Steven_Yang Thesis

NAVAL
POSTGRADUATE
SCHOOL
MONTEREY, CALIFORNIA
THESIS
Further dissemination only as directed by President, Code
261 (September 2012) Naval Postgraduate School, Monterey,
CA 93943–5000, or DoD authority.
EXPERIMENTAL VERIFICATION OF THE ACOUSTIC
RADIATION FORCE
by
Steve Yang
September 2012
Thesis Advisor: B. Denardo
Second Reader: G. Karunasiri

THIS PAGE INTENTIONALLY LEFT BLANK

i
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Acoustic Radiation Force
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6. AUTHOR(S) Steve Yang
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Naval Postgraduate School
Monterey, CA 93943–5000
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13. ABSTRACT
A radiation force is the time-averaged force due to waves on a body. The objective is
to experimentally test the theoretically predicted acoustic radiation force on a body
that is small compared to the wavelength of the sound. Because the effect is nonlinear,
the amplitude of the sound must be sufficiently large for the force to be significant.
Applications include the use of high-intensity ultrasound to separate unwanted
particles from a liquid. The experiment consists of measuring the acoustic radiation
force on a solid aluminum ball that lies along the axis of symmetry of a high-amplitude
loudspeaker. The experiment is conducted in a walk-in anechoic chamber, so that only
traveling waves occur symmetrically about the axis of the loudspeaker. The distance
between the loudspeaker and the ball is varied. Experimental data are gathered and
compared to the theoretical prediction, which is based on pressure and velocity
measurements with an acoustic intensity probe. Approximate agreement between theory and
experiment occurs if an account is made of the outward jetting or “wind” from the
loudspeaker.
14. SUBJECT TERMS
Acoustic Radiation Force, Nonlinear Acoustics
15. NUMBER OF
PAGES
83
16. PRICE CODE
17. SECURITY
CLASSIFICATION OF
REPORT
Unclassified
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CLASSIFICATION OF THIS
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ABSTRACT
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ABSTRACT
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ii

iii
Further dissemination only as directed by President, Code
261 (September 2012) Naval Postgraduate School, Monterey,
CA 93943–5000, or DoD authority.
EXPERIMENTAL VERIFICATION OF THE ACOUSTIC RADIATION FORCE
Steve Yang
Lieutenant, United States Navy
B.S., Oregon State University, 2006
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE IN ENGINEERING ACOUSTICS
from the
NAVAL POSTGRADUATE SCHOOL
September 2012
Author: Steve Yang
Approved by: Bruce Denardo
Thesis Advisor
Gamani Karunasiri
Second Reader
Daphne Kapolka
Chair, Engineering Acoustics Academic
Committee

iv

v
ABSTRACT
A radiation force is the time-averaged force due to waves
on a body. The objective is to experimentally test the
theoretically predicted acoustic radiation force on a body
that is small compared to the wavelength of the sound.
Because the effect is nonlinear, the amplitude of the sound
must be sufficiently large for the force to be significant.
Applications include the use of high-intensity ultrasound
to separate unwanted particles from a liquid. The
experiment consists of measuring the acoustic radiation
force on a solid aluminum ball that lies along the axis of
symmetry of a high-amplitude loudspeaker. The experiment is
conducted in a walk-in anechoic chamber, so that only
traveling waves occur symmetrically about the axis of the
loudspeaker. The distance between the loudspeaker and the
ball is varied. Experimental data are gathered and compared
to the theoretical prediction, which is based on pressure
and velocity measurements with an acoustic intensity probe.
Approximate agreement between theory and experiment occurs
if an account is made of the outward jetting or “wind” from
the loudspeaker.

vi

vii
TABLE OF CONTENTS
I.
INTRODUCTION............................................ 1

A.
ACOUSTIC RADIATION FORCE AND APPLICATIONS.......... 1

B.
PREVIOUS WORK...................................... 1

II.
THEORY.................................................. 5

A.
ACOUSTIC RADIATION FORCE........................... 5

B.
FORCE CAUSED BY A SPHERICAL WAVE ON A BODY......... 6

III.
DEMONSTRATION........................................... 9

A.
UNBAFFLED DRIVER.................................. 11

B.
BAFFLED DRIVER.................................... 12

IV.
EXPERIMENTAL APPARATUS................................. 17

A.
ANECHOIC CHAMBER.................................. 17

B.
DRIVER ASSEMBLY................................... 20

C.
ANALYTICAL BALANCE................................ 21

D.
NEXUS CONDITIONER AND INTENSITY PROBE............. 23

E.
ELECTRONIC DATA ACQUISITION EQUIPMENT AND SETUP... 25

F.
ALUMINUM ROD AND INTENSITY PROBE CLAMP............ 29

G.
CHAIN LINK SPACERS AND BUSS WIRE.................. 31

H.
KANOMAX VANE ANEMOMETER........................... 32

I.
KANOMAX HOTWIRE ANEMOMETER........................ 33

V.
EXPERIMENT............................................. 35

A.
PRESSURE MEASUREMENTS............................. 35

B.
MASS MEASUREMENTS................................. 39

C.
DATA INTERPRETATION............................... 41

VI.
CONCLUSION............................................. 53

A.
AXIS OF SYMMETRY.................................. 53

B.
REFLECTIONS....................................... 54

C.
FUTURE WORK....................................... 57

APPENDIX.................................................... 59

A.
VOLTAGE MEASUREMENTS WITH THE INTENSITY PROBE..... 59

B.
KINETIC AND POTENTIAL ENERGY DENSITIES VS.
DISTANCE.......................................... 60

C.
MASS DATA FROM THE ANALYTICAL BALANCE............. 61

D.
HOTWIRE AND VANE ANEMOMETER READINGS.............. 61

E.
MASS READINGS WITH CELLOPHANE SHIELD.............. 62

F.
ROUGH MASS READINGS AT FURTHER DISTANCES.......... 62

G.
MATLAB CODE....................................... 62

LIST OF REFERENCES.......................................... 68

INITIAL DISTRIBUTION LIST................................... 70

viii

ix
LIST OF FIGURES
Figure 1.
Demonstration setup. ........................... 10

Figure 2.
Unbaffled and baffled driver. .................. 13

Figure 3.
Volume determination with glass spheres. ....... 14

Figure 4.
Data acquisition equipment for demonstration. .. 16

Figure 5.
Anechoic chamber. .............................. 17

Figure 6.
Fiberglass wedges inside the anechoic chamber. . 18

Figure 7.
Control room. .................................. 19

Figure 8.
15-inch diameter electrovoice driver and
enclosure volume. .............................. 20

Figure 9.
Analytical balance. ............................ 22

Figure 10.
Analytical balance setup. ...................... 22

Figure 11.
Nexus conditioner and 2-channel intensity
probe. ......................................... 24

Figure 12.
250 Hz pistonphone. ............................ 24

Figure 13.
Data acquisition equipment in control room. .... 28

Figure 14.
ACO reference microphone. ...................... 29

Figure 15.
Clamp and 1 cm incremented aluminum rod. ....... 30

Figure 16.
Chain links. ................................... 31

Figure 17.
Kanomax Vane anemometer. ....................... 32

Figure 18.
Vane-Anemometer set up to measure jetting from
the loudspeaker. ............................... 33

Figure 19.
Hotwire anemometer. ............................ 34

Figure 20.
Curvature of the intensity probe. .............. 36

Figure 21.
Relevant distances associated with the
intensity probe with d = 8.5 mm spacing. ....... 38

Figure 22.
Sixth-Order Polynomial Curve Fits of the
Kinetic and Potential Energy Densities as a
function of distance from the loudspeaker. ..... 39

Figure 23.
Theoretical acoustic radiation force curve
plotted with mass data. ........................ 41

Figure 24.
Flame test close and far away from the dome. ... 44

Figure 25.
Flame test off axis and moving the flame from
far away to close in. .......................... 45

Figure 26.
Theoretical acoustic radiation force curve
plotted along with mass data and shielded mass
data. .......................................... 46

Figure 27. Cellophane shield and setup...................... 47

Figure 28.
Pressure amplitude as a function of distance
sketch for a spherical wave. ................... 49

Figure 29.
Graph of time-average kinetic energy density
vs. distance from the loudspeaker dome for a
spherical wave. ................................ 51

Figure 30.
Rectangular absorbing material. ................ 54

x
Figure 31.
Wooden platform supporting the balance and
assortment of aluminum bars. ................... 56

Figure 32.
Fire suppression system in the anechoic
chamber. ....................................... 57

xi
ACKNOWLEDGMENTS
First and foremost, I would like to thank Valerie for
all her support in assisting me through my academics at the
Naval Postgraduate School. Her understanding and endless
love were crucial to my success.
I wish to express my sincere gratitude to Dr. Bruce
Denardo. His vast knowledge and guidance made this research
possible. He has taught me about the difficulties and
complexities of experimental research. His invaluable
mentorship will never be forgotten.

xii

1
I. INTRODUCTION
A. ACOUSTIC RADIATION FORCE AND APPLICATIONS
The acoustic radiation force is a time-averaged force
exerted on a particle by a sound wave. The radiation force
can either be attractive or repulsive depending on the
object’s distance from the acoustic source, wavelength, and
size of the object.
The acoustic attractive portion of the radiation force
due to a traveling wave is thoroughly investigated in this
thesis. The objective is to experimentally verify the
attractive nature of the acoustic radiation force.
There are many applications of the acoustic radiation
force in biology, astrophysics, medicine, and particle
separation. The particle separation research has
significant naval relevance because it is the basis of a
new possible means to remove undesired particles in oil and
to separate oil and water by using ultrasonic methods. The
current method of oil filtration is a centrifuge system
that is antiquated, requires constant attention and is
loud, which is detrimental to stealth on board U.S. Navy
submarines. An acoustic method of filtration would be able
to alleviate many of the previously stated issues, most
notably, loud metallic noises.
B. PREVIOUS WORK
L.V. King (1934) first calculated expressions for the
acoustic radiation force by an incident traveling plane
wave and a standing wave. He used spheres and relatively
long wavelengths in his experiments to test the theory of

2
the acoustic radiation force. T.F.W. Embleton (1951)
continued King’s work by developing a theory and
experimentally testing the acoustic radiation force for
spherical waves by using pendulum deflections. Embleton’s
breakthrough was finding the inverse fifth power
relationship of the force at short distances from a source.
In 1967, W.L. Nyborg (1967) derived a general form for the
acoustic radiation force by assuming the object is located
in the axis of symmetry from the sound field. The result of
his findings is that the force is the difference of the
gradient of the kinetic and potential energy densities,
with a dimensionless factor of the kinetic energy involving
the relative densities of the body and fluid.
Research into various aspects of the acoustic
radiation force has been on-going at the Naval Postgraduate
School since 2004. Stanley Freemyers (2004) attempted to
measure the acoustic radiation force. He used a 15-inch
diameter loudspeaker and varied the voltage amplitude from
5 volts to 40 volts and the frequency from 50 Hz to 200 Hz.
Freemyers’ experimental results showed many deviations from
theory and experimental data; nevertheless, there was very
rough agreement. Spherical wave theory was used, although
this was a poor fit of the sound field.
Michael Schock and Scott Sundem (2005) furthered
Freemyers’ research by observing a significant difference
in an unshielded and shielded acoustic radiation force. A
cellophane “wind shield” was used to enclose the aluminum
sphere to check the acoustic transparency. Their data
indicated an attractive acoustic force on the air.

3
Mario Bentivoglio and James Rochelle (2009)
extensively tested many loudspeaker drivers for acoustic
radiation force measurements. They found the Electrovoice
EVX-155 was best suited for quantitative testing. The
driver was very stable at high amplitudes. Bentivoglio and
Rochelle developed and used a local spherical wave
approximation but had conflicting results between their
experimental data and theory.
Eric Oviatt and Konstantinos Patsiaouras (2009) forced
a spherical wave by placing a 4 foot circular wooden baffle
with a 2.75-inch hole in the center on top of the driver.
They observed strong jetting from the loudspeaker, which
produced an upward drag on the aluminum sphere ball.
Justin Ivanic and Mohamed Akram Zrafi (2011) continued
the work on experimentally verifying the acoustic radiation
force with theory. They used an acoustic intensity probe to
measure the acoustic particle velocity. They found a
systematic error that has not been resolved. Some of the
suspected sources were air currents in the anechoic
chamber, errors in theoretical assumptions, errors in the
theoretical calculation curve, standing wave and
reflections in anechoic chamber and issues with the
accuracy of the intensity probe.
There have been some breakthroughs at the Naval
Postgraduate School but also a number of systematic errors.
This current research is the first to yield reliable
measurements of the acoustic radiation force and to probe
the systematic errors.
The goal of this research is publishable measurements
and predicted values of the acoustic radiation force. The

4
force is measured on an axis of symmetry of the high-
amplitude loudspeaker and measurements of the force on a
small sphere as a function of distance from the source. The
experimental data taken is compared to the theoretical
predictions to determine the extent to which the two agree.
The experiment is be done in the anechoic chamber in
the basement of Spanagel Hall at the Naval Postgraduate
School. The analysis is done in two stages. The first stage
is the experimental calculation of the gradients of kinetic
and potential energies using an intensity probe and a high
amplitude 100 Hz loudspeaker at different ranges. The data
gathered are used to calculate the theoretical radiation
force curve. The second stage involves suspending a solid
aluminum ball suspended from a highly precise analytical
balance at different distances from the loudspeaker. The
acoustic radiation force is measured directly. An analysis
is done comparing the outcome of the two experiments.

5
II. THEORY
A. ACOUSTIC RADIATION FORCE
Wesley Nyborg (1967) found an expression relating the
acoustic radiation force per unit volume with the gradients
of the potential and kinetic energy densities. The equation
assumes a sphere in an acoustic field along an axis of
symmetry (z-axis), where the sphere is small compared to
the wavelength:
(2.A.1)
where is the acoustic radiation force per unit volume,
is the time averaged kinetic energy density, is
the time averaged potential energy density, is the
density of the sphere, is the ambient density of the
fluid and unit vector is along the z axis from the
source. The time averaged kinetic and potential energy
densities, with as particle velocity, as the speed of
sound in the medium and as acoustic pressure are,
(2.A.2)
f
!"
rad = 3
1!
"o
"
2 +
"o
"
#
$
%
%
%
&
'
(
(
(
)
)z
*ek + !
)
)z
*ep +
,
-
.
.
.
.
/
0
1
1
1
1
z#
f
!"
rad
!ek " !ep "
!
!o
z!
u c
p
!ek " =
1
2
#o !u2
"

6
(2.A.3)
From Nyborg’s expression (2.A.1), the density of air
(1.2 kg/m3
) is much less than the density of aluminum (2800
kg/m3
) so the approximation ρo/ρAl ≈ 0 can be made in the
equation. The acoustic radiation force equation then
reduces to
(2.A.4)
This equation will be used to test the theory
experimentally.
B. FORCE CAUSED BY A SPHERICAL WAVE ON A BODY
We now consider the acoustic radiation force in the
special case of a traveling spherical wave. Kinsler, Frey,
Coppens and Sanders (2000) express the linear acoustic
pressure and particle velocity of a spherical wave as:
(2.B.1)
(2.B.2)
ep =
1
2!oc2
p2
.
f
!"
rad =
3
2
!
"#
$
%&
'
'z
ek (
'
'z
ep
)
*
+
,
-
.z#.
p =
A
r
cos(!t " kr)
u
!
=
A
!ocr
cos("t # kr)+
1
kr
sin("t # kr)
$
%&
'
()r",

7
where is a constant amplitude, is the frequency and
is the distance from the center of the source.
Substituting equations (2.A.4) and (2.A.5) into
equations (2.A.2) and (2.A.3) yields
(2.B.3)
(2.B.4)
Substituting the kinetic and potential energy
equations (2.A.6) and (2.A.7) respectively into the
acoustic radiation force equation (2.A.1) results in
(2.B.5)
For a spherical wave near a small source kr<<1, the
kinetic energy term then dominates over the potential
energy in equation (2.A.8). The radiation force from a
traveling spherical wave on a small sphere is proportional
to 1/r5
.
A ! r
!ek " =
A2
4#oc2
r2
1+
1
kr( )2
$
%
&
&
'
(
)
)
ep =
A2
4!oc2
r2
.
f
!"
rad = !
3A2
k3
2"oc2
(kr)5
r#.

8

9
III. DEMONSTRATION
The attractive acoustic radiation force due to a
diverging traveling acoustic wave can be observed using a
commercial off-the-shelf loudspeaker and a suspended
styrofoam ball (Fig. 1). The driver is a JL Audio 6W3v3–8,
which is rated for 150 Watts and has a nominal impedance of
8 ohms. The JL speaker is capable of handling 35 Vrms
according to the equation
2
( )
P= .rmsV
R
(3.1)
The driver is enclosed in a 25 cm x 13 cm x 18 cm
sealed wooden box to improve speaker performance. The
loudspeaker and power amplifier are set to run up to 35
Vrms. The function generator (Fig. 4) is fixed at a
frequency of 100 Hz, which equates to a wavelength of 3.4
m. The styrofoam ball has diameter 3.81 cm (1.5 in) and
mass 0.78 grams. The length of the string that is
suspending the ball is 1.0 m. The ball is suspended fairly
high to reduce the effect of the table on the sound field
and so that the deviation from equilibrium can be clearly
seen. The acoustic radiation force attracts the ball
towards the center of the dome as the voltage is increased
from zero to the maximum voltage. The distance from the
styrofoam ball to the center of the inverted dome is varied
to see how quickly the force falls off with distance.

10
Jetting (Faber, 1995) occurs due to an asymmetry of
pushing a fluid through an orifice as opposed to sucking.
The boundary layer typically quickly separates in the first
case but not in the second. The resultant flow for a
loudspeaker is thus an outward jet. The jetting is more
apparent with a baffled driver and clearly seen using a lit
match or the styrofoam ball. The flame from the lit match
indicates the force due to the jetting. The flame is a
reasonable test to approximate the magnitude of flow at
different distances away from the driver.
Figure 1. Demonstration setup.

11
A. UNBAFFLED DRIVER
The center of the ball is located 10 cm from the
middle of the speaker centered on the inverted dome dust
cover. The ball is suspended close to the driver to
visually see the effects of the acoustic radiation force.
As the voltage is increased from 0 to 35 Vrms, a small
deviation from equilibrium is clearly seen at 15 Vrms and
the styrofoam ball is approximately 7.5 cm from the dome.
As the voltage is increased even further, the ball is
attracted towards the center of the loudspeaker. At 28 Vrms,
there is sufficient force and instability that styrofoam
ball hits the center of the driver which knocks the ball
away. Beyond 10 Vrms, the nonlinearity causes a larger
change in the distance from equilibrium. The acoustic
radiation force is an example of a nonlinear acoustics
effect. The force rapidly decreases as 1/r5
from the source
for a spherical wave.
An unbaffled driver (Fig. 2) forces fluid through the
6-inch diameter speaker so the jetting effects are small.
If a lit match is held a few centimeters away from the
driver at 28 Vrms, the flame is forced outward due to the
loudspeaker’s jetting. Most of the time, the flame is
extinguished by jetting at 2–3 cm from the dome. The
effects of jetting at 10 cm away from the center of the
inverted dome are negligible compared to the acoustic
radiation force.
An approximate value of the acoustic radiation force
can be calculated by using the effective spring constant of
a pendulum using equation (3.A.1). The length ( ) is 1.0 m,
mass ( ) of the styrofoam ball is 0.78 g, gravity ( ) is
L
m g

12
9.8 m/s2
and the observed distance ( ) of the deflected
ball from equilibrium is 2.5 cm. The mass-equivalent
radiation force is about 20 mg. The calculation is shown
below:
(3.A.1)
B. BAFFLED DRIVER
A baffle (Fig. 2) is a plate with a hole in the
center. An acrylic plate is machined to fit over the outer
frame of the driver. The acrylic circular baffle has
diameter of 21.3 cm with a 3 cm diameter hole in the
center. The thickness of the baffle is 1.3 cm. The distance
from the outer edge of faceplate of the baffle to the
center of the inverted dome is 4.5 cm. The baffle causes
more jetting and also more acoustic radiation force due to
a greater acoustic amplitude as a result of the baffle. The
spherical wave solution of the acoustic radiation force is
proportional to the square of the amplitude as seen in
equation (2.B.1).
When the styrofoam ball is placed 10 cm from the
center of the driver, the ball is attracted towards the
driver at 4 Vrms. The acoustic radiation force is clearly
x
Frad = kx
k =
mg
L
mrad =
F
g
=
mx
L
mrad =
F
g
=
mx
L
=
0.78g*.025m
1m
= 0.0195g ! 20mg

13
much stronger with the baffle. At 14.5 cm from the dome (10
cm from the face of the baffle), jetting dominates and no
acoustic radiation force is observed. When slowing
increasing the voltage to 35 Vrms, the ball is forced away
from the driver at 6.5 Vrms. The jetting significantly
increases due to reducing the size of the orifice from 6 in
bare speaker face (15.24 cm) down to a 3 cm baffle hole.
When a flame is placed at 10 cm or 14.5 cm, the flame is
immediately extinguished when the voltage is increased.
Figure 2. Unbaffled and baffled driver.
1. Helmholtz Resonator
The jetting effect can be increased because the baffle
acts to create a Helmholtz resonator. Using the unflanged

14
effective length equation from KFCS (2000), the resonance
frequency can be calculated from the geometry.
One quantity that is required is the volume of the
baffle-enclosed area (Fig. 3). We found this volume by
using numerous miniscule precision glass spheres to fill
half the area. The spheres are then poured into a graduated
beaker to find the volume. Initially, the baffled area was
completely filled with glass spheres. As more and more
spheres filled the area, the driver cone compressed which
allowed the volume of the baffled area to increase. This
caused the measured volume to be inaccurate. The half
volume filling allows for a more accurate equilibrium
volume calculation.
Figure 3. Volume determination with glass spheres.

15
The calculation of the Helmholtz frequency is shown below:
(3.B.1)
(3.B.2)
Experimentally the Helmholtz resonance frequency is
found to be 402–410 Hz using an oscilloscope. The
oscilloscope shows a significant phase difference around
400 Hz and a maximum peak amplitude. The calculated and the
experimental frequency are close.
Because all the acoustic radiation force experiments
are conducted at 100 Hz the baffle creating a Helmholtz
resonator is not a major contributor to the jetting at this
lower frequency.
V1/2 = 200ml = 0.2l = 0.2x10!3
m3
LNECK = 0.5in
Radius = a =
d
2
=
3.0cm
2
= 1.5x10!2
m
Leff = L +1.4a =
2.54
2
x10!2
m +1.4x1.5x10!2
m
S = !a2
f =
c
2!
S
LeffV
f = 343
m
s
x
1
2!
x
!(1.5x10"2
m)2
2.54
2
x10"2
m +1.4x1.5x10"2
m
#
$%
&
'( 2x0.2x10"3
( )
= 395Hz

16
Figure 4. Data acquisition equipment for demonstration.

17
IV. EXPERIMENTAL APPARATUS
A. ANECHOIC CHAMBER
The NPS anechoic chamber (Fig. 5) is an 8 m by 9 m
room with 7 m high walls. The walls, ceiling and floor are
covered with fiberglass triangle wedges (Fig. 6) that point
towards the center of the room to absorb sound. The floor
is suspended approximately 1.5 meters from the ground by
wire mesh. The mesh allows a person to walk into the
anechoic chamber without stepping on the upward pointing
fiberglass wedges.
Figure 5. Anechoic chamber.

18
The anechoic chamber minimizes traveling wave
reflections produced by a source of sound. The P.F. 612
fiberglass wedges are grouped into three and are
perpendicular to the neighboring group to allow for maximum
absorption.
Figure 6. Fiberglass wedges inside the anechoic chamber.
The control room (Fig. 7) adjacent to the anechoic
chamber is where all the analysis equipment is stored with
the exception of the analytical balance, reference
microphone and conditioner associated with the reference
microphone. The door from the control room into the
anechoic chamber is also lined with the same fiberglass

19
material to ensure uniformity. There are small window ports
to allow cables into and out of the chamber.
Figure 7. Control room.

20
B. DRIVER ASSEMBLY
The loudspeaker (Fig. 8) is a 15-inch diameter
Electro-Voice EVX-155 that has been re-coned at Santa Cruz
Sound Company in Santa Cruz, Ca. The power handling
capability is 600 W (continuous) and the nominal impedance
is 8 ohms.
The driver is contained in a 0.46 m x 0.46 m x 0.56 m
wooden enclosure. The speaker and enclosure are oriented
vertically so that the sound field travels upward towards
the analytic balance. This allows the aluminum sphere to
hang symmetrically above the peak of the cone. The driver’s
wooden enclosure is on a platform built to hold the driver
just above the wire mesh of the anechoic chamber.
Figure 8. 15-inch diameter electrovoice driver and
enclosure volume.

21
C. ANALYTICAL BALANCE
The high precision balance (Fig. 9) is an AND GR-202
Analytical Balance. The AND balance has a built-in hook
underneath the balance to allow for precision measurements
to be made outside the weighing chamber. A hook holds the
aluminum sphere. The acoustic radiation force ( F ) due to
the loudspeaker can be easily calculated from the mass
reading ( m ) of the balance as F mg= , where g is the
acceleration due to gravity. There is a secondary display
and a push button to zero the balance in the control room.
An operator does not have to physically stand in the
anechoic chamber while gathering measurements (Fig.10). The
specifications of the balance are as follows:
SPECIFICATIONS
Weighting Capacity 210 g/42 g
Min. weighing value (1 digit) 0.1 mg/0.01 mg
Repeatability (Std dev) 0.1 mg/0.02 mg
Stabilization time 3.5 sec/ 8 sec
Calibration weight Built-in
Net weight Approx. 6.0 kg

22
Figure 9. Analytical balance.
Figure 10. Analytical balance setup.

23
D. NEXUS CONDITIONER AND INTENSITY PROBE
The NEXUS conditioner (Fig. 11) is a Bruel and Kjaer
(B&K) model number 2691 that can be used for a multitude of
applications. The conditioner has phase-match channels and
allows for sensitivity matching with the two microphones on
the intensity probe. Phase matching is required to
accurately determine acoustic particle velocity from the
instantaneous pressure gradient between the two microphones
of the B&K sound intensity probe type 3599 (Fig. 11). The
intensity probe and conditioner convert a sound pressure
into a voltage signal for analysis. An initial transducer
setup was required to adjust the sensitivity of the two
microphones to ensure precise equivalence. This calibration
was done using a B&K type 4228 250Hz Pistonphone (Fig. 12)
and Stanford Research System SR785 Dynamic Signal Analyzer.
The pistonphone is a precise calibration tool used on
microphones to ensure their accuracy. The sensitivity was
adjusted until the difference of the output signals of
Channel 1 and 2 produced a minimum voltage. The
specifications of the pistonphone and conditioner are
stated below:

24
Figure 11. Nexus conditioner and 2-channel intensity probe.
Figure 12. 250 Hz pistonphone.

25
PISTONPHONE
Sound Pressure Level 124.08 re 20uPa
Nominal Frequency 250Hz +/- 0.1%
Calibrated 25 APR 2011
NEXUS CONDITIONER SETUP
Amplifier Setup
CH 1/CH 2 3.16mV/Pa / 3.16mV/Pa
Transducer Setup Sensitivity
CH 1/CH 2 11.933mV/Pa / 11.967mV/Pa
Transducer Supply
Supply Voltage Polarization Cable length
CH 1 14V 200V 8m
CH 2 14V 200V 8m
NOTE: Channel 1 is Channel A
Channel 2 is Channel B
E. ELECTRONIC DATA ACQUISITION EQUIPMENT AND SETUP
Most of the electrical equipment (Fig. 13) required
for the experiment was inside the anechoic chamber control
room. The driving circuit has an HP 33120a function
generator and a QSC MX3000a Dual Monaural Amplifier. The
two-channel intensity probe signals are fed into the B&K
NEXUS conditioner and then into the Stanford Research

26
System SR560 Low Noise Preamplifier. The amplified signal
is sent to Stanford FFT Spectrum Analyzer to obtain voltage
output data.
The Stanford preamp is able to take both channels and
to output Channel A, Channel B or more importantly the
difference between the two channels. The difference output
gives the instantaneous pressure difference between Channel
A and B from the intensity probe. The FFT analyzer is able
to isolate the voltage output signal.
An ACO half-inch microphone (Fig. 14), which serves as
a reference microphone is connected to a HP34401A digital
multimeter and is located about 1 meter above and to one
side the driver. To ensure accuracy and consistency in the
data, the reference microphone voltage is held constant as
the height of the intensity probe and aluminum ball varies.
The reference microphone output voltage is seen on the
multimeter, which is located in the control room.
The balance has a remote display with zeroing
capability inside the control room. This allows for
streamlining data taking process because the analytical
balance is suspended on a platform attached to the ceiling
of the anechoic chamber. A stool must be used to locally
read the scale in the anechoic chamber. The specifications
of the preamplifier and microphone are shown below:

27
STANFORD PRE-AMPLIFIER SETTINGS
Filter Cutoffs 1–10 kHz
Highpass 6 dB/oct
Lowpass 6 dB/oct
Coupling AC
Gain Mode Low Noise
Gain 5
ACO MICROPHONE
Reference Voltage 180 mV (constant)

28
Figure 13. Data acquisition equipment in control room.

29
Figure 14. ACO reference microphone.
F. ALUMINUM ROD AND INTENSITY PROBE CLAMP
A 1/2-inch diameter 8-ft long aluminum rod (Fig. 15)
was machined with forty grooves that were ink-marked. The
grooves are in 1 cm increments from one end of the rod.
This alleviated the ambiguity in measuring the exact
distance from the peak of the cone in the center of the
driver. The markings sped up data taking when many distance
measurements were required. The only measurement that
required a ruler was the initial distance from the center
of the cone to the closest microphone on the intensity

30
probe. Every groove has a mark with permanent black ink
with the exception of every fifth one which is delineated
with red.
The intensity probe clamp (Fig. 15) was machined to
allow the probe to be raised in 1 cm increments. The clamp
has a small thin brass plate, which fits inside the grooves
of the aluminum rod. A plastic insert was also placed on
the end of the screw clamp to prevent scoring of the
intensity probe extender rod.
Figure 15. Clamp and 1 cm incremented aluminum rod.

31
G. CHAIN LINK SPACERS AND BUSS WIRE
For ease of data collection and reproduction,
individual chain links (Fig. 16) were cut from a single
steel chain to obtain nearly exact distances. Several
different mathematical methods of individual link
calculations were used to obtain the distance. The
calculated distance is 1.2655 cm of each link. Twenty links
is sufficient to obtain the acoustic radiation force
measurements. The aluminum sphere distance to the
loudspeaker dome is incremented 1.2655 cm at a time to
obtain the force on axis of the loudspeaker.
Thirty-two gauge buss wire connects the aluminum
sphere to the chain link spacer. Buss wire allows the ball,
wire and chain to be electrically grounded to ensure there
are no electrostatic effects.
Figure 16. Chain links.

32
H. KANOMAX VANE ANEMOMETER
The Kanomax model 6812 volume flow anemometer (Fig.
17) is used to measure the amount of jetting from the
Electrovoice loudspeaker. The anemometer uses an extremely
low friction vane-type probe to measure the flow rate
through the vane in meters per second. The flow rate in
meters per second can be used to calculate into milligrams
of drag force to account for the jetting (Fig. 18). The
specifications of the anemometer are as follows:
SPECIFICATIONS
Propeller Vane Diameter 70 mm
Sensitivity 0.20–40.00 m/s
Sampling Rate 2 sec
Figure 17. Kanomax Vane anemometer.

33
Figure 18. Vane-Anemometer set up to measure jetting from
the loudspeaker.
I. KANOMAX HOTWIRE ANEMOMETER
The Kanomax model A041 hotwire anemometer (Fig. 19) is
used to measure the amount of jetting from the Electrovoice
loudspeaker. The platinum wire is heated to a high
temperature. As the air flows across the wire, the wire is
cooled. This causes the resistance of the wire to decrease;
therefore, the current is changing proportionally with a
constant voltage source. The change in current is then
measured to output an air velocity. A slight change in room
temperature is sensed by the hotwire so a temperature
compensating circuit is added to ensure stability. The
hotwire anemometer is able to measure only actual airflow.

34
Figure 19. Hotwire anemometer.

35
V. EXPERIMENT
The experiment is divided into two distinct phases.
The first phase is obtaining the theoretical prediction
curve by taking pressure measurements of the acoustic
field. The pressure measurements are used to find the
gradient of the kinetic and potential energy densities
resulting in the acoustic radiation force. The second phase
is taking actual mass-equivalent force measurements to
compare with the theoretical prediction.
A. PRESSURE MEASUREMENTS
The pressure measurements are taken first to obtain
the theoretical acoustic radiation force curve. All the
analyzing equipment is energized an hour and a half prior
to taking data to establish thermodynamic equilibrium. Due
to the curvature of the bottom of the intensity probe (Fig.
20), the lower microphone (“A”) is placed 3.6 cm above the
dome of the loudspeaker. This is the closest distance
without the loudspeaker dome contacting the bottom of the
probe as the dome oscillates. As the voltage of the driver
is increased to approximately 60 VAC, the reference
microphone input voltage is adjusted to maintain 180 mV to
ensure a constant acoustic field and reproducibility. When
the voltage is stable at 180 mV at approximately 30
seconds, microphone A rms voltage and Stanford pre-
amplifier A–B rms voltage is taken using the FFT spectrum
analyzer, which is set up to read the voltage amplitude at
100 Hz. After the two voltages are recorded, the driver is
turned off and the intensity probe distance is increased by
one centimeter from the driver on the aluminum rod.

36
Figure 20. Curvature of the intensity probe.
A five-minute wait on the stopclock is started to
allow for the driver to return to equilibrium. After the
five-minute wait, the driver voltage is increased to
approximately 60 VAC and the reference microphone is
maintained at 180 mV then the second voltage data points
are taken. This data-taking process is repeated thirty
times to ensure there are enough data points to fit a
curve.
The voltages in section A of the Appendix are
converted into pressures by using 3.16 mV/Pa conversion
factor displayed on the Nexus conditioner. The gain from
the Stanford pre-amplifier is divided out. The kinetic
energy density uses the linearized Euler’s equation for the
amplitude of the particle velocity

37
(5.A.1)
where is the frequency of 100 Hz, is the density of
air, =8.5 mm is the distance between the two microphone
membranes and is the amplitude of the instantaneous
difference in the pressures of microphones A and B. The
particle velocity amplitude is then used to calculate the
kinetic energy density (2.A.2) and the potential energy
density (2.A.3). The results of the calculations are in
section B of the Appendix.
The distance calculations for the KE and PE densities
for the intensity probe are located in Figure 21. The
gathered data from section B is plotted in Matlab. A curve
fit tool in Matlab fits the data points using a sixth order
polynomial. The curve fits are shown in Figure 22.
U=
Pdiff
2! f "d
,
f !
d
Pdiff

38
Figure 21. Relevant distances associated with the intensity
probe with d = 8.5 mm spacing.

39
Figure 22. Sixth-Order Polynomial Curve Fits of the Kinetic
and Potential Energy Densities as a function of distance
from the loudspeaker.
In order to obtain the theoretical acoustic radiation
force, the gradient of kinetic and potential energy
densities must be calculated according to equation (2.A.4).
B. MASS MEASUREMENTS
All equipment is energized prior to taking data, as in
the case of the pressure measurement data. The balance is
used to directly measure the force on the ball including
the acoustic radiation force and jetting from the
loudspeaker. Buss wire is used to hang the aluminum ball to
the under hook of the analytical balance with thirty-five

40
chain links in between. The initial ball height from the
dome is 2.07 cm with an added 1.905 cm (radius of the ball)
for a total of 3.98 cm.
The voltage is increased on the driver to
approximately 60 VAC at 100 Hz. The reference microphone is
maintained at 180 mV. When the balance reading is steady in
approximately 30 seconds, the balance reading is read off
the remote display in the control room. After the balance
weight is taken, the loudspeaker is turned off. The
anechoic chamber is entered to remove a single link off the
chain. A five-minute wait on the stopclock is started to
allow for equilibrium.
After the five-minute wait, this process is repeated
twenty-three times to give a final distance of 33.08 cm
(1.2655 cm x 23 links + 3.975 cm). The final mass data is
located in section C of the Appendix. Figure 23 shows the
plot of the final mass data with the theoretical curve.

41
Figure 23. Theoretical acoustic radiation force curve
plotted with mass data.
C. DATA INTERPRETATION
The data in Figure 23 clearly shows a difference
between the theoretical curve and the direct mass
measurements.
1. Jetting
The difference is at least partially explained by
jetting from the loudspeaker. The vane anemometer shows
there is more outward jetting at 30 cm than there is at 10
cm from the data in section D in the Appendix. The jetting
effect causes the observed force to be less than the
attractive acoustic radiation force. The upward drag force

42
on the ball can be calculated. Cengel and Cimbala (2010)
give the general drag force equation as
(5.C.1)
where is the density of air of 1.204 kg/m3
, is the
cross-sectional area, is the velocity and is the
drag coefficient of the volume associated with a specific
Reynolds number R . The Reynolds number for a sphere is
(5.C.2)
where is the diameter and is the kinematic viscosity of
air at room temperature and atmospheric pressure. At a
distance of 0.30 m from the loudspeaker dome the vane
anemometer reads 0.33 m/s, so
( ) 5 2
m 0.0254
R=U 0.33 * 1.5 * * 838.
1.516 10
d m s
in
v s in x m−
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
A Reynolds number of 838 yields a drag coefficient of
approximately 0.6 from the CD vs. Reynolds number graph from
Cengel and Cimbala(2010). The drag force is then
FD =
1
2
!oU2
SCD (R),
!o S
U CD
(R)
R=U
d
v
,
d v

43
2
5
2
3
1 1.204 0.33 0.0254
* * * * 0.75 * *0. 4.48 10 ,6
2
D
kg m m
F in
m s in
x Nπ −⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
6 2
1 10
* 4.574 .
9.8
DF x mg s
mg
g kg m
⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
The calculated drag force of 4.6 mg accounts for 70%
of the difference between the direct force-mass measurement
and the theoretical curve. The points for the mass data
should be higher and closer to the theoretical curve. The
vane anemometer is averaging over a wider area of due to
the diameter of the vane being 6.98 cm (2.75 in) while the
ball is 3.81 cm (1.5 in). The difference in area could
cause the deviation from theory to actual data. This
discrepancy requires further research.
A hotwire anemometer is also used to find the airflow
rate of the jetting. The wire of the hotwire anemometer is
a much smaller sensor than the vane anemometer. The hotwire
is able to output a more localized airflow reading. The
hotwire anemometer senses a higher jetting flow rate
approaching the dome shown in section D of the Appendix,
which is contradictory to the vane anemometer. The two
anemometers approximately agree at 30 cm but deviate closer
in.
Since the vane anemometer is averaging over a larger
area, we conclude the jetting is very turbulent near the
dome. The vane anemometer is reading both upward and
downward flow to cause cancellations approximating zero.

44
Unfortunately, the hotwire anemometer is not able to output
flow direction so an inward flow towards the driver could
not be measured.
Since the fluid dynamics around the dome of the driver
is so complex, the jetting force and direction is crudely
measured by using a flame (Fig. 24). A lit match is able to
check the relative jetting force, as the flame is taken
closer to the dome. In theory, the direction of the flame
on and off axis can also be roughly seen (Fig. 25). The
flame is erratic on and off axis near the dome. The result
of these tests proves to be inconclusive.
Figure 24. Flame test close and far away from the dome.

45
Figure 25. Flame test off axis and moving the flame from
far away to close in.
In attempt to nullify the amount of jetting around the
ball, a cubical cellophane shield was created (Fig. 27).
The cube is 3.0 inches on a side with a hole at the top to
just allow for the 3.81 cm (1.5 in) diameter aluminum ball
to enter. A mass measurement is taken at 30 cm from the top
of the dome with and without the cube. Without the cube,
the analytic balance measures negative 4.4 mg, which is
consistent with the previous mass data. With the cube, the
analytic balance measures a positive 1.6 mg at 30 cm from
the dome shown in section E of the Appendix. The theory
line at 30 cm is 1.4 mg. There is agreement between theory
and shielded mass data. The cube is able to divert most of
the jetting flow around the aluminum ball.
At 7 cm, there is another discrepancy from the
original mass data (Fig. 26). Surprisingly, the mass

46
reading with the shield is 230 mg, whereas the mass reading
without the shield is 45.7 mg. The difference in mass at 7
cm is approximately 184 mg. Using the drag force equation
(5.C.1) and a velocity of 1.6 m/s, only 109 mg is accounted
for due to jetting. The drag force equation assumes a
laminar steady flow. The flow is very turbulent near the
source. Some of the mass equivalent force of 75 mg could be
due to turbulent flow but that is only a speculation. The
discrepancy is unknown and requires further investigation.
Figure 26. Theoretical acoustic radiation force curve
plotted along with mass data and shielded mass data.

47
The shielded mass force is much greater than the
theory. The theory should reasonably agree with the
shielded results. There are many assumptions made in the
calculation of theoretical curve. The next step is to
verify all the assumptions made in the theory equation are
correct. This verification of assumptions will not be
analyzed in this thesis.
Figure 27. Cellophane shield and setup.
2. Spacing of the Intensity Probe
Another possible source of deviation from the mass
data and the theoretical acoustic radiation force curve is
the finite-difference error due to the 8.5 mm spacing
between the two microphones, because the pressure of the
sound field is falling off so quickly with distance. The
spacing causes the particle velocity to be approximate. The

48
spherical wave fit of the pressure amplitude is used to
verify if the finite spacing is a source of error:
,
A
P
x b
=
+
(5.C.3)
where, in SI units, x is the distance from dome, A = 24.44
and b = 0.04771 by using a curve-fit tool software. The
peak amplitudes of the gradients are determined by
instantaneous pressure between x0 and x0±d (Fig. 28). From
the gradient values, the velocity amplitudes are determined
and then the average kinetic energy densities. These
average kinetic energy density values are plotted against
the exact theoretical curve for the average kinetic energy
of a spherical wave. The error between the two is analyzed.
The instantaneous pressure at x0 using equation (2.B.1)
is
[ ]0 0
0
cos ( ) .
A
p k x b t
x b
ω= + −
+
(5.C.4)

49
Figure 28. Pressure amplitude as a function of distance
sketch for a spherical wave.
The instantaneous pressure at x0±d using equation
(2.B.1) is
[ ]0
0
cos ( ) .
A
p k x d b t
x d b
ω± = ± + −
± +
(5.C.5)
The instantaneous gradients of the pair of points are then
[ ] [ ]0 0
0 0
cos ( ) cos ( )
.
k x d b t k x b tA
p
d x d b x b
ω ω
±
⎧ ⎫± + − + −
∇ = −⎨ ⎬
± + +⎩ ⎭
(5.C.6)

50
The peak particle velocity amplitudes can be found from the
Euler-equation relationship
,
2
p
U
fπ ρ
±
±
∇
= (5.C.7)
where f is 100 Hz and ! is 1.204 kg/m3
for the density of
air. The time-averaged kinetic energy densities are
2
.
4
k
U
e
ρ ±
±
= (5.C.8)
The exact instantaneous particle velocity for a spherical
wave is equation (2.B.2). The exact time-averaged kinetic
energy density is
2
2 2 2 2
1
1 .
4
k
A
e
c r k rρ
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
(5.C.9)
The curves ke and ke ±
vs. x are plotted on the same
curve to determine the error (Fig. 29). The curve is exact
and the points correspond to a finite-difference
approximation that is used for the intensity probe. There
is very little error caused by the 8.5 mm spacer on the
intensity probe. This is not a source of error in our
experiment.

51
Figure 29. Graph of time-average kinetic energy density vs.
distance from the loudspeaker dome for a spherical wave.
3. Data Approaching Zero
The mass data curve in the tail should go to zero as
in the theory. The mass data does approach zero but at much
farther distance, as shown in section F of the Appendix. At
125 cm, the analytical balance reads nearly zero.
0.05 0.10 0.15 0.20 0.25 0.30 0.35
0
1
2
3
4
5
time-averagedkineticenergydensity<ek
>(J/m
3
)
distance from dome x (m)

52

53
VI. CONCLUSION
There is approximate agreement between the calculated
theory curve and the directly measured mass data shown in
Figure 23. The disparity comes with the shielded cube data
and the theory. The source of the disagreement requires
further research.
Many issues exist in the experiment such as:
complexity in fluid dynamics near the dome, finite-size
effects and thermoviscous losses. There are other minor
issues, which could possibly contribute to the deviation
between theory and experiment. Some of these are explained
below.
A. AXIS OF SYMMETRY
One source of deviation could be due to the lack of
symmetry. The driver’s enclosure is a wooden box that has
four distinct corners instead of a circular base. The
theory states an axis of symmetry must be preserved. The
absorption material below the balance platform is also
rectangular (Fig. 30). The theory states that symmetry is
crucial to the experiment. An extensive and careful probing
of the acoustic field off-axis would establish the degree
to which the field is symmetric about the axis. Quick
measurements show only at most a 1% asymmetry of the field
over the diameter of the sphere, so a lack of symmetry of
the field does not appear to be playing a role.

54
Figure 30. Rectangular absorbing material.
B. REFLECTIONS
Unwanted reflections could also be a source of
disagreement. Reflections from the wooden platform holding
the analytical balance (Fig. 31) could cause errors in the
data. The absorption material might not be absorbing all
sound generated from the loudspeaker but rather allowing
some of the sound to reflect. The rigid metal bars can also
cause reflections in the anechoic chamber (Fig. 31). There
are aluminum bars holding the balance and the incremented
rod. The bars do not have sound absorbing or suppression

55
materials on or around them. A new fire extinguishing
system was installed in the anechoic chamber at the Naval
Postgraduate School in 2010 (Fig. 32). The piping is not
insulated by sound. The metal piping lining the left side
of the anechoic chamber could be also causing reflections.
The reflections would cause errors in both the mass and
pressure data.
It should be noted that reflections themselves do not
invalidate the theory, which is valid even for standing
waves (Nyborg, 1967). However, reflections here would
probably degrade the symmetry of the acoustic field about
the axis (sec. A), which violates the theory.

56
Figure 31. Wooden platform supporting the balance and
assortment of aluminum bars.

57
Figure 32. Fire suppression system in the anechoic chamber.
C. FUTURE WORK
A next step forward in acoustic radiation force
research is to resolve all the systematic errors and
understand the complexity as much as possible. Researching
the literature on thermoviscous effects in the acoustic
radiation force should be done. Fully understanding the
fluid dynamics associated with jetting near the dome of the
driver is crucial to completing this research. The finite-
size effect experiments can be done with different diameter
aluminum balls. With further research and calculations, the
theory should match the experimental data.

58
However, we have shown approximate agreement between
theory and experiment, which allows us to move forward to
performing ultrasonic filtration experiments. The first
step is a simple proof-of-principle experiment with
neutrally-buoyant particles in water, which would bring us
one step closer to actual shipboard application of the
acoustic radiation force phenomena.

59
APPENDIX
A. VOLTAGE MEASUREMENTS WITH THE INTENSITY PROBE
Increment
(cm)
Microphone
A
(mVrms)
Microphone
A-‐B
(mVrms)

1
2635
139

2
2503
131.1

3
2349
124.2

4
2206
116.4

5
2072
108.5

6
1945
100.8

7
1823
93.49

8
1715
86.73

9
1610
79.87

10
1507
73.07

11
1443
68.99

12
1353
63.55

13
1284
58.34

14
1223
54.68

15
1163
50.53

16
1106
47.02

17
1050
43.98

18
1000
40.62

19
954.6
37.6

20
912.2
35.17

21
873.9
32.95

22
840
31.03

23
800.4
29.07

24
769.2
27.24

25
740
25.6

26
713
24.14

27
698.3
22.85

28
663.4
21.51

29
638.8
20.41

30
619.1
19.42

60
B. KINETIC AND POTENTIAL ENERGY DENSITIES VS.
DISTANCE
Distance
(m)
KE
(J/m3
)
Distance
(m)
PE
(J/m3
)

0.0539
1084917.992
0.036
98502.57492

0.0639
965100.8415
0.046
88880.81812

0.074
866184.6888
0.056
78280.28134

0.0839
760804.8484
0.066
69039.45597

0.094
661038.5503
0.076
60906.80896

0.104
570542.9918
0.086
53669.25362

0.1139
490792.1757
0.096
47147.60991

0.124
422382.5849
0.106
41726.75319

0.134
358207.5042
0.116
36773.76499

0.1439
299809.5935
0.126
32219.05876

0.154
267263.4688
0.136
29540.57922

0.1639
226776.6945
0.146
25970.59726

0.1739
191117.3701
0.156
23389.25516

0.184
167889.8042
0.166
21219.70092

0.194
143372.5163
0.176
19188.70975

0.2039
124145.9512
0.186
17353.88033

0.2139
108611.9875
0.196
15641.01536

0.2239
92650.36809
0.206
14186.862

0.2339
79385.83201
0.216
12927.93632

0.2439
69456.36449
0.226
11805.01328

0.254
60964.65898
0.236
10834.52368

0.2639
54066.82439
0.246
10010.24983

0.2739
47452.3117
0.256
9088.673543

0.2839
41665.97198
0.266
8393.921346

0.2939
36799.9511
0.276
7768.725632

0.3039
32722.15085
0.286
7212.160849

0.3139
29318.36314
0.296
6917.838649

0.3239
25980.5375
0.306
6243.631725

0.3339
23391.24406
0.316
5789.168085

0.3439
21177.06463
0.326
5437.608707

61
C. MASS DATA FROM THE ANALYTICAL BALANCE
Link
distances
(m)
Mass
(mg)

0.03975
53.5

0.052405
49.2

0.06506
45.7

0.077715
42.9

0.09037
37.9

0.103025
32.4

0.11568
27.2

0.128335
22.4

0.14099
17.5

0.153645
13.3

0.1663
9.8

0.178955
6.9

0.19161
3.7

0.204265
2.3

0.21692
0.9

0.229575
-‐0.7

0.24223
-‐1.8

0.254885
-‐2.3

0.26754
-‐2.5

0.280195
-‐3.1

0.29285
-‐4

0.305505
-‐4

0.31816
-‐4

D. HOTWIRE AND VANE ANEMOMETER READINGS
Distance
(cm)
Vane
Anemometer
(m/s)
Hotwire
Anemometer
(m/s)

30
0.3
0.33

25
0.31
Not
Taken

20
0.31
0.35

15
0.26
0.66

10
0
1.18

6
0
1.61

62
E. MASS READINGS WITH CELLOPHANE SHIELD
Distance
(cm)
Mass
w/
shield
(mg)

30
1.7

25
4.6

20
10.9

15
35.4

10
96.7

7
230

F. ROUGH MASS READINGS AT FURTHER DISTANCES
Distance
(cm)
Mass
(mg)

50
-‐4.4

75
-‐2.9

100
-‐2

125
-‐1

G. MATLAB CODE
clc;
clear;
%voltages
A =
[2635;2503;2349;2206;2072;1945;1823;1715;1610;1507;1443;1353;1284;1223;
1163;1106;1050;1000;954.6;912.2;873.9;840;800.4;769.2;740.0;713;698.3;6
63.4;638.8;619.1];
AminusB=[139;131.1;124.2;116.4;108.5;100.8;93.49;86.73;79.87;73.07;68.9
9;63.55;58.34;54.68;50.53;47.02;43.98;40.62;37.6;35.17;32.95;31.03;29.0
7;27.24;25.6;24.14;22.85;21.51;20.41;19.42];
%distance meters 30 points
zAm=[0.036:0.01:0.3260];
%middle of mic 0.036m+0.01795 m=0.05395m to 0.34395 (8.5mm spacer)
zAminusBm=[0.05395:0.01:0.34395];
%pressure conversion
P_A=A./3.16e-3./5;
P_AminusB=AminusB./3.16e-3./5;
figure (1)
hold on;
%plot of A

63
plot(zAm,A,’-.*r’)
%plot of A minus B
plot(zAminusBm,AminusB,’:og’)
%PE
PEz=P_A.^2./(2*1.2*343^2);
%KE
U=P_AminusB./(2*pi*100*1.225*.0085); % 8.5mm spacer
KEz=0.5.*(1.2.*U.^2);
%cftool
%
% Linear model Poly6:
% f(x) = p1*x^6 + p2*x^5 + p3*x^4 + p4*x^3 + p5*x^2 +
% p6*x + p7
% Coefficients (with 95% confidence bounds):
% p1 = -1.052e+09 (-1.477e+09, -6.267e+08)
% p2 = 1.175e+09 (7.12e+08, 1.638e+09)
% p3 = -4.96e+08 (-6.942e+08, -2.978e+08)
% p4 = 9.121e+07 (4.899e+07, 1.334e+08)
% p5 = -3.756e+06 (-8.412e+06, 8.995e+05)
% p6 = -1.095e+06 (-1.342e+06, -8.474e+05)
% p7 = 1.397e+05 (1.348e+05, 1.445e+05)
%
% Goodness of fit:
% SSE: 1.261e+06
% R-square: 0.9999
% Adjusted R-square: 0.9999
% RMSE: 234.1
%
%
% Linear model Poly6:
% f(x) = p1*x^6 + p2*x^5 + p3*x^4 + p4*x^3 + p5*x^2 +
% p6*x + p7
% Coefficients (with 95% confidence bounds):
% p1 = -1.421e+10 (-1.943e+10, -8.991e+09)
% p2 = 1.85e+10 (1.226e+10, 2.474e+10)
% p3 = -9.456e+09 (-1.242e+10, -6.492e+09)
% p4 = 2.324e+09 (1.614e+09, 3.035e+09)
% p5 = -2.489e+08 (-3.388e+08, -1.59e+08)
% p6 = 6.729e+05 (-4.956e+06, 6.302e+06)
% p7 = 1.478e+06 (1.343e+06, 1.613e+06)
%
% Goodness of fit:
% SSE: 1.897e+08
% R-square: 0.9999
% Adjusted R-square: 0.9999
% RMSE: 2872

64
cfPE=(-1.052e+09.*zAm.^6 + 1.175e+09.*zAm.^5 -4.96e+08.*zAm.^4 +
9.121e+07.*zAm.^3 -3.756e+06.*zAm.^2 -1.095e+06.*zAm.^1 + 139700);
cfKE=(-1.4205e+10.*zAminusBm.^6 + 1.85e+10.*zAminusBm.^5 -
9.456e+09.*zAminusBm.^4 + 2.324e+09.*zAminusBm.^3 -
2.489e+08.*zAminusBm.^2 + 672900.*zAminusBm.^1 + 1.478e+06);
figure (2)
hold on;
box on;
set(gcf,’color’,’w’);
plot(zAm,PEz,’-.or’);
plot(zAminusBm,KEz,’:og’);
leg=legend(‘PE Density’,’KE Density’);
set (leg,’location’,’NorthEast’);
xlabel(‘Distance (m)’);
ylabel(‘Energy (J/m^3)’);
title(‘Kinetic and Potential Energy Densities vs. Distance’);
%plot(zAm,cfPE);
%plot(zAminusBm,cfKE);
hold off
syms zm;
cfPE=(-1.052e+09.*zm.^6 + 1.175e+09.*zm.^5 -4.96e+08.*zm.^4 +
9.121e+07.*zm.^3 -3.756e+06.*zm.^2 -1.095e+06.*zm.^1 + 139700);
gradPE=diff(cfPE)
cfKE=(-1.4205e+10.*zm.^6 + 1.85e+10.*zm.^5 -9.456e+09.*zm.^4 +
2.324e+09.*zm.^3 -2.489e+08.*zm.^2 + 672900.*zm.^1 + 1.478e+06);
gradKE=diff(cfKE)
m=[0.05:.0001:.33];
gPE =(- 6312000000.*m.^5 + 5875000000.*m.^4 - 1984000000.*m.^3 +
273630000.*m.^2 - 7512000.*m - 1095000);
gKE =(- 85230000000.*m.^5 + 92500000000.*m.^4 - 37824000000.*m.^3 +
6972000000.*m.^2 - 497800000.*m + 672900);
ARF=(3./2.*gradKE-gradPE)
ARF =(- 121533000000.*m.^5 + 132875000000.*m.^4 - 54752000000.*m.^3 +
10184370000.*m.^2 - 739188000.*m + 4208700./2);
ARF=abs(ARF);
ARF=ARF.*2.895833243042200e-05./9.8; %(volume)
%ARF=ARF.*2.895833243042200e-05./9.8; (volume of sphere)

65
%Al ball measurements
links=[0.03975:.012655:0.3182]’;
mass=[53.5; 49.2; 45.7; 42.9; 37.9; 32.4; 27.2; 22.4; 17.5; 13.3; 9.8;
6.9; 3.7; 2.3; 0.9; -.7;-1.8;-2.3;-2.5;-3.1;-4.0;-4.0;-4.0 ];
figure (3)
box on;
grid on;
hold on;
plot(links,mass,’ ob’);
ylabel(‘Force Equivalent Mass (-mg)’);
hold off;
a=0;
b=[0:.01:.33];
% 3in sphere data
ball=[230;96.7;35.4;10.9;4.6;1.7];
distance=[0.07;0.10;0.15;0.20;0.25;0.30];
figure(4)
box on
grid on;
hold on;
%plot(links,mass,’ ob’);
plot(m,ARF);
plot(b,a);
title(‘Acoustic Radiation Force Curve vs. Distance Graph’);
leg=legend(‘Theoretical Force Curve’);
hold off;

66
figure(5)
box on
grid on;
hold on;
plot(links,mass,’ ok’);
plot(m,ARF);
plot(b,a);
leg=legend(‘Mass Data’,’Theoretical Force Curve’);
hold off;
figure(6)
box on
grid on;
hold on;
plot(links,mass,’ ok’);
plot(m,ARF);
plot(distance,ball,’ ok’);
plot(b,a);
leg=legend(‘Mass Data’,’Theoretical Force Curve’,’Mass Data with
Shielded Cube’);
hold off;

68
LIST OF REFERENCES
Bentivoglio M., & Rochelle, J. (2009). Measurement and
analysis of the acoustic radiation force (Master’s
thesis). Naval Postgraduate School, Monterey, CA.
Cengel, Yunus A. and John M. Cimbala. (2010). Fluid
mechanics: Fundamentals and applications, 2nd
ed. New
York: McGraw-Hill, Ch. 11.
Embleton, T. F. W. (1954). Mean force on a sphere in a
spherical sound field. I. (Experimental). Journal of
Acoustic Society of America, 26: 46–50.
Faber, T. E. (2010). Fluid dynamics for physicists. New
York: Cambridge University Press.
Freemyers, S. (2004). Radiation pressure due to a localized
acoustical source (Master’s thesis). Naval Postgraduate
School, Monterey, CA.
Ivancic J., & Zrafi, M. (2011). Experimental Investigations
of the Acoustic Radiation Force (Master’s thesis).
Naval Postgraduate School, Monterey, CA.
Karakikes, V. (2006). Particle velocity transduction and
feedback control for acoustic radiation force
experiments (Master’s thesis). Naval Postgraduate
School, Monterey, CA.
Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J.
V. (2000). Fundamentals of acoustics, 4th ed. New York:
John Wiley & Sons.
Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics,
2nd ed. Oxford: Butterworth-Heinemann.
Nyborg, W. L. (1967). Radiation pressure on a small rigid
sphere. Journal of Acoustic Society of America, 42: 948–
952.
Oviatt, E., & Patsiaouras, K. (2009). Acoustic attraction
(Master’s thesis). Naval Postgraduate School, Monterey,
CA.

69
Sundem S., & Schock, M. (2005). Radiation force due to
diverging acoustic waves (Master’s thesis). Naval
Postgraduate School, Monterey, CA.

70
INITIAL DISTRIBUTION LIST
1. Defense Technical Information Center
Ft. Belvoir, Virginia
2. Dudley Knox Library
Monterey, California
3. Professor Bruce Denardo
4. Professor Gamani Karunasiri
5. Professor Daphne Kapolka

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