical bench assembly, or removed from it for any reason.

The interferometer cavity is brought into resonance by

10

 

11

2975 kHz QUARTZ QUARTZ
TRANSDUCER REFLECTOR \ —1

 

 

 

 

 

 

TANK

 

 

 

MATCHING
CIRCUIT

 

 

 

 

OSCILLOSCOPE

 

 

 

 

PULSED
OSCILLATOR

Figure 3.1 . Electronic pulse alignment arrangement.

12

adjustment of the transmitter frequency for a maximum in the sound
intensity in the cavity, which occurs when there is a maximum in
the number of orders in the diffraction pattern. The frequency change
between adjacent maxima is a measure of fo, the fundamental cavity
resonant frequency (see Table 1.1).

When searching for subharmonic generation, the characteristic
Splitting of the diffraction pattern is used as the primary indication
of the existence of subharmonics. Detailed analysis can then be made
electronically. Spurious responses in the spectrum analyser can be
identified by their continued existence, when the subharmonics are
extinguished by slightly altering the transmitter frequency.

The frequencies and thresholds of all attainable subharmonics
were measured for various cavity configurations. A systematic study
of the effects of reflector type and thickness, fundamental frequency,
and separation of transducers was conducted. The acoustical pressures
of the fundamental and subharmonic waves at threshold were measured

for several cavity configurations using Optical methods.

CHAPTER IV

RESULTS

Analysis of the waveforms present in the interferometer with
the spectrum analyser indicates that whenever the optical diffraction
pattern splits, there are always new frequencies occurring in pairs,

such that

is“) + 158(2) =f (11.1)

where fs(1) and fs(2) are the frequencies of the pair, and f is
the fundamental frequency. Usually a single pair appears, but
occassionally several pairs appear simultaneously. In every case,

each subharmonic frequency always obeys the relation
f = n f (h.2)

where fo is the fundamental cavity resonant frequency, and n is
an integer. These results agree completely with Korpel and Adler.
Since the interferometer is also at resonance for the fundamental

frequency f, the relation
f = m f (LI-.3)

holds, where m is also an integer. Combining Eqs. h.2 and h.3, we

obtain the condition
fs = (n/m)f (h.h)

The accuracy of Eq. h.h will be discussed later. Table h.l shows
a small sample of data, illustrating Eq. h.l and h.h.

Below a certain energy density of the fundamental frequency,

13

Fundamental
Frequency (kHz)

3115.9

3130-3

~31115.2

1h

TABLE h.1

Typical subharmonic data sample

f0 = 15.0“ kHz
Reflector: 1.0 MHz quartz

Cavity length: 5.0 cm

Subharmonic
m Frequency Pairs (kHz)

207 1069.0
20h7.

208 1069.
2061.

108h.
20h6.

47H NU) UL)

209 1083.
2061.

1099.
20h5.

-41» and

.71
136

71
137
72
136
72
137

73
136

15

a given subharmonic will not appear. Slightly above this threshold,
the subharmonic signal amplitude is relatively constant. The thres-
hold level for each subharmonic depends on several factors, including
accuracy of alignment of the interferometer itself. A representative
curve is given in Fig. h.l.

Not all frequencies satisfying cavity resonance produce
subharmonics, so the frequency dependence was investigated. All sub-
harmonics with attainable thresholds were measured over a fundamental
frequency range covering 2.h MHz to 3.6 MHz. A quartz transducer with
a thickness corresponding to a 2.0 MHz resonance was used as a reflector,
and the separation distance was 5.0 cm. Figure h.2 shows the result
in terms of the integers m and n defined in Eqs. h.2 and h.3, with

each subharmonic plotted as a dot. The symmetry around a line through

Fundamental frequency: 2716.7 kHz

 

 

 

Subharmonic frequency: 935.7 KHz
Reflector: 1.0 MHz quartz
Cavity length: 5.0 cm.
-10 —
RELATIVE .
VOLTAGE _20 _
(db)
-30 _
-h0 _
L I I 47' L J
200

0 100
QUARTZ POTENTIAL (VOLTS)
Figure h.l . A typical subharmonic threshold curve.

300

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n = m/2 illustrates Eq. h.1. Two items of importance should be
pointed out. First, there is a gap around the resonant frequency

of the driven quartz within which no subharmonics occur. Second, all
of the lower frequency subharmonics of each pair are within a few
percent of 1/3 or l/h of the pump frequency.

The thresholds of each subharmonic, measured in terms of the
potential across the driven transducer, are shown in Fig. h.3. Each
curve in the figure is a cross section of Fig. h.2 for a fixed value
of n. Only a few cross sections for n values in the neighborhood
of 60 to 80 are plotted. The cross sections for n values from 1M0
to 180 will yield redundant threshold information, since all sub-
harmonics occur in pairs, but the shapes of the individual curves
will be quite different, of course.

Figures h.h and h.5 show the results of changing the reflector,
everything else held constant. A quartz with a 1.0 MHz resonance was
used at a distance of 5.0 cm from the driven quartz. Again the lack
of subharmonics around the resonant frequency of the driven quartz
is evident. However, the presence of subharmonics around f/2 is
new. Also, the subharmonics are spread over a much larger frequency
band and frequencies around f/h are absent.

With subharmonics near f/2, and even m, a single subharmonic
of frequency f/2 appears most frequently, although a pair of sub-
harmonics, both near f/2, occasionally appear. It m is odd, often
several pairs appear simultaneously, all with the same threshold.

Every subharmonic with a frequency near f/2 is present, as in Fig. h.6.

18

 

 

 

 

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Other quartz reflectors were examined in less detail.
Thicknesses corresponding to resonant frequencies of 1.2 MHz, 1.5 MHZ,
3.0 MHz, 3.5 MHz and h.0 MHz were used. All exhibit the phenomenon
of no subharmonics around the resonant frequency of the driven quartz.
Results similar to the preceding examples were obtained. although
thresholds were very high for the 3.0 MHz reflector and subharmonics
only occurred in a narrow frequency region just above 3 MHz. Figure
h.7 summarizes this survey, showing all regions where subharmonics

were obtained in terms of the driving frequency. These ”bands"

 

appear to lie in regions around 1/2, 1/3, and l/h of the pump

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m-3 m-l m+l m+
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RELATIVE fl
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0 .5f f
'l-aa...l__l__1_.1..l .1
I I f
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Figure h.7 . Bands of observed subharmonic frequencies

in terms of the driving frequency f.

22

frequency, although a more extensive survey might indicate otherwise.
Investigations into the range of cavity lengths for which
subharmonics will occur reveals that at small distances (2 cm and
less), it becomes more difficult to produce subharmonics, as the
thresholds become very high. The same thing happens with distances
Of 8 to 10 cm and more. For any particular reflector, the curve will
be different, but in general, A to 6 cm produced minimum thresholds,
everything else held constant. Hence, most work was done around this
region.
All of the air-backed quartz reflectors introduce a fre-
quency dependent phase shift upon a wave during reflection. A
reflector with a constant phase shift was constructed in an attempt
to determine what effect phase changes have on the system. A brass
cylinder was constructed with saw cuts angled into the rear surface
to scatter any sound reaching it, as in Fig. h.8. Treating the reflec-
tion as a single boundary infinite medium problem, the ratio of reflec-
ted to incident sound intensity is 0.8h7. The angle of the cuts is

determined to minimize reflections [9]. A search was conducted at

- _ — I

I

3.3 cm

-I-

“*i 2.5 cm:‘—'

 

Figure h.8 . Brass reflector.

23

frequencies from 2.5 MHz to 3.5 MHz and with cavity lengths of 3.0 cm
to 9.0 cm. No subharmonics were obtained under any circumstances,
even with a potential of 600 volts or more across the driven quartz
transducer.

The subharmonics are extremely sensitive to small changes
in frequency around the resonance of the interferometer. Changes of
less than 10 Hz in the driving frequency will extinguish a subharmonic
for an energy density slightly above threshold. Considerably above
threshold level, changes of 5 to 10 Hz will often "coax" the system
to jump to an adjacent mode, i.e. the subharmonic n/m will extinguish
and be replaced by another, say (n + l)/m. Occasionally, when well
above threshold, two or three subharmonics with nearly the same
frequency may exist simultaneously (each has a corresponding sub-
harmonic obeying Eq. h.l, of course). In Fig. 1.9, a typical cavity
resonance illustrates the presence Of subharmonics around flu and

f/3, somewhat above the minimum thresholds.

NUMBER OF
DIFFRACTION
ORDERS 5 - 30 kHz

 

    

50 - 100 Hz

  

 

\v
V

 

I \
FREQUENCY

Figure h.9 . Typical cavity resonance, showing
regions for which subharmonics occur.

2h

Heating effects in the cavity often interfere with subharmonic
generation; drifting of the cavity length will necessitate continual
corrections to be made in the fundamental frequency if subharmonics
are to be maintained. This problem is more pronounced under experi-
mental conditions where thresholds are high. Hence, most work was
done using conditions where subharmonic thresholds are moderate.

Previously, it was stated that parallelism of the transducers
comprising the interferometer is confined to about 0.05 degree. To
this accuracy, checks were made of the effect of misalignment on the
system. Gross misalignment, of the order of 0.5 degree, is enough
to extinguish all subharmonics under any conditions. However, within
the range of 0.5 and 0.05 degree the only significant change observed
is to raise the threshold curves upward in Fig. h.3 and Fig. h.5.

The general features remain about the same. This characteristic
gives a double check on the pulse method of alignment.

The relative sound pressure in the cavity as a function of
frequency, due to the fundamental frequency standing wave, was
measured optically, using a photomultiplier to measure the intensity
of the zeroth order in the diffraction pattern (see Appendix). Figure
h.10 shows the result for a 1.0 MHz quartz reflector and a cavity
length of 5.0 cm. The curve is essentially discrete, since very
little sound pressure is produced unless the cavity is excited at a
multiple of f0. Also, the energy density of the fundamental wave is
well below the subharmonic threshold level. A response curve for the
driven quartz transducer (taken under progressive wave conditions) is

superimposed on the same figure for comparison purposes later.

25

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The magnitudes of the fundamental and subharmonic pressures
in the cavity at the threshold energy density were determined by a
combined optical-electronic method. The fundamental pressure is
determined optically by measuring the light intensity of the zeroth
order in the diffraction pattern on the low side of threshold. The
subharmonic pressure is measured on the spectrum analyser just above
threshold, after calibration of the analyser by a similar optical
technique. Assuming the subharmonic depletes a negligible amount of
energy from the fundamental wave, a pressure comparison can be made.

Under typical conditions, the peak pressure of the funda-
mental wave at threshold is around 1.5 atmosphere, corresponding to a
peak density variation of Ap/p = 6.h x 10.5 . The subharmonic pres-
sures are typically about one tenth of the fundamental pressure; the
assumption is therefore valid, since the energy is only 10-2 times
that of the fundamental wave.

In Fig. h.ll, several photographs display diffraction
patterns resulting from the presence of subharmonics in the cavity.
The first three illustrate subharmonic generation at frequencies
near 1/2, 1/3, and l/h of the pump frequency; in the fourth photo-
graph, the subharmonics deviate sufficiently from integral sub-
multiples of the pump frequency, so that the detailed substructure
is visible.

Equation A.l6 predicts multiple diffraction. That is, each
diffraction order due to one wave will, in turn, be re-diffracted by
the other waves. That this is indeed the case, can be seen in Fig.

4.11, where diffraction orders of the fundamental and the lower

27

n7... FREQ. (kHz)

 

 

 

 

 

 

 

 

 

 

 

 

 

31.... IN“ I J I J
.1198 . . I .
1559.8 summomc ' F l l '
-2 -1 O 1 2
...... 1 _ l f l 1 I
.3118 . ”', ‘ It
11311.7 -2r I (I3 1| 2'
3192.6 I I I I J
.252 _ ., l? . '
8011.2; 13 I21 (I) l1 «1» '3
3193.9 L I I I 1
.29h . A“ A I. A "In .-.
937.8 1414‘ 12-11 I I 21 3: '1 j
0

Figure h.ll . Diffraction pattern photographs obtained
during subharmonic generation.

28

frequency subharmonic wave are marked above and below each photograph,
respectively. Only the subharmonic orders diffracted from one funda-
mental order are marked to avoid confusion in the figure. The photo-
graphs also illustrate that the intensities of the subharmonic orders
are nearly equal to the fundamental intensities. This is to be expect-
ed, even though the subharmonic pressures are only one-tenth of the
fundamental pressure, since both pressure values are large enough

to lie on the relatively flat portions of the curves in Fig. A.l.

CHAPTER V

DISCUSSION

At the present time, a theory to explain the experimental
observations made in this study is not available. Difficulties arise
when attempts are made to solve the equations of motion for a fluid
[10] with standing waves, unless the inherent nonlinearity is neglected.
However, a qualitative understanding of a few aspects of the system
may be obtained.

A standing wave in an acoustic interferometer is subject to

several loss mechanisms, including absorption of the wave by the liquid,

 

spreading of the sound field, and nonlinear effects which couple energy
into other frequencies. These losses are just cancelled out for the
fundamental frequency wave by the energy supplied by the quartz trans-
ducer at one end of the cavity. However, for a subharmonic wave in
this cavity, the loss can only be made up by a coupling of energy at
other frequencies into the subharmonic frequency by some nonlinear
mechanism. Since all energy initially came from the fundamental fre-
quency, there will be a certain level of energy at the fundamental
frequency for which the loss at the subharmonic frequency is just
cancelled. Above this energy level, the subharmonic wave will grow
to an intensity for which the gain-loss balance is regained. Hence,
a threshold curve similar to Fig. h.l is expected. Also, unless the
subharmonic wave obeys Eq. n.2, it will not be at a cavity resonance
frequency, causing additional losses and raising the threshold to a
very high energy level.

High thresholds at large separation distances are most likely
due to the appreciable beam spread and consequent loss of energy in

the system. Absorption losses in water are small, being around

29

 

30

0.02 db/cm, and therefore will not contribute significantly. At

small distances, high thresholds may be due to less coupling between
the two waves because of the small distances involved, although the
losses should also become much smaller. The "near field" of the
transducer may also influence the thresholds, due to the inhomogeneity
in the sound field near the transducers. In any case, the liquid
appears to be the nonlinear mechanism, rather than the transducers,

or the transducer-liquid interface, although more evidence certainly
is needed.

The relative phase changes of fundamental and subharmonic
waves upon reflection appear to strongly influence which n/m ratios
are permitted to exist in the cavity and in what their thresholds will
be. The brass reflector may be treated as a single boundary problem,
since the rear surface will scatter most of the sound reaching it.

The ratio of reflected to incident amplitude, denoted the reflection
coefficient, is
p2V2
R = WI (5.1)
szz
p1"1

where p1 and v are the density and sound velocity respectively in

l

the incident material, and p2 and v refer to the same quantities

2
in the reflecting material. Similarly, a water-quartz-air combination,

treated as a two boundary problem with the pc of air assumed to be

negligibly small, yields the complex reflection coefficient

31

pzcz
(

P1°1

l

 

R (k) = exp [ -21 tan- tan k d)] (5.2)
where the subscripts 1 and 2 refer to water and quartz, respecé-
tively, k is the wave constant for longitudinal waves in the
x-direction in quartz, and d is the quartz thickness. It is

important to note that Eq. 5.2 exhibits a frequency dependent phase I...
shift, whereas Eq. 5.1 does not. The complete absence of subharmonic
generation using a brass reflector seems to indicate that a frequency
dependent phase shift is a requirement for subharmonic generation.
Additional evidence is provided by Fig. n.9, where we see that slight
frequency deviations from cavity resonance may allow a subharmonic to
exist by satisfying certain phase requirements, even though the funda-
mental frequency is not quite in resonance in the cavity.

The above discussion implies that Eq. h.h does not hold
strictly, since m and n are not exactly integers. However, the
deviation in m is very small, as can be seen from Fig. n.9, being
less than one part in 100. The frequency measurements leading to
Eq. h.2 indicate that n is an integer to about one part in 10, which
will completely mask the deviations in m, so Eq. h.h is correct to
within about one part in 10.

The dip in the center of the cavity response curve in Fig. h.10
is a common feature in coupled oscillator problems in many fields. The
lack of subharmonics around the driven quartz transducer's resonant
frequency is at least partially due to this phenomenon. Due to the

low sound intensity in this region, even for high potential across the

32

driven quartz transducer, it was not possible to reach an energy
density sufficient to produce subharmonics. However, the possibility
of a feature inherent in the system which prevents subharmonic gener-
ation at the resonant frequency of the driven quartz should not be

eliminated.

.’

APPENDIX

Ultrasonic Light Diffraction Theory

The diffraction of light by ultrasonic waves is a convenient
method of determining much about the ultrasonic wave and the medium
through which it travels. Analysis of the diffraction pattern pro-
duced by the sound field in an acoustic Fabry-Perot interferometer
yields information about the frequencies and pressures of the various
waves that are present.

The sound field produces periodic changes in the index of
refraction of the medium around its undisturbed value when the sound
is not present. A plane wave of light, impinging upon this distur-
bance, will be in some way altered. For moderate frequencies, sound
intensity, and path length of the light through the sound beam, the
light will be modulated in phase with a periodicity of the sound
field when it emerges on the other side of the sound beam. The
phase modulated light can be resolved into a series of plane waves,
each traveling in a slightly different direction, which will form
a diffraction pattern if focused on a screen.

From the diffraction integral, it is known that the amp-
litude distribution of the diffracted light due to a phase grating
is given by

D
A(9) = C eiwtfexp { i [ 21“}- sin 9 + v(x,t)]} dx, (A.l)

-D

where e is the angle of deviation of the light,(n is the angular

frequency of the light, A is the wavelength of the light, 2D is

33

311

the width of the light beam, and C is a normalization constant.
The parameter v(x,t) is a dimensionless parameter describing the
change of the index Of refraction, and is related to the acoustical

pressure p(x,t) by

v<x.t> = 31% p(x,t) . (1.2)

where L is the width of the sound beam, and K is the piezo-
optic coefficient of the medium.
For a single stationary wave in the cavity, the acoustical

pressure may be written as
* * . * *
p(x,t) = p0 [sinfiu t - k=x) + sin.fin t + k x)] , (A.3)

where wf is the angular frequency of the sound, and k* is the
wave constant of the sound.

Solving Eq. A.l, leads to a discrete light distribution
with light occurring only at angles

n A
sin 9n = * : (A'h)

A

 

where n is an integer. The time averaged light intensities are

given by
+oo
- 2 2
I..- E Jr (v0) JM (v0) . (15>
r=-oo
where Jr is the rth order Bessel Function, and v0 = §£k%_§ po .

 

35

Equation A.h is more general than the above indicates, and holds for
any periodic sound field provided the light impinges normally.

Figure A.l shows how the first few diffraction orders vary
in intensity with pressure [11]. Note that if I; is measured experi-
mentally, and the piezo-optic coefficient is known, the acoustic
pressure may be determined.

When subharmonics are present, Eq. A.3 must be generalized.

For the common case of only two subharmonics with

f (1) + f (2) = f
s s
we may write

* * *

p(x,t) = E pj[sin(mj + - kJ x) + sinQfi: t + kj x)], (A.6)
j=O

where the subscripts 0, 1, and 2 refer to the fundamental, sub-

harmonic fa(1), and subharmonic f (2), respectively. Combining
a s

Eqs. A.6 and A.2, results in

2 ‘x7_1
v(x,t) = E vj [ sin aj + sin Bj] (A.7)
J=O'
where
2n LrK
J ' A j ’
* *
(1 = <0. t — k x ,
J J J
and
* *

.o>o3 unwcowuwum a you Amuovuo aofluomuMMHo
Bow umuww osu OH mofiufimcmucw unwafi owouo>s mo sumac . H.< mesmHh

 

 

36

 

 

'H

 

 

 

 

 

0.0

H.0

N.0

m.o

:6

m.o

H

II—I

37

Using the identity

Exp [iha sin C] = E Jr (a) exp[i r O], (A.8)

r=-°o

where Jr is the rth order Bessel Function, and inserting Eq. A.7

into Eq. A.1, we obtain

+00 +00
A(e) = C e100t Z Z Jm(vo) Jfi(vo) Jq(v1) Jr(v1) J8(v2) Ju(v2)

m: n: q r: 3:“
=-m =-oo

D/2
X 61:1) [i ( 2:11

-D/2

 

sin 9 + m as + n 80 + qai + r81 + 80% + u82)] dx .

(A-9)

Evaluating the integral in Eq. A.9, we obtain

+00 +00
11(9) = c em}: ZJm(vo) Jn(vo) ngl) Jr(v1) JSIV2) Ju(v2)

m) n) q r: 3) u
=-00 =-m

2 sin{[§£ sin e + k: (n-m) + k: (r-q) + k; (11-3)] D}

X

 

»2fl . 'X- * * ‘
A sin 6 + k0 (n-m) + k1 (r-q) + k2 (u-s).

X exp {it [(1): (m+n) + a): (q+r) +5; (s+u)]} . (A.10)

38

1 : V2 = O, and A(e) = 8(9).

Using this to evaluate the normalization constant, we obtain

With no sound present, v0 = v

C = —— . (A.11)

If the quantities j = m~- n, k = q - r, .2 = s - u, and

'X'
sin[(gflsine-ij-kki-Xk2)D]

 

3k! —
2n * * *
[ A sin 9 j ko k k1 -.2 k2 ] D

are defined, Eq. A.10 reduces to

+m

Me) = E Jm(vo> J.” (vonq (v1) Jq_k (v1) 13 (v2) 1“ (v2)
J9m)k:Q:

1 9 s='m

ijk‘ exp {ithi—w: (2m- j) +039]? (Zq - k) +u): (28 ‘2) I} .

'- (A.12)
For the limiting case as D approaches infinity, ij1 1G 0 only
if

2n . * * *
A sin 6 - Jko - kk1 -‘R k2 — 0

Rewriting Eq. A.13 as

sin 9 = -A:- j + -A; k + —;;.2 ,
"o "1 >‘2

and using the fact that

1 l
— —_¥’ + *
A A

' l-‘
.nl

39

finally gives the relation

 

sin 6"k' = -L; j' + 5* k' , (A.1h)
J KO >"1

where j' = j +32 and k' = k.-.2 . The amplitudes of the discrete

orders are

Ajkl := 2E:: Jm(vo) Jm-j(vo) Jq(vl) Jq-k(vl) Js(v2) J8-1(v2)

m) (I) s
=-m

X exp {1th. + 1): (2m - 1') + w: (2q-k) + w: (2s -.¢ )1] . (11.15)

The time averaged light intensities are

.fj-kl = EJ:(VO) “Ti-j (v0) 2 J:(v1) Ji-k(vl)
m=-°o qzooo
X E 1:072) 1:162) . (A. 16)

Equations A.1h and A.16 describe the Fraunhofer diffraction
pattern which results from the interference of light with a funda-
mental and two subharmonic standing waves which obey Eq. h.1. Equation
A.16 is true only under a phase grating assumption. This assumption

is not entirely valid for the acoustical pressures observed in the

cavity. Therefore, intensities calculated with Eq. A.16 are only
approximate. For a detailed discussion of the validity of the

phase grating assumption, refer to Klein and Cook [12].

[1]

[2]

[3]

[It]

[6]

I7]

[9]

[10]

[11]

[12)

BIBLIOGRAPHY

Lord Rayleigh, The Theory of Sound, Volume l, London,
Macmillan and CO., 1877, pp.78-85.

 

 

J. J. Stoker, Nonlinear Vibrations, Interscience Publishers,
Inc., N. Y., 1950.

C. Hayashi, Nonlinear Oscillations 12_Physical Systems, McGraw-
Hill, N. Y., 196A.

 

M. Kuljis, "Experimental Study of Acoustic Parametric Open
Cavity Array-Amplifier", Electrotehnika, l, 1965, pp.3-12.

A. Korpel and R. Adler, "Parametric Phenomena Observed on
Ultrasonic Waves in Water", Appl. Phys. Lett., 1, 15 August
1965, pp. 106-1080

W. R. McCluney, "An Investigation of Subharmonic Generation
in an Ultrasonic Resonant Cavity", M.S. Thesis, University of
Tennessee, 1966. (Also published as Technical Report No. 2,
AD 6hl 60h, Office of Naval Research.)

M. A. Breazeale and W. R. McCluney, "Subharmonic Generation
in an Ultrasonic Resonant Cavity", J. Acoust. Soc. Am. £0,
November, 1966, p. 1262.

C. V. Raman and N. S. N. Nath, "The Diffraction of Light by
High Frequency Sound Waves: Part II", Proc. Indian Acad.
Sci., 2, 1935, h13-h20.

W. G. Mayer, "Reflection and Refraction of Mechanical Waves
at Solid-Liquid Boundaries. II", J. Appl. Phys. 33, November,
1963, pp. 3286-3288.

W. Keck and R. T. Beyer, "Frequency Spectrum of Finite Amplitude
Ultrasonic Waves in Liquids", Phys. Fluids, 3, May-June, 1960,

pp. 3h6-3h7.
B. D. Cook and E. A. Hiedemann, "Diffraction of Light by
Ultrasonic Waves of Various Standing Wave Ratios", J. Acoust.

Soc. Am., 33, July, 1961, pp. 9h5-9h8.

W. R. Klein and B. D. Cook, to be published in IEEE Trans.
on Sonics and Ultrasonics.

hl

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