This presentation was given to my PhD advisory committee in May 2010

1. NC STATE UNIVERSITY
Glenwood Garner
SIAMES Research Group
North Carolina State University
Linear & Nonlinear Acoustic
Modeling for Standoff Analysis
Ph.D. Preliminary Examination
May 20, 2010
10:00 am, MRC 463
ERL Anechoic Chamber Audio Spotlight
Polytec PDV-100

2. NC STATE UNIVERSITY
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Presentation Overview
• Motivations, Objectives
• Original Contributions
• Third-Order Nonlinear Scattering
• Fractional Calculus Spatial Power Law Model
• Switched Tone Probing
• Shear Wave Elasticity Imaging (SWEI)
• Future Work

3. NC STATE UNIVERSITY
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Linear & Nonlinear Acoustic Modeling for Standoff Analysis
Table of Contents (abridged)
I. Introduction
1. Overview
2. Original Contributions
3. Dissertation Outline
4. Published Works
I. Literature Review
1. Linear Acoustic Theory
2. Nonlinear Acoustic Theory
3. Piston Radiators
4. Sound Scattering
5. Fractional Power Law Theory
6. Acoustic Fluid-Solid Interaction
7. Transient Effects
8. Shear Wave Elasticity Theory
I. Third-Order Sound Scattering
1. Second Harmonic Characterization
2. Receiver Characterization
3. Numerical Techniques
4. Measurement
5. Results & Analysis
IV. Fractional Diffusion Acoustic Piston Radiation
1. Finite Difference Method
2. Derivation of Spatial Power Law
Dependence
3. Measurements
4. Results & Analysis
IV. Switched Tone Transient Analysis
1. Long-Tail Transients
2. Log-Decrement Linear Metrology
3. Nonlinear Switched-Tone Model
4. Switched-Tone Measurement
5. Results & Analysis
IV. Standoff Shear Wave Elasticity Imaging
1. Comparison Between Contacting & Non-
contacting Methods
2. Application
3. Measurements
4. Image Processing
5. Results & Analysis
IV. Conclusion & Future Work

4. NC STATE UNIVERSITY
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Standoff Acoustic Analysis: Motivations
8
9
3
3 10
8.6
35 10
343
8.6
40 10
mm
mm
λ
λ
×
= =
×
= =
×
24,000 people killed or injured annually by over 100 million worldwide
landmines, unexploded ordnance, and IEDs.
• Human Prodders
• Dogs
• Metal Detectors
• Infrared
• Neutron Backscatter
• Millimeter Wave Detection
• Ground Penetrating Radar
• Acoustics
Lives endangered
High false alarm rate
Intolerant to soil
moister/dielectric
constant
Better for detecting AT mines
Mechanical effects, limited
jamming, shielding, spoofing
Gros (1998)
Can we used acoustics to probe our
environment for abnormalities?
Mazzaro (2009)

5. NC STATE UNIVERSITY
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Standoff Acoustic Analysis: Motivations
Two-tone probing has been demonstrated in finding AP & AT mines.
http://www.acoustics.org/press/154th/fillinger.html
Current Technology Drawbacks
• Requires ground shakers…or
• Large amplitude loudspeakers
• Not a standoff technique
• Inherent nonlinearity of air
• Acoustic beam spreading Research Objectives
• Develop third-order nonlinear air
model
• Model thermal diffusion of piston
sound beam
• Explore nonlinear probing
techniques
• Adapt current techniques to
standoff applications.
Donskoy (2002), Korman (2004), Sabatier (2003)

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Standoff Acoustic Analysis: Original Contributions
• Developed third-order nonlinear air model, demonstrated
use of cascaded second-order systems.
• Demonstrated improved directivity of third-order nonlinear
parametric array.
• Fractional calculus spatial power law model for piston
beams.
• Evaluation of long-tail transients in geologic materials
• Use of switched tones to generate third-order
intermodulation.
• Standoff application of shear wave elasticity imaging
(SWEI) to locate inhomogeneities in targets.

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Standoff Acoustic Analysis: Publications
Glenwood Garner, Jonathan Wilkerson, Michael M. Skeen, Daniel F. Patrick,
Ryan D. Hodges, Ryan D. Schimizzi, Saket R. Vora, Zhi-Peng Feng,
Kevin G. Gard, and Michael. B. Steer
“Acoustic-RF Anechoic Chamber Construction and Evaluation”
Radio & Wireless Symposium, Orlando, FL, 2008.
Glenwood Garner III, Jonathan Wilkerson, Michael M. Skeen, Daniel F. Patrick,
Hamid Krim, Kevin G. Gard, and Michael B. Steer
“Use of Acoustic Parametric Arrays for Standoff Analysis and Detection”
Government Microcircuit Applications Conf., Las Vegas, NV, 2008.
Glenwood Garner III, Marcus Wagnborg, and Michael B. Steer
“Standoff Acoustical Analysis of Natural and Manmade Objects”
Government Microcircuit Applications Conf., Orlando, FL, 2009.
Glen Garner, Jonathan Wilkerson, and Michael B. Steer
“Third-Order Distortion of Sound Fields”
Government Microcircuit Applications Conf., Reno, NV, 2010
Glenwood Garner,and Michael Steer
“Nonlinear Propagation of Sound in Air”
IEEE Transactions on Geoscience and Remote Sensing (unpublished)
Conference
Journal

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Third-Order Nonlinear Scattering: Overview
*Taken from: http://spie.org/Images/Graphics/Newsroom/Imported/0569/0569_fig1.jpg
When two sound beams (plane
waves) interact at angle θ, phase
interference produces Moiré
bands*.
Westervelt (1957)
22
2 2
02
ij
i j
T
c
t x x
ρ
ρ
∂∂
− ∇ =
∂ ∂ ∂
2
2
02 2
0
1
, wheres
s
p q
p
c t t
ρ
∂ ∂
∇ − = −
∂ ∂ 0
2
20
2 4 2 2
0 0 0
1 1
cos
2
T
P
q p
c c tρ ρ
ρ
θ
ρ ρ =
  ∂ ∂
= +  ÷
∂ ∂   
Lighthill (1952)
c: sound speed
p: pressure (total, scattered, incident)
ρ: density
a: transmitter radius
q: simple source density
Mixing Volume ( , )f a θ=

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0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
σ=βωr/c
0
2
HarmonicAmplitudeB(n)
(σ)
1st Harmonic
2nd Harmonic
3rd Harmonic
Third-Order Nonlinear Scattering: Overview
Ingard & Pridmore-Brown (1956)
1 1 1
2 2 2
cos( )
cos( )
Tp p t k x
p t k y
ω
ω
= −
+ −
1 1 1
2 2 2 2
cos( )
cos( cos sin )
Tp p t k x
p t k x k x
ω
ω θ θ
= −
+ − −
( ) ( )
( )
2
0
1
1
sin cos
2 ( sin )
sin
sin
ir
T i i i i
i
i
i
i
r
p D k a p e t k r
r
J k a
D k a
k a
α
θ ω
θ
θ
θ
−
=
= −
=
∑
Muir & Willette (1972)
Lauvstad &Tjotta (1962)
Garner & Steer (2010)
( ) ( )
( ) ( )
(1) (2)0
1 1 1 1 1 1 1 1
(1) (2)
2 2 2 2 2 2 2 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
T
r
p p B D p B D
r
p B D p B D
σ θ σ θ
σ θ σ θ
= +
+ + 
r0/r=1
Lockwood, Muir, &
Blackstock (1972)
( )0 0( ) lnD kr r rσ βε θ=

10. NC STATE UNIVERSITY
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0 1 2 3 4 5 6 7 8 9
0
20
40
60
80
100
120
Intermodulation Power Sweep f1
= 55 kHz, f2
= 65 kHz
Amplitude(dBSPL)
Input Voltage (dB)
f
1
55 kHz
f
2
65 kHz
IM3L 45 kHz
IM3U 75 kHz
f
1
attenuated
f2
attenuated
IM3L attenuated
IM3U attenuated
4.5 5 5.5 6 6.5 7 7.5
x 10
4
0
2
4
6
8
10
Ultrasonic Attenuation
Attenuation(dBSPL)
Frequency (Hz)
Single Tone
Intermodulation Tones
Third-Order Nonlinear Scattering: Distortion
Transmitter and receiver nonlinearity must be characterized in all
nonlinear measurements.
p = 90 dB, r = 2m, f = 40kHz
a = 0.22 m a = 0.016 m
IM3 Power sweep for 55kHz & 65 kHz
input frequencies, with & without acoustic
attenuator (melamine foam) over
microphone

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Third-Order Nonlinear Scattering: Measurement
( ) ( )
2 2
0 0
1 0
0
1 0
0
' sin cos
cos
cos for 0
'
'
cos
cos for 0
'
m m
m
m m
m
m m
r r
r
r
r
r
ϕ ϕ
ϕ
ϕ ϕ
ϕ
ϕ
ϕ ϕ
−
−
= + + −
  + −
− + <  ÷
  
= 
 + −− − > ÷
 
l l l
l l
l l
• l = 2 meters
• θ = 20º
• f1 = 55 kHz, f2 = 65 kHz
• p1 ≈ p2 ≈ 125 dB SPL
• r0 = 7 meters
• Independent signal generators and
amplifiers to eliminate electrical
intermodulation.
• Conducted outside to prevent
standing waves
• Absorber used to mitigate side
lobe interaction.

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Third-Order Nonlinear Scattering: Measured Fields
Very good agreement between measured and theoretical IM3. Scattered
third-order 3 dB beam widths of 1.7 degrees. To be included in 1st
journal paper (IEEE GRS), demonstrate directivity of third-order
nonlinear air model.

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Westervelt (1963), second-order scattering (red)
• f1 = 64 kHz, f2 = 66 kHz
• p1 = p2 = 111 dB SPL
• a = 0.1 meters
• ~18º beam width
-1.5 -1 -0.5 0 0.5 1 1.5
0
5
10
15
20
25
30
Amplitude(dBSPL)
φ (radians)
Directivity Comparison
IM3 Lower Theory
Diff. Freq. Theory
Third-Order Nonlinear Scattering: Directivity
( )
2 2 2
4
0 0 04 2 cos
si t
s
s
s s
p a e
p
r c ik ik
ω
β ω
ρ α θ
−
=
− +
Third-order scattering (black)
• f1 = 33 kHz, f2 = 64 kHz
• p1 = p2 = 127 dB SPL
• a = 0.1 meters
• ~1.7º beam width
At 2 kHz, you get a significant gain in directivity with third-order
scattering compared to second-order scattering

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Fractional Diffusion Model: Overview
Used to model long-tail effects (stretched exponentials), power-law
attenuation, and fractal geometries.
1( )( )
0
2 2
1
( ,0) j t krj t kr
p r p e e
r r a
ωω −−
 = − 
= +
0 0.5 1 1.5
0
1
2
On-axis pressure, f = 80000, a = 0.016, r0
= 0.18758
distance (m)
amplitude(Pa)
10
-4
10
-3
10
-2
10
-1
10
0
0
1
2
distance (m)
amplitude(Pa)
Blackstock (2000), Zemanek (1971)

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Fractional Diffusion Model: Overview
K
Fractional order element, Schiessel (1995)
( )
1
0
1 ( )
( )
( ) ( )
t m
m
d f f
D f t d
dt m t
α
α
α α
τ
τ
α τ + −
= =
Γ − −∫
• Fractional derivatives produce memory
effects
• Allow for smooth transistion between
diffusive and wave phenomenon.
• Good at capturing derivatives of varying
scale.
2
2
2
u u
b
t x
α
α
∂ ∂
=
∂ ∂
• α = 1: Diffusion
• α = 2: Wave Equation
Caputo Fractional Derivative
allows traditional initial & boundary conditions
Agrawal (2002)
(0, ) ( , ) 0, 0
( ,0) ( ), 0
u t u L t t
u x f x x L
= = ≥
= < <
(0, ) cos( ), 0
( , ) 0
u t t t
u t
ω= ≥
∞ =
Typical
boundary
conditions
Piston source
boundary
conditions
(extend to 3D)
Can we find an efficient, fractional order solution to model thermal
diffusion of piston source?
Mainardi (1994), Podlubny (1999)

16. NC STATE UNIVERSITY
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Fractional Diffusion Model: Overview
Current fractional models aim to capture fractional order power-
law attenuation.
( )
0
0
0
2
2 2
2 2
0
1
0
z
z
z
u
u u
c t t
τ
∂ ∂
∇ − + ∇ =
∂ ∂
( ) 0
2
2
2
0
1
1
z
k
c i
k i
ω
ωτ
β α
=
+
= −
Begin by adding a fractional viscous loss term to wave equation
Apply Fourier time/space fractional
derivatives to obtain the following dispersion
relation
where and using the Szabo
smallness approximation
cβ ω=
0 1
0
0 0
0
0
sin
2 2
z
z z
c
α α ω
τ π
α
+
=
=
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
500
550
600
650
700
750
ω/c = 732.733
propagationfactor(β)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-15
-10
-5
0
fractional derivative power (z
0
)
attenuation(αNp/m)
40 kHz
No viscosity Classical viscosity
Chen (2004), Holm (2010)

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Can a spatial fractional inverse power law be derived (similar to
computing a fractional dispersion relation) to model beam spreading
from baffled vibrating piston source?
0 1 2 3 4 5 6 7 8 9 10
x 10
5
0
50
100
150
200
Frequency (Hz)
Attenuation(dB/m)
tem p. = 68, hum idity = 50
Analytic Attenuation
Power Law Fit, α 0
= 6.0156e-012, z0
= 0.837
Fractional Diffusion Model: Method
0
y
α α ω=
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0
0.5
1
1.5
2
2.5
f = 80000, theta = 0, p0
= 0.18758, y = 1
distance (meters)
amplitude(Pa)
Exact field pressure
Fractional power law fit
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0
0.1
0.2
0.3
0.4
f = 80000, theta = 21.15, p
0
= 0.008, y = 1.3
distance (meters)
amplitude(Pa)
Exact field pressure
Fractional power law fit
0
y
p p r−
=

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Switched-Tone Probing: Overview
Can the problems associated with standoff nonlinear analysis be avoided
using switched tones?
f1
f1f2
f2
• Inherent nonlinearity of air and
non-perfect collimation cause
intermodulation.
• Due to lack of spatial (velocity)
dispersion, switched tones
remain separated in time and
no intermodulation is
generated.
• If target has long-tail
“ringing”, intermodulation is
generated on the target’s
surface.
• Surface velocity is measured
with laser doppler vibrometer
(LDV).

19. NC STATE UNIVERSITY
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Switched-Tone Probing: Physical Model
Negative Stress -TJ Voltage V
Particle Velocity vi Current I
Body Force per unit
volume
Fi Source Voltage per unit
length
Vs
Mass per unit volume ρ Inductance per unit length L
Inverse stiffness coefficient cJJ
-1
Capacitance per unit length C
Inverse damping coefficient ηJJ
-1
Conductance per unit length G
2
L
zδ 2
L
zδ
C zδ
z →
V
+
−
I →
zδ¬ →
( )i
JJ JJ J
v
c T
t z t
η
∂∂ ∂ 
+ = − − ÷
∂ ∂ ∂ 
Auld (1990)
0 0.2 0.4 0.6 0.8 1
-30
-20
-10
0
10
20
30
Time (sec)
AcousticPressure(Pa)
(a) Incident Mic Response
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
(b) LDV Response: Tau=0.17631, Q=1988.6441, fo
=3590.2713
Time (sec)
Velocity(mm/s)
velocity
peak detection
exponential fit
1
2
2
2
ln
1
n d
x
T
x
πς
δ ςω
ς
= = =
−
1
2
Q
ωτ
=
Log Decrement
Walker (2007), RF continuous excitation
Mazzaro (2009), RF switched tones
Parker (2005), biological tissues

20. NC STATE UNIVERSITY
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Switched-Tone Probing: Physical Model
ADS Circuit Model
Surface Velocity (Current) response to 25
Pa, 200 ms input signals
590 Hz 840 Hz
Can you model physical resonance
as an RF filter?
• Probe a target using switched tones to
characterize material type, density,
composition.
• Try using electrical diode model to
capture physical nonlinearity.
• Goal: Demonstrate on uniform
samples shapes to verify this
phenomenon is a material property
• Goal: Demonstrate invariance to
sample shape.

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Switched-Tone Probing: Current Results
• Third-order intermodulation generated by
switched tones in metal, plywood, and
fiberglass.
• Did not see this response in microphone placed
next to targets.
• Need to employ windowed FFT and shorter
duration tones to eliminate possibility of
standing wave error & parametric effect.
• Applied Physics Letters draft completed.

22. NC STATE UNIVERSITY
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Shear Wave Elasticity Imaging: Overview
SWIE is a linear metrology technique that relies on a localized change in
stiffness to indicate an inhomogeneity.
2
2
2 2
0
1
0
c t
ξ
ξ
∂
∇ − =
∂ ( )
2
2
2 2
0
1
0
1 ( )c x t
ξ
ξ
γ
∂
∇ − =
+ ∂
r
( )
2
2 1
E
c
ρ ν
=
+
x
r
( )0,0
Modify the general linear shear wave
equation to have piecewise Young’s
modulus of elasticty.
Shear waves typically excited using low
frequency shaker and measured using
Doppler ultrasound (contacting method).
Propose using low frequency speaker (sub-
woofer) and LDV to measured surface
velocity/displacement (standoff method).
Gao (1995), Parker (1996) – Sonoelasticity Imaging
0E
'( )E x

23. NC STATE UNIVERSITY
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Shear Wave Elasticity Imaging: Application
Can we use SWEI to model and measure the vibration pattern of
materials at standoff ranges?
Expanded polystyrene
model (StyrofoamTM
)
3
10kg mρ =
0.03 0.07ν = −
0.830 GPaE =
1,1 690 Hzf =
0.225 m
0.19 m Predicted (1,1) mode (converted to velocity)
and velocity measured with LDV
To do: Verify predicated amplitudes are correct, measure higher modes,
measure different materials.

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Shear Wave Elasticity Imaging: Application
Can we use SWEI to detect dense objects hidden within soft targets?
0.225 m
0.19 m 5 cm 7.5 cm
inhomogeneity
×
580 Hz excitation, 2.2 m standoff distance
Medical Imaging
0
0 0
'( ) 8
', ' ,
E x E
ρ ν ρ ν
≈ ×
≈
Standoff Detection
0
0
'( ) 241,000
' 700
E x E
ρ ρ
≈ ×
≈ ×
To do: Implement inhomogeneous lossy model, determine maximum
detection depths in various materials.
Incredible difference in
stiffness & density
compensate for poor
solid-air matching.

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Standoff Acoustic Analysis: Remaining Tasks
• Implement numerical integration for third-order scattering model?
• 90% MATLAB code finished
• Submit to IEEE Geosciences & Remote Sensing
• Develop fractional diffusion model
• Complete switched-tone probing measurements
• Determine best time domain methods and signal processing
• Use shorter pulses to mitigate parametric effect
• Submit to Applied Physics Letters
• Complete standoff SWEI measurements
• Different materials/inhomogeneities
• Geologic Materials/non-parallelepiped shapes
• Submit to IEEE Geosciences & Remote Sensing
Questions?