DSpace at VNU: Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom and in vitro bone studies

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Ultrasonics xxx (2013) xxx–xxx
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Contents lists available at ScienceDirect

Ultrasonics
journal homepage: www.elsevier.com/locate/ultras
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Excitation of ultrasonic Lamb waves using a phased array system
with two array probes: Phantom and in vitro bone studies

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a

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Department of Radiology and Diagnostic Imaging, University of Alberta, Edmonton, Alberta T6G 2B7, Canada

Department of Biomedical Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam
c
Department of Surgery, University of Alberta, Edmonton, Alberta T6G 2B7, Canada
b

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a r t i c l e

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Kim-Cuong T. Nguyen a,b, Lawrence H. Le a,⇑, Tho N.H.T. Tran a, Edmond H.M. Lou c

i n f o

Article history:
Available online xxxx

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Keywords:
Ultrasound
Phased array
Beam steering
Osteoporosis
Cortical bone

a b s t r a c t
Long bones are good waveguides to support the propagation of ultrasonic guided waves. The low-order
guided waves have been consistently observed in quantitative ultrasound bone studies. Selective excitation of these low-order guided modes requires oblique incidence of the ultrasound beam using a transducer-wedge system. It is generally assumed that an angle of incidence, hi, generates a specific phase
velocity of interest, co, via Snell’s law, hi = sinÀ1(vw/co) where vw is the velocity of the coupling medium.
In this study, we investigated the excitation of guided waves within a 6.3-mm thick brass plate and a
6.5-mm thick bovine bone plate using an ultrasound phased array system with two 0.75-mm-pitch array
probes. Arranging five elements as a group, the first group of a 16-element probe was used as a transmitter and a 64-element probe was a receiver array. The beam was steered for six angles (0°, 20°, 30°, 40°,
50°, and 60°) with a 1.6 MHz source signal. An adjoint Radon transform algorithm mapped the time-offset
matrix into the frequency-phase velocity dispersion panels. The imaged Lamb plate modes were identified by the theoretical dispersion curves. The results show that the 0° excitation generated many modes
with no modal discrimination and the oblique beam excited a spectrum of phase velocities spread asymmetrically about co. The width of the excitation region decreased as the steering angle increased, rendering modal selectivity at large angles. The phenomena were well predicted by the excitation function of
the source influence theory. The low-order modes were better imaged at steering angle P30° for both
plates. The study has also demonstrated the feasibility of using the two-probe phased array system for
future in vivo study.
Ó 2013 Elsevier B.V. All rights reserved.

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1. Introduction
Osteoporosis is a systemic skeletal disease characterized by
gradual loss of bone density, micro-architectural deterioration of
bone tissue, and thinning of the cortex, leading to bone fragility
and an enhanced risk of fractures. Cortical thickness of long bone
measurement has been investigated for the incidence of osteoporosis. Loss of cortical bone involves an increase of intracortical
porosity due to trabecularization of cortical bone [1,2] and cortical
thinning due to the expansion of marrow cavity on the endosteal
surface [3]. The cortical thicknesses at distal radius and tibia in
postmenopausal women with osteopenia were found to be thinner
than those of normal women in an in vivo study using highresolution peripheral quantitative computed tomography [4]. Recently, a high correlation was demonstrated between proximal
humeral cortical bone thickness measured from anteroposterior

shoulder radiographs and bone mineral density measured by

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⇑ Corresponding author. Tel.: +1 (780)4071153; fax: +1 (780)4077280.
E-mail address: (L.H. Le).

Dual-energy X-ray absorptiometry in an in vivo study for osteoporosis diagnosis [5].
Ultrasound has been exploited to study long bones using the socalled axial transmission technique, where the transmitter and the
receiver are deployed as a pitch–catch configuration with the receiver moving away from the transmitter. Since the acoustic
impedance (density  velocity) of the cortex is much higher than
those of the surrounding soft-tissue materials, the cortex is a
strong ultrasound waveguide. The propagation of ultrasound is
guided by the cortical boundaries and its propagation characteristics depend on the geometry (thickness) and material properties

(elasticities and density) of the cortex and the surrounding tissues.
Ultrasonic guided waves (GWs) propagate within long bone in
their natural vibrational modes, known as guided modes at different phase velocities, which depend on frequency. The GWs travel
longer distance and suffer less energy loss than the bulk waves because the boundaries keep most of the GW energies within the
waveguide.
The application of GWs to study long bones is quite recent but
the results so far are quite interesting. Nicholson et al. found the
velocity of the fundamental Lamb mode A0 differed by 15%

0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
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Kim-Cuong T. Nguyen et al. / Ultrasonics xxx (2013) xxx–xxx

between eight healthy and eight osteoporotic subjects (1615 m/s
versus 1300 m/s) [6]. The same group studied a population of
106 pubertal girls and also found the velocity of a slow-traveling
wave (1500–2300 m/s) consistent with that of the fundamental
A0 mode [7]. Protopappas et al. identified four low-order modes,
S0, S1, S2, and A1 in an ex vivo study of an intact sheep tibia [8].
Lee et al. found a strong correlation between the phase velocities

of A0 and S0 modes with cortical thicknesses in bovine tibiae [9].
Ta et al. found that the L(0, 2) mode was quite sensitive to the
thickness change in the cortex [10]. Basically in most studies, the
first few low-order guided modes have been consistently observed
and further studied for their potential to characterize long bones.
Guided modes are dispersive and might come close together,
posing a challenge for their identification. The ability to isolate
the guided modes of interest is the key for a successful analysis
of ultrasound data. Post-acquisition signal processing techniques
such as singular value decomposition [11], s À p transform [12],
group velocity filtering [13], dispersion compensation [14], and
the joint approximate diagonalization of eigen-matrices algorithm
(JADE) [15] are viable methods to separate wavefields. Guided
modes can also be selectively excited by using angle beam.
Preferential modal excitation and selectivity using angle beam
is widely used in ultrasonic non-destructive testing and material
characterization. It is generally assumed that given the compressional wave velocity of the angle wedge and an incident angle, only
a phase velocity is generated via Snell’s law. However in practice,
the ultrasound beam has a finite beam size and does not generate
just a single phase velocity for a given wedge angle. The element
size of the transducer and the incident angle influence the excitation of the GWs within the structure, which is generally known as
the source influence [16–18]. Instead of being excited with a definitive phase velocity (single excitation), GWs with a spectrum of
phase velocities are generated at oblique incidence. For normal
incidence, the phase velocity spectrum is very broad and dispersive, which implies infinite phase velocities to be excited, thus
making mode isolation difficult. For a fixed size transducer,
increasing the beam angle decreases the width of the phase velocity spectrum, thus generating fewer guided modes.
The use of angle beam to study long bone is very limited. Le et al.
used a 51° angle beam to study bulk waves at receivers deployed
downstream from the point of excitation [19]. Ta et al. used various
angle beams to excite low-order longitudinal modes and was the

first to mention briefly the concept of phase velocity spectrum in
the bone community without much details [10,20]. Although a pair
of transducers is still the most common means to acquire bone data,
ultrasound array system has been used in axial transmission bone
study [21]. The array system or multi-transmitter–multi-receiver
system has many advantages over single-transmitter–single-receiver system. The former has better resolution because of the smaller
element footprint, fast acquisition speed, accurate coordination of
the receivers, and less motion-related problems. In case the system
is a phased array (PA) system, beam steering is possible.
The objective of this work is to investigate the use of a PA system
to excite guided waves in brass and bone plates. The system has two
multi-element array probes with one acting as a transmitter and the
other as a receiver. The acquired data are processed and transformed
to the dispersion maps via an adjoint Radon transform. The theoretical dispersion curves based on plate models are used for modal
identification. We attempt to explain the variation of guided-wave
excitation with the incident angle using the source influence theory
(SIT). The novelties of our work lie in our employment of two phased
array probes and the use of Radon transform to estimate dispersion
energy. To our knowledge, these have never been done in the bone
community. While the SIT has been studied for a circular disk
transducer, we find it interesting to apply the theory to our data
acquired by a PA system.

2. Materials and methods

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2.1. Preparation of samples

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We performed experiments on a brass plate and a bovine bone
plate. The brass plate was 6.3 mm thick with a 255 mm  115 mm
surface dimension. We prepared a bone plate from a fresh bovine
tibia. The skin and soft tissue were removed. Using a table bandsaw, both ends of the tibia were cut and then the diaphysis was
cut along the axial direction to make a plate. Both surfaces of the
plate were sanded and smoothed by a disk sander. The resultant
bone plate had a relatively flat (190 mm  48 mm) surface area
with a thickness of approximately 6.5 mm. The top face was polished further to prepare the surface ready for the placement of
the probes.

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2.2. Ultrasound phased array system

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We used an Olympus TomoScan FOCUS LTTM Ultrasound PA
system (Olympus NDT Inc., Canada) with two array probes as
shown in Fig. 1a. The system has the following specifications:
0.5–20 MHz bandwidth, 20 kHz pulsing rate, 10-bit A/D converter,
and up to 100 MHz sampling frequency. Real-time data compression and signal averaging are also available. The scanner has a
high-speed data acquisition rate of 4 MB/s with maximum 1 GB file
size and 8192 data point per A-scan (or time series). The unit is
connected to a computer via Ethernet port. The Windows XP-based
computer was loaded with the TomoviewTM software (Version 2.9
R6) to control the data acquisition process and to modify the
parameters of the ultrasound beam such as scanning mode, beam
angle, focal position, and active aperture. The acquired data can
be exported to the computer for further post-acquisition analysis.

The scanner also supports multi-probe operations such as single-transducer-multi-element probe combination or two multi-

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Fig. 1. The ultrasound phased array system: (a) The TomoScan FOCUS LTTM phased
array acquisition system (1), the Windows XP-based computer with the TomoViewTM software to control the acquisition process (2), and the probe unit. (b) The
housing with the 16-element and 64-element probes. The P16 was the transmitter
array while the P64 was the receiver array.

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Kim-Cuong T. Nguyen et al. / Ultrasonics xxx (2013) xxx–xxx

element probes up to 128 elements. Beam steering and focusing
(transmit focus) at oblique angle can be achieved by electronically
delaying the firing of the elements without mechanical movement.
Receive-focusing is also possible. The recorded echoes are stored,
delayed, and then summed to produce an ultrasound signal. The
scanner was previously used to study scoliosis [22].
The two array probes used are the 16-element (2.25L16) and
64-element (2.25L64) array probes with a central frequency of
2.25 MHz (Fig. 1b). Here we denote them as P16 and P64 respectively. The two probes sat tightly within a housing, which was designed and built in-house to ensure the probes were stabilized and
the relative distance between them were fixed during data acquisition. The active areas of the P16 and the P64 probes are respectively 12 mm  12 mm and 48 mm  12 mm. Both have the same
pitch of 0.75 mm. Pitch is defined as the distance between the centers of two adjacent elements.
2.3. Data acquisition
The data were acquired using an axial transmission configuration. The experiment setup is schematically shown in Fig. 2. The
setup shows the arrangement of two probes (within housing) on
ultrasound coupling media, which were in contact with the underlying plate. The plate was a brass plate or a bone plate in our case.
Two pieces of 5-mm thick ultrasound gel pad, acting as coupling
medium, were cut from a commercial ultrasound gel pad (Aquaflex, Parker Laboratories, Inc., USA) with surface areas slightly larger so that the probes rested comfortably on the pads. The whole
set up was held in place by the 3MTM transpose medical tape.
The ultrasound gel (Aquasonic 100, Parker Laboratories, Inc.,
USA) was applied to all contact surfaces to ensure good coupling.
The experiments were performed at room temperature of 22 °C.
We chose to use five transducer elements as a group due to the
compromise between maximum steering angle and frame size
(number of acquired A-scan). The first five elements of P16 probe

were used as the transmitter. For the receivers, five elements
worked as a group and each group was spaced by one pitch
(0.75 mm) increment, that is, 1-2-3-4-5, 2-3-4-5-6, etc. The offset
spanned from 22.75 mm to 67 mm with an aperture of
44.25 mm. The scanner had an option to select source pulse of different dominant period, thus controlling the central frequency of
the incident pulse. We chose a pulse with 1.6 MHz central frequency. The calculated near field length L, was around 3.7 mm,
as given by L = kA2f/4v [23], where the aspect ratio constant k,
which is the ratio between the short and long dimensions of the
transmitter, is 0.99; the transmitter aperture, A is 3.75 mm for a
five-element source; the frequency, f, is 1.6 MHz; v is 1500 m/s,
the sound velocity in the ultrasound gel pad. The axial resolution
of the beam was 0.24 mm based on one-half of the pulse length.

There were 60 A-scans and each A-scan was 2500-point long with
a sampling interval of 0.02 ls. The data filled a 60 Â 2500 timeoffset (t À x) matrix of amplitudes. In our experiments, we steered
both the transmitter and receiver in sync at six angles: 0° (normal
incidence), 20°, 30°, 40°, 50°, and 60°. The synchronization at the
same inclination enhanced the sensitivity of the receivers to record
the guided waves traveling at the phase velocity of interest [24].
Depending on the steering angle, the calculated lateral resolution
ranged from 0.23 mm to 0.78 mm [25]. In this paper, beam was
steered at an incident angle and thus we use the terms, steering
angle and incident angle, interchangeably.

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2.4. Beam steering

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When an ultrasound beam incidents on the bone surface at an
angle, hi, a guided wave traveling with a phase velocity, co, between
the transmitter and receiver and along the bone structure (parallel
to the interface within the bone structure) will be generated
according to Snell’s law (Fig. 2):

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À1



hi ¼ sin

vw


ð1Þ

co

or

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ko ¼ kw sin hi ¼

x sin hi

vw

ð2Þ

where vw is the ultrasonic velocity of the coupling medium,
kw = x/vw is the incident wavenumber in the coupling medium,
ko = x/co is the horizontal wavenumber of the guided wave in the
cortex, and x is the radial frequency. Based on Eqs. (1) or (2), only
phases with a single phase velocity or wavenumber are generated,
which corresponds to a horizontal excitation line at co for all frequencies in the f À c panel. However, we observed more phase
velocities in our experimental data and thus Snell’s law was not

adequate to explain the phenomenon.
Based on the SIT [16,17], there exists an excitation zone where
guided waves traveling with phase velocities around co are excited.
The excited phase velocity spectrum is mainly governed by the size
of the transducer element and the incident angle and can be
approximated by the excitation function F for a piston-type source
of width, A [17],



ro jRðhi Þj
Aðk À ko Þ
Fðf ; cÞ ¼
sin
2ðk À ko Þ
2 cos hi

ð3Þ

where k = x/c and ro is the uniform pressure on the source surface.
The factor jR(hi)j accounts for the change in traction at the interface
and more detail about this factor is referred to [16,17]. In our work,
we assume ro and jR(hi)j take the values of unity and A = 3.75 mm
for a five-element source. For the six steering angles we considered
and using 1.6 MHz as the central frequency of the incident pulse,
their excitation spectra are shown in Fig. 3. We also follow Ditri
and Rose [16] to define a À9 dB phase velocity bandwidth, rÀ9dB,
for the source array:

rÀ9dB

cþ À cÀo
ðDco ÞÀ9dB
¼ o
¼
co
co
cÀ
o

Fig. 2. A cross-section of the experiment setup. The housing hosted two ultrasound
probes in place: a 16-element (P16) probe as the transmitter and a 64-element
probe as the receiver. The probes rested on the ultrasound gel pads, which acted as
coupling media. The pads then overlaid the plate. Only one group (five elements) in
P16 was used as source generator and 60 groups in P64 as receivers. The receivers
were steered at the same inclination as the transmitting beam to enhance the
receiving sensitivities to propagating guided waves with phase velocity, co related
to the inclination, hi by Snell’s law, sin hi = vw/co where vw is the velocity of the
coupling medium.

3

ð4Þ

cþ
o

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where

and
are the phase velocities smaller and larger than co
respectively when jFj drops by À9 dB of the maximum.

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2.5. Adjoint radon transform

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Following [12], we consider a series of ultrasonic time signals
d(t, xn) acquired at different offsets, x0, x1, . . . , xNÀ1 where t denotes
time and the x-axis is not necessarily evenly sampled. We write the
time signals as a superposition of Radon signals, m(s, p):

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Fig. 3. The normalized excitation spectra for six different steering angles. The velocity value shown above each figure is the phase velocity determined by Snell’s law (Eq. (1)
þ
in the text). The phase velocity determined by Snell’s law is denoted by co; The phase velocities, cÀ
o and c o , are defined at the values of jFj equal to À9 dB of the maximum; The
À
cÀ
p1 and c p2 refer to the phase velocities (
Kim-Cuong T. Nguyen et al. / Ultrasonics xxx (2013) xxx–xxx

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dðt; xn Þ ¼

KÀ1
X
mðs ¼ t À pk xn ; pk Þ;

n ¼ 0; . . . ; N À 1

ð5Þ

k¼0

where the ray parameter or slowness, p is sampled at p0, p1, . . . , pKÀ1,
and the intercept, s, also known as the arrival time at zero-offset, is
also discretized as s = gDt where g = 0, 1, . . . , G À 1. Taking the temporal Fourier transform of Eq. (5) yields

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Dðf ; xn Þ ¼

Mðf ; pk ÞeÀi2pfpk xn :

Rewriting Eq. (6) using matrix notation for each frequency gives

D ¼ LM

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and

2

eÀi2pfp0 x0
6.
L¼6
4 ..
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305

ð6Þ

k¼0

298
300

302

K À1
X

eÀi2pfp0 xNÀ1

ð7Þ

ÁÁÁ
..
.

eÀi2pfpKÀ1 x0
..
.

Á Á Á eÀi2pfpKÀ1 xNÀ1

3
7
7:
5

ð8Þ

The adjoint Radon solution is given by

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308

M ¼ LH D

309

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where LH is the adjoint of the operator L and H denotes the complex
conjugate transpose. The adjoint Radon solution is slightly better
than the Fourier transform, yields better results at fixed number
of records, and is less susceptible to aliasing problem. Finally, the
dispersion curve or f À c panel is obtained by replacing p = 1/c in
M. To avoid aliasing, sampling in slowness, Dp, must be smaller
than 1=ðr aper fmax Þ where raper is the acquisition aperture and fmax is
the maximum frequency of the data [26].

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3. Results and discussion

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3.1. Brass plate

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Prior to dispersion analysis, the acquired t À x data underwent
some simple but essential signal processing steps. First, the triggers were muted. Second, the mean amplitude was subtracted
from the data to remove the background. There was insignificant

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321
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ð9Þ

energy beyond 1.0 MHz and the data were band-pass filtered with
a trapezoidal window (0.1, 0.15, 0.9, 1.0 MHz).
Fig. 4 shows the Radon panels or dispersion panels of the brass

plate for the six steering angles: 0° (normal incidence), 20°, 30°,
40°, 50°, and 60°. Also indicated on the figures are the excited
phase velocities, co, as predicted by Snell’s law (Eq. (1)) given the
corresponding steering angles. We also superimposed the theoretical dispersion curves of the Lamb modes on the figure. We used
the commercial DISPERSE software [27] to simulate the dispersion
curves based on an elastic plate model. The material properties of
the brass plate are listed in Table 1. Before we discuss each case in
detail, several general observations can be made when the steering
angle changes from normal incidence to 60°. First, the excitation
does not generate GWs of mono phase velocity as predicted by
Snell’s law. Instead a spectrum of phase velocities is excited. Second, as the steering angle increases, less high-velocity GWs are excited and more low-velocity GWs are generated and focused. Third,
the phase velocity spectrum becomes smaller and more selective
as the steering angle increases.
As shown in Fig. 4, the normal beam (Fig. 4a) lacks the focusing
power and gives rise to a wide spectrum of GW energies of all frequencies. The normal beam excites more energetic high-velocity
GWs than the low-velocity GWs. The first three antisymmetric
A-modes (A0, A1, and A2) and the first four symmetric S-modes
(S0, S1, S2, and S3) can be identified. The majority of the energies lies
above 4 km/s and between 0.5 MHz and 0.9 MHz. The three strongest modes are A2, S2 and S3. The low-velocity GW energies are very
weak. The beam excites the higher frequency portion of the loworder modes A0 and S0 at 0.8 MHz where they converge into a small
energy cluster around 1.95 km/s. At 0°, the Snell’s law-predicted
phase velocity is a very large value or, theoretically speaking, infinity, indicating that the high-velocity GWs are favorably excited. It
is quite challenging to identify and isolate guided modes using
normal beam excitation. At 20° incidence (Fig. 4b), the beam excites guided waves around the predicted phase velocity of
4.38 km/s. All seven previously-identified guided modes are present but their cluster peaks show up at different (higher) frequencies. The low-velocity GWs below 4.38 km/s are energetic. The
beam excites strong S0 energy at low 0.2 MHz and beyond. However above 0.4 MHz, the dispersion curves of the A0 or S0 come together and it is difficult to tell which mode the energy clusters

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5

þ
À
À
Fig. 4. The dispersion panels of the brass plate data for the six different steering angles. Superimposed are the theoretical dispersion curves. The co ; cÀ
o ; c o ; c p1 , and c p2 are
referred to Fig. 3 for their definitions.

Table 1
Parameters used to simulate dispersion curves for the brass plate and bone plate. The
compressional wave velocity (vp) and shear wave velocity (vs) of the brass plate were
taken from Table A-2 of [23] while the density (q) was measured. The vp, vs, and q of
the bone plate were taken from [28] while the attenuation coefficients, ap and ap were
from [19]. We also measured the vp of the brass and bone plates and the
measurements were 4.56 km/s and 4.09 km/s respectively, which are very close to
the reported values in the literature [23,28].

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365
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Sample

Brass plate

Bone plate

Thickness (mm)
vp (km/s)
vs (km/s)
q (kg/m3)
ap (dB/MHz cm)
as (dB/MHz cm)

6.3
4.43
2.12
8440

6.5
4.0
1.8
1850
5.0
11

between 0.6 and 0.8 MHz belong to. When the steering angle increases to 30° (Fig. 4c), the predicted phase velocity is 3.0 km/s.
The number of GW modes, especially the high-velocity modes
above 4.0 km/s, reduce and the low-velocity modes around
1.95 km/s are more enhanced. At this angle, we only identify five
significant modes, A0, S0, A1, S1, and A2 with the majority of the
GW energies lying between 0.4 and 0.9 MHz. For the A0, S0, A1,
and S1, the simulated dispersion curves match very well with the
corresponding modal clusters with the curves going through the
first two low-order modal clusters on their tracks. It is interesting
to note from these dispersion curves that the same modal energies

are not continuous on their respective dispersion curves. This
might imply that the modes experience strong attenuation at certain frequencies. The first two low-order modes consistently show
their presence between 0.5 MHz and 0.9 MHz with progressively

stronger energies than those at smaller steering angles. The last
three modes start to lose their strength as the steering angle increases. The weakening of the A1 mode is obvious. When the steering angle increases from 40° (Fig. 4d) to 60° (Fig. 4f), the predicted
imaged phase velocities drops from 2.33 km/s to 1.73 km/s and all
high-velocity GWs become less visible. Only two low-velocity A0
and S0 modes exist and are imaged with enhanced focus and resolution. In addition, more energy is excited at frequencies lower
than 0.4 MHz. At 60° incidence, there is some 0.4 MHz energy traveling around 1.8 km/s, which seems to belong to the A0 mode.
Overall, the dispersion curves match the excited Lamb modes reasonably well.

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3.2. Bovine bone plate

391

We investigated further the guided mode selectivity and focusing by beam-steering on a bovine bone plate. The data was processed using the same procedures and parameters as those for
the brass plate data. The dispersion panels (Fig. 5) show the same
general observations as those in the brass plate. Notably, the ultrasound beam energizes a spectrum of phase velocities and at smaller steering angles, the beam favors the excitation of high-velocity
GWs. As the steering angle is more oblique, the spectrum is
narrower and the beam is more selective toward low-velocity
excitation.
The predicted phase velocities, co, for the six steering angles remain the same as those for brass plate (Table 2). The normal beam
excites a wide spectrum of phase velocities with an emphasis on
high-velocity GWs (Fig. 5a). The energetic GWs travel at velocities
above 3.5 km/s. There are a few weak and small energy clusters.
We calculated the dispersion curves using a plate model with

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Q1 Please cite this article in press as: K.-C.T. Nguyen et al., Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom
and in vitro bone studies, Ultrasonics (2013), />
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þ
À
À
Fig. 5. The dispersion panels of the bone plate data for six different steering angles. Superimposed are the theoretical dispersion curves. The co ; cÀ
o ; c o ; c p1 , and c p2 are referred
to Fig. 3 for their definitions.

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absorption (Table 1). The bone plate model is different from the
brass plate model in shear wave velocity and density. The match
is not perfect with some clusters lying between dispersion curves.
We managed to identify eight Lamb modes: 4 antisymmetric
modes, A0 À A3 and 4 symmetric modes, S0 À S3. The two highintensity clusters are the fast-traveling A2 and S2 modes. The A1
curve goes through a low-frequency cluster ($0.32 MHz) and a
high-frequency cluster ($0.75 MHz). In the neighborhood of
1.9 km/s, there is a noticeable fragmented band of weak energies
at 0.25 MHz and between 0.7 and 0.8 MHz, which is better imaged
in the 20° panel (Fig. 5b). Except the A3 mode, which is absent in
the brass plate dispersion panel, the modes in the brass plate
and bone panel are very similar with A2 and S2 being the two strongest modes. As the steering increases, this group of low-velocity
energies receive enhanced focusing and imaging resolution while
the high-velocity phases become weaker, say the A2 and S2 modes
for example. At 30° (Fig. 5c), only GWs traveling below 4.2 km/s are
energized. As the steering angle increases from 30° (Fig. 5c) to 60°
(Fig. 5f), only the bundle of GW energies bounded between A0 and

S1 and over 0.5 MHz are imaged while the rest of the GWs lose
their intensity and become hardly visible. At 50° (Fig. 5e) and 60°

(Fig. 5f), this band of low-order energies experience enhanced
excitation. It is worth mentioning here that the adjoint Radon
transform is able to resolve these closely packed energy clusters.
The steered beam greatly enhances and focuses the slow-traveling
(around 1.75 km/s) small energy cluster around 0.3 MHz, which lie
between the A0 and S0. Quite similar to the brass plate, the A0 and
S0 modes persist in all steering angles.

429

3.3. Excitation function

436

The SIT predicts that the loading size of the transducer influences the range of phase velocities generated by the transmitting
source [16,17]. For a given angle of incidence, the beam does not
generate a single phase velocity (based on Snell’s law) but a spectrum of phase velocities. The GW modes with dispersion curves
passing through the phase velocity zones for a given frequency
has greater ‘‘chance’’ to be excited as compared to the portion of
the curves, which are far away from the zone. This excitation probability is provided by the excitation function defined by Eq. (3).
However, the strength of the excitation at that particular frequency
is governed by the excitability function of the theory. As shown in

437

Table 2
À

Parameters for the À9-dB phase velocity bandwidth for five steering angles. The cÀ
p1 and cp2 refer to the phase velocities (hi (°)

co (km/s)

cþ
o (km/s)

cÀ
o (km/s)

(Dco)À9dB (km/s)

rÀ9dB

cÀ
p1 (km/s)

cÀ
p2 (km/s)

20
30
40
50
60

4.38
3.00

2.33
1.96
1.73

8.54
4.32
2.95
2.30
1.93

2.95
2.29
1.93
1.71
1.57

5.59
2.03
1.02
0.59
0.36

1.28
0.68
0.44
0.30
0.21

2.22
1.84

1.62
1.50
1.42

1.62
1.44
1.34
1.30
1.27

Q1 Please cite this article in press as: K.-C.T. Nguyen et al., Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom
and in vitro bone studies, Ultrasonics (2013), />
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484

the results, even though the dispersion curves of the modes run
through the À9 dB phase velocity bandwidth, the modal energies
are sporadic and not continuous along the dispersion tracks. It is
uncertain that the absence of some frequency components is due
to the attenuation of those components or the preferential modal
excitation. Studying the excitability function is beyond the scope
of this study. Here, we attempt to use the excitation function of
the source influence theory to explain the observed behaviors of
the dispersion energies with steering angles.
The À9 dB bandwidth defines a range of phase velocities within
which significant excitation may happen. The bandwidth parameters are tabulated in Table 2. An example of how to delineate the
bandwidth of a spectrum and other relevant parameters is provided by the 20° case in Fig. 3b. As shown in Fig. 3, the normal
beam excitation function exceeds the À9 dB value over an ‘‘infinitely’’ wide phase velocity range and there is basically no selectivity to phase velocity. This is evident in both data sets
(Figs. 4 and 5a). The selectivity to phase velocities improves when
the steering angle increases. The bandwidth decreases from 128%
to 21% of co when the incident angle increases from 20° to 60°,
offering enhanced modal selectivity. The bandwidth-narrowing,
as predicted by the excitation function, is supported by our experimental data. While the upper boundary of the observed GW region is well defined by cþ

o , the lower boundary of the region is
less so by the cÀ
o . For the two largest steering angles, 50° (Figs. 4
þ
and 5e) and 60° (Figs. 4 and 5f), the cÀ
o and co are the lower and
upper boundaries of the observed phase velocity spectra. For the
þ
brass plate, the cÀ
o and co are sufficient to define the 40° phase
À
velocity spectrum (Fig. 4d). However the cÀ
p1 and c p2 are required
to define the lower bounds of the observed phase velocity regions
for the 30° (Fig. 4c) and 20° (Fig. 4b) respectively. For the bone
plate, the cÀ
p1 fixes the lower bounds of the 30° (Fig. 5c) and 40°
(Fig. 5d) while the cÀ
p2 outlines the lower bound for the 20°
(Fig. 5b). Although the À9 dB bandwidth defines the dominant
phase velocity region where the excitation may happen, the excitation can happen in the sidelobes of the excitation function as illustrated by the dispersion panels of the experiment data.

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4. Concluding remarks

486

In this study, we investigated the use of a commercial nonmedical PA system to excite GWs within a brass plate and a bone
plate with two array probes. Acquisition with two probes not only

eliminated the crosstalk between transducer elements but also
allowed adjustable long offset for GW buildup. By using a fixed
five-element of a 16-element probe as the loading and a 64element receiver probe, many energetic fast and slow GWs of a
wide frequency range were excited and observed in both plates.
We also studied the effects of modal selectivity within the plates
by beam-steering. By varying the angle of the steering beam, the
excited bandwidth of the phase velocity spectrum changed, in
good agreement with the prediction by the excitation function of
the source influence theory. Consequently, modal selectivity is
possible by choosing larger steering angle. The results of this study
allow us to consider a PA system for clinical work. The acquisition
time by a PA system is reduced significantly by 100 fold without
any mechanical probe movement as compared to a single-transmitter-single-receiver system and problems relating to inaccurate
recording transducer’s coordination, patient discomfort, and
patient motion are minimized if used in clinical settings. The
low-order low-velocity GW modes have been consistently
observed in both plates. The adjoint Radon transform has successfully extracted the dispersion energies with good resolution. Using
the PA system in combination of low frequency toneburst for
in vivo study will be the next avenue for our future studies.

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Acknowledgment

511

KC Nguyen wants to thank Ministry of Education and Training
of Vietnam for the award of a 322 scholarship to make the study

possible. We also thank Devlin Morrison for building the transducer housing.

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Q1 Please cite this article in press as: K.-C.T. Nguyen et al., Excitation of ultrasonic Lamb waves using a phased array system with two array probes: Phantom
and in vitro bone studies, Ultrasonics (2013), />