Wave Motion 46 (2009) 451–467
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Wave Motion
journal homepage: www.elsevier.com/locate/wavemoti
Understanding a time reversal process in Lamb wave propagation
Hyun Woo Park a, Seung Bum Kim b, Hoon Sohn c,*
a
b
c
Department of Civil Engineering, Dong-A University, Busan 604-714, Republic of Korea
Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA
Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea
a r t i c l e
i n f o
Article history:
Received 8 May 2008
Received in revised form 27 April 2009
Accepted 30 April 2009
Available online 9 May 2009
Keywords:
Lamb wave
Time reversal process (TRP)
Within-mode dispersion
Multimode dispersion
Reflections
Amplitude dispersion
Regular waveguide
Piezoelectric transducer
Reference-free NDT
a b s t r a c t
This study investigates the time reversal process (TRP) of Lamb wave signals which are
transmitted and received by piezoelectric transducers bonded on plate-like structures. A
number of previous studies have paid attention to spatial and temporal refocusing capability of an original excitation through the TRP in highly dispersive and complex media. However, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, the
refocusing capability is limited due to permanent residual side bands even if the duration
of the time reversed signal increases. Based on the reciprocity of elastodynamics and linear
piezoelectricity, theoretical interpretation is conducted for the main and residual side
bands of the reconstructed signal in the time domain. In particular, the interpretation
includes the temporal effect of velocity and amplitude dispersions, the existence of multimodes, and the reflections from boundaries during the TRP. Then, numerical and experimental tests are conducted to validate the theoretical findings of this paper. Practical
issues for the successful implementation of the TRP of Lamb waves are briefly addressed
as well.
Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
According to conventional time reversal acoustics, an input signal can be focused at an excitation point if an output signal
recorded at another point is reversed in the time domain and emitted back to the original source point [1]. This time reversibility is based on the spatial reciprocity and time-reversal invariance of linear wave equations [2].
Time reversal acoustics was first introduced by the modern acoustics community and applied to many fields such as lithotripsy, ultrasonic brain surgery, active sonar and underwater communications, medical imaging, hyperthermia therapy,
bioengineering, and non-destructive testing (NDT) [1,3–5]. Then, Ing and Fink adopted the TRP to Lamb waves based NDT
in order to compensate for the dispersion of Lamb waves and to detect defects in a pulse-echo mode [6–9]. The main interest
of these studies was refocusing energy in the time and spatial domain by compensating for the dispersive characteristics of
Lamb waves.
The authors advanced this TRP concept to develop a NDT technique where defects can be identified without requiring
direct comparison with previously obtained baseline data [10–12]. The authors’ work intends to reconstruct the
‘‘shape” of the original input signal during the TRP. When nonlinearity is caused by a defect along a direct wave path, the
shape of the reconstructed signal is reported to deviate from that of the original input signal. Therefore, examining the deviation of the reconstructed signal from the known initial input signal allows instantaneous identification of damage without
requiring the baseline signal for comparison.
* Corresponding author. Tel.: +82 42 869 3625; fax: +82 42 869 3610.
E-mail addresses: hwpark@dau.ac.kr (H.W. Park), skim210@asu.edu (S.B. Kim), hoonsohn@kaist.ac.kr (H. Sohn).
0165-2125/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.wavemoti.2009.04.004
452
H.W. Park et al. / Wave Motion 46 (2009) 451–467
However, when the TRP is applied to Lamb waves in a homogeneous regular waveguide, the refocusing capability is limited due to permanent residual side bands even if the duration of the time reversed signal increases. Draeger and Fink
demonstrated this limitation of the TRP through the cavity equation for the one-channel time reversal on chaotic cavities
[13]. This approach utilizes the approximate eigenmode decomposition of wave fields in chaotic cavities which have many
advantageous properties for modal analysis. Unfortunately, most structural waveguides used in civil, mechanical and aerospace engineering usually are not chaotic cavities. Therefore, the interpretation of the TRP based on eigenmode decomposition is rather difficult because the modal properties vary according to the geometric shape and boundary conditions of the
structural waveguides. In this respect, interpretation of the TRP in terms of wave propagation is more appropriate because
dispersion characteristics are independent of the geometry and boundary conditions of a waveguide and determined by the
product of the driving frequency and the thickness of a waveguide.
In this study, the time reversibility of Lamb wave signals generated and sensed using piezoelectric transducers is theoretically investigated in the time domain, limitation in terms of full reconstruction of the input signal is discussed, and techniques to restore the shape of the input waveform are developed. In particular, attention has been paid to understanding of
the following effects on the TRP in the time domain: (1) velocity and amplitude dispersion characteristics of Lamb waves, (2)
the existence of multiple Lamb wave modes, and (3) reflections from structural boundaries. Numerical simulations and
experimental tests are conducted to validate the theoretical findings of this study.
2. A time reversal process for Lamb waves
2.1. Introduction to Lamb waves
All elastic waves including body and guided waves are governed by the same set of partial differential equations [14]. The primary difference is that, while body waves are not constrained by any boundaries, guided waves need to satisfy the boundary conditions imposed by the physical systems as well as the governing equations. Lamb waves are one type of guided waves that exist
in thin plate-like structures, and they are plane strain waves constrained by two free surfaces [15]. The advances in sensor and
hardware technologies for efficient generation and detection of Lamb waves and the increased usage of solid composites in loadcarrying structures have led to an explosion of studies that use Lamb waves for detecting defects in composite structures [16–24].
Because Lamb waves are guided and constrained by two free surfaces, the Lamb waves can propagate a relatively long distance
without much attenuation. This long sensing range makes Lamb waves attractive for damage diagnosis.
Unlike body waves, the propagation of Lamb waves is complicated due to their dispersive and multimode characteristics
[25]. Fig. 1 illustrates two distinct velocity dispersion characteristics of Lamb waves. The first dispersion characteristic is
velocity dispersion within a single mode, and it is referred to as the within-mode dispersion (often referred to as group velocity
dispersion) in this paper. This within-mode dispersion is caused by the frequency dependency of a single Lamb wave mode.
That is, the different frequency components in a single mode travel at different speeds, and this within-mode dispersion results in the spreading of the wave packet as it propagates. The second dispersion characteristic is velocity dispersion among
multiple modes and referred to as the multimode dispersion (often referred to as modal dispersion) hereafter. This multimode
dispersion exists because different modes at a given frequency travel at different speeds. Therefore, when an input waveform
with a discrete driving frequency is applied to a thin medium, it is separated into multiple modes, traveling at different speeds.
Finally, the amplitude attenuation of a Lamb wave is also frequency dependent, and it is called amplitude dispersion. Due to
these unique dispersion characteristics of Lamb waves, time reversibility of Lamb wave signals can be complicated.
2.2. Introduction to time reversal acoustics
The propagation of body waves in elastic media is a classical topic covered in many elasticity textbooks [14,26,27]. By
definition, the body waves propagate throughout a medium, which is not constrained by boundaries. Body wave character7
Group velocity (m/ms)
6
S
0
5
4
3
2
A
1
S
A
0
2
S
1
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency Thickness (MHz mm)
Fig. 1. A typical dispersion curve of Lamb waves in a thin aluminum plate.
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
istics are described by the Navier governing equations [26], and they can be further divided into longitudinal and shear
waves. In time reversal acoustics, an input body wave can be refocused at the source location if a response signal measured
at a distinct location is time-reversed (literally the time point at the end of the response signal becomes the starting time
point) and reemitted to the original excitation location. This phenomenon is referred to as time reversibility of body waves,
and this unique feature of refocusing has found applications in lithotripsy, ultrasonic brain surgery, active sonar and underwater communications, medical imaging, hyperthermia therapy, bioengineering, and non-destructive testing (NDT) [1,3–5].
2.3. Extension of time reversal acoustics to Lamb waves
While the TRP for non-dispersive body waves has been well-established, the study of the TRP for Lamb waves is still relatively new [10]. Previous works mainly investigated spatial and temporal focusing of the reconstructed input signal in the
presence of dispersive characteristics of waves in theoretical and experimental fashions [13,28–31]. Among the previous
works on interpreting the TRP of Lamb waves, the work done by Draeger et al. [13,28] is the most relevant to this study.
In particular, the TRP was theoretically investigated through the cavity equation using the approximate eigenmode decomposition of wave fields in a strategically designed waveguide called chaotic cavities [13]. The refocusing capability of the TRP
could be simply expressed by the energy ratio of the main mode to the residual side bands in the reconstructed signal thanks
to the modal properties of chaotic cavities.
However, most structural waveguides used in civil, mechanical and aerospace engineering are usually not chaotic cavities. Therefore, the interpretation of the TRP based on eigenmode decomposition is rather difficult because the modal properties vary according to the geometric shape of the structural waveguides. To alleviate this difficulty, interpretation of the
TRP in terms of wave propagation is more desirable because dispersion characteristics are independent of the shape of a
waveguide and determined by the product of the driving frequency and the thickness of a waveguide. Concerning the
TRP of Lamb waves, the fundamental property of Lamb wave propagation is well known as described in Section 2.1. Because
of dispersion and multimodal characteristics of Lamb waves, the interpretation of the time reversibility of Lamb waves becomes complicated on the regular homogeneous plate, and it has limited the applicability of the TRP to Lamb waves. This
subsection intends to describe how the TRP works for Lamb wave propagation.
The TRP is first formulated in the frequency domain incorporating the PZTs used for excitation and measurement of Lamb
waves (Fig. 2). Based on the piezoelectricity of PZT materials, an electrical voltage V(x) applied to a PZT wafer is converted to
a mechanical strain eðxÞ through the following electro-mechanical efficiency coefficient ka ðxÞ [32]. Note that the dynamics
of a PZT wafer are neglected and it is assumed that the dynamic behavior of a plate is uncoupled from that of the PZT wafer
[33]
eðxÞ ¼ ka ðxÞVðxÞ
ð1Þ
On the other hand, a voltage is generated when a PZT wafer is subjected to a mechanical strain. This conversion from the
mechanical strain to the voltage output is related by the other mechanical-electro efficient coefficient, ks ðxÞ. Note that all
field variables in Eq. (1) are frequency dependent, and x denotes an angular frequency. Hereafter, the angular frequency
is omitted from the entire field variables for simplicity unless stated otherwise.
When an excitation voltage is applied to PZT A as shown in Fig. 2a, the corresponding response voltage at PZT B can be
represented by the following equation:
V B ¼ ks Gka V A
ð2Þ
where G is the structure’s transfer function relating an input strain at PZT A to an output strain at PZT B. Here, the input voltage applied at PZT A, VA, is first converted to a mechanical strain via ka. Then, the corresponding response strain at PZT B is
converted to the output voltage at PZT B, V B through ks (Fig. 2b).
(b) Response
Signal
(d) Restored
Signal
PZT B
a
rw
Fo
Compared
(a) Input Signal
PZT A
rd
P
on
ati
ag
p
ro
rd
wa
ck
Ba
P
ati
ag
rop
on
Time Reversed
(c) Reemitted
Signal
Fig. 2. A schematic outline of the time reversal process applied to a plate structure [10].
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
In the second step of the TRP, the measured response voltage, VB, is reversed in the time domain before reemitted back to
PZT B (Fig. 2c). This TRP in the time domain is equivalent to taking a complex conjugate of the signal in the frequency domain. In the final step, the reversed version of VB is applied back to PZT B, and the corresponding response is measured at PZT
A. This final response at PZT A is referred to as the ‘‘reconstructed” signal. The reconstructed voltage signal at PZT A, VR, can
be related to the response voltage, VB, in the previous step as follows:
V R ¼ ks Gka V B
ð3Þ
where the superscript * denotes the complex conjugate operation. Note that the transfer function G in Eq. (3) is assumed to
be identical to the one in Eq. (2) based on the reciprocity of elastodynamics [32,34]. By inserting Eq. (2) into Eq. (3), the
reconstructed signal, VR, is related to the original input signal, VA
V R ¼ ks Gka V B ¼ ks Gka ks G ka V A ¼ CKK V A
ð4Þ
*
where K = kska, and C is the time reversal operator defined as C = GG .
If the time reversal operator and the mechanical-electro coefficients are assumed to be constant over the frequency range
of interest, Eq. (4) indicates that the reconstructed signal, VR, is a ‘‘time-reversed” and ‘‘scaled” version of the original input
signal VA (Fig. 2d). That is, the ‘‘shape” of the original input signal should be at least reproduced by the reconstructed signal
during the TRP.
In reality, the time reversal operator is frequency dependent for Lamb waves [10,32] because signal components at different frequencies travel at different speeds and varying attenuation rates. Therefore, the shape of the reconstructed signal
will not be identical to that of the original input signal for Lamb wave propagation.
3. Understanding various effects on the TRP of Lamb waves
3.1. Frequency dependency of the TRP
In time reversal acoustics, the time reversal can be considered an adaptive filter [35,36]. When it comes to Lamb waves,
the main characteristics of this time reversal adaptive filter is its frequency dependency, causing amplitude dispersion of
Lamb waves. Park et al. [10] investigated the time reversibility of Lamb waves on a composite plate and introduced the time
reversal operator into the Lamb wave equation based on the Mindlin plate theory [37]. Because of the amplitude dispersion
of Lamb waves, the time reversal operator varies with respect to frequency as shown in Fig. 3, and wave components at different frequency values are non-uniformly amplified during the TRP. Due to this amplitude dispersion of the TRP, the original
input signal cannot be properly reconstructed if the input signal consists of multiple frequency components such as a broadband input signal. A narrowband excitation has been used to avoid this issue. Note that Park et al. [10] considered only the
fundamental anti-symmetric mode. In reality, the existence of multimodes complicates the TRP of Lamb waves [38]. The effects of within-mode dispersion, multimode dispersion, and reflections on time reversal are subsequently studied in this
paper.
Park et al. [10] described the frequency bandwidth effects of input signals on the TRP and justified the use of narrowband
excitation as shown in Fig. 4. A Gaussian pulse (Fig. 4a) and a 100 kHz tone burst (Fig. 4d) input signals are employed to
investigate the frequency dependency of the TRP. The distance between the PZT pair is assumed to be long enough so that
the within-mode dispersion of Lamb waves can be observed at the response PZT (Fig. 4b and e). When the response signals
are reversed in the time domain and reemitted to the input PZT, the within-mode dispersion of Lamb waves is compensated
(Fig. 4c and f).
As demonstrated here, the within-mode dispersion can be compensated during the TRP. Some wave components within
the single Lamb wave mode travel at higher speeds and arrive at a sensing point earlier than those traveling at lower speeds.
1
Normalized magnitude
Magnitude of the time reversal operator
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
Frequency (KHz)
Fig. 3. Normalized time reversal operator of the A0 mode [10].
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
1
1
Normalized Amplitude
Normalized Amplitude
(d)
(a)
0.8
0.6
0.4
0.2
0
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0
0.1
Time (msec)
0.2
1
1
(e)
0.5
Normalized Amplitude
Normalized Amplitude
(b)
0
-0.5
-1
0.5
0
-0.5
-1
4.7
4.8
4.9
5
4.7
4.8
4.9
5
Time (msec)
Time (msec)
1
1
Normalized Amplitude
Normalized Amplitude
0.3
Time (msec)
0.5
0
(c)
Discrepancy
-0.5
0.5
0
-0.5
(f)
-1
0
0.1
0.2
Time (msec)
0.3
0
0.1
0.2
0.3
Time (msec)
Fig. 4. Reconstruction of input signals using broadband and narrowband input signals through a numerically simulated time reversal process: (a) original
broadband input signal-Gaussian pulse, (b) response signal-Gaussian pulse, (c) original input (dotted) and reconstructed input (solid) signal-Gaussian pulse,
(d) original narrowband input signal-100 kHz tone burst, (e) response signal-100 kHz tone burst, and (f) original input (dotted) and reconstructed input
(solid) signal-100 kHz tone burst [10].
However, during the TRP at the sensing location, the wave components, which travel at slower speeds and arrive at the sensing point later, are reemitted to the original source location first. Therefore, all wave components traveling at different
speeds concurrently converge at the source point during the TRP, compensating for the within-mode dispersion of Lamb
waves.
Due to the amplitude dispersion, the shape of the original pulse is, however, not fully recovered when the Gaussian input
is used (Fig. 4c). The various frequency components of the Gaussian input are non-uniformly scaled and superimposed during the TRP as indicated in Fig. 4c. On the other hand, the shape of the reconstructed tone burst waveform is practically identical to that of the original input tone burst within a certain tolerance level because the amplification of the time reversal
operator is almost uniform for a limited frequency band (Fig. 4f).
This subsection illustrates that the frequency dependency of the TRP can be minimized by using a narrowband excitation signal rather than a broadband excitation. Although the use of a narrowband input can enhance the TRP, the shape of
the reconstructed signal still will not be identical to that of the original input signal due to the multimode dispersion and
reflections from the structure’s boundaries. Their effects on the TRP are theoretically described in the following
subsections.
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
3.2. Understanding the effect of the within-mode dispersion on the TRP
In Section 3.1, it is shown that the amplitude dispersion of the TRP can be minimized by using a narrowband excitation
signal rather than a broadband excitation. In this subsection, the effect of within-mode dispersion on the TRP is mathematically described by considering a single symmetric or anti-symmetric mode induced by a narrowband input excitation. In
Section 3.3, this description is extended for general Lamb wave propagation where multiple symmetric and anti-symmetric
modes exist.
Considering the amplitude and within-mode dispersion of a single symmetric/anti-symmetric mode, the transfer function
G in Eq. (2) can be simplified as follows [39]:
G ¼ c expðikrÞ
ð5Þ
where c, i and k denote the amplitude dispersion function, the imaginary number and the wave number of a specific mode
while r denotes a distance between actuating and sensing PZT wafers, respectively. In turn, the time reversal operator C in
Eq. (4) can be expressed as follows:
C ¼ GG ¼ ½c expðikrÞ½c expðikrÞ ¼ cc
ð6Þ
Eq. (6) shows that the within-mode dispersion in the forward wave propagation process, expðikrÞ, is automatically compensated during the time reversal operation, resulting in only amplitude attenuation at the end. By substituting Eq. (6) into Eq. (4), the relationship between the reconstructed signal and the original input signal can be
simplified
V R ¼ cc KK V A
ð7Þ
, the amplitude dispersion (attenuation) funcWhen a narrowband tone burst signal is applied around a central frequency x
tion and electro-mechanical transduction coefficients can be assumed to be constant over the narrow frequency band [10].
Then, the reconstructed input signal in the time domain can be obtained by taking an inverse Fourier transform of Eq. (7)
V R ðtÞ ¼
1
2p
Z
1
V R expðixtÞdt ¼
1
1
2p
Z
1
1
V A ðT tÞ
cc KK V A expðixtÞdx ¼ C j
ð8Þ
Þc ðx
Þ, j
ÞK ðx
Þ, and T denotes the total time duration of the measured reconstructed signal. Note that
¼ Kðx
where C ¼ cðx
the complex conjugate of the input signal in the frequency domain is equivalent to the time reversed version of the original
input in the time domain after the inverse Fourier transform.
Eq. (8) confirms again that, as long as a single Lamb wave mode is concerned, the reconstructed signal is simply a ‘‘timereversed” and ‘‘scaled” version of the original input signal as previously described in Eq. (4). Note that the within-mode
dispersion of a single mode signal is compensated during the TRP, and it does not affect the time reversibility as also demonstrated in the previous works [8,10,11]. In the next subsection, the time reversibility is extended considering the multimode dispersion of multiple symmetric and anti-symmetric modes.
3.3. Understanding the effect of multimodes on the TRP
The effects of two independent wave modes (P and S modes) on the TRP have been theoretically investigated by Draeger
et al. for the first time when both wave modes are simultaneously generated in an isotropic solid [28]. According to their
work, the TRP produces not only the original input signal at the expected time but also sidebands that arrive before and
afterwards. Similar to this case, the multimodal characteristics of Lamb waves complicate the TRP. The effect of the multimode characteristic is schematically shown in Fig. 5. When a tone burst signal is exerted to PZT A (Fig. 5a), multimodes are
received at PZT B (Fig. 5b). In Fig. 5, the narrowband input frequency is selected so that only the first symmetric (S0) and antisymmetric (A0) modes are generated. When the response signal is reversed in the time domain and reemitted to PZT B
(Fig. 5c), each reemitted signal associated with the A0 or S0 modes at PZT B creates both S0 and A0 modes producing a total
of four modes in the reconstructed signal (Fig. 5d and e). In Fig. 5, S0/A0 denotes the S0 mode signal measured at PZT A due to
the A0 mode input at PZT B. A0/S0, S0/S0, and A0/A0 are similarly defined. After superposition of signals in Fig. 5d and e, the
reconstructed signal consists of the main mode at the middle and two sidebands around the main mode (Fig. 5f). Note that
the main mode signal in the middle is the superposition of the A0/A0 and S0/S0 and ‘‘symmetric” sidebands are produced as a
result of A0/S0 and S0/A0. Finally, the shape of the main mode will be practically identical to that of the original input signal.
Considering this multimode effect on the TRP, the reconstructed signal will be composed of the following four mode
groups:
AA
SA
AS
V R ðtÞ ¼ V SS
R ðtÞ þ V R ðtÞ þ V R ðtÞ þ V R ðtÞ
ð9Þ
V SA
R ðtÞ
where
represents the symmetric mode signal in the reconstructed (one) generated by the anti-symmetric mode signal
AA
AS
in the forward (one), and V SS
R ðtÞ, V R ðtÞ, and V R ðtÞ are defined in a similar fashion. Consequently, the time reversal operator C
in Eq. (4) is also decomposed to those associated with symmetric and anti-symmetric modes.
C ¼ CSS þ CAA þ CSA þ CAS
ð10Þ
H.W. Park et al. / Wave Motion 46 (2009) 451–467
457
Fig. 5. The effect of multimodes on the time reversal process. (Note: S0/A0 denotes S0 mode produced at PZT A due to A0 mode input at PZT B. A0/S0, S0/S0,
and A0/A0 are similarly defined.)
SS
Initially, the coupling effect among symmetric mode signals (V SS
R ðtÞ and C ) is explained, and this description is extended
for multiple symmetric and anti-symmetric mode signals. For brevity, the superscription ‘‘S” denoting the symmetric mode
is omitted until all multiple modes are included at the end of this subsection. Considering the coupling among symmetric
mode signals, Eq. (6) can be extended as follows:
C¼
nS X
nS
X
g p g q
ð11Þ
p¼1 q¼1
where g p denotes the transfer function of the pth symmetric mode, and nS, represents the total number of symmetric modes
.
at a given excitation frequency x
Similar to Eq. (5), g p can be expressed as a function of the amplitude and velocity dispersions
g p ¼ cp expðikp rÞ
ð12Þ
where cp and kp denote the amplitude dispersion function and the wave number of the pth symmetric mode. Substituting Eq.
(12) into Eq. (11) results in
C¼
nS X
nS
X
g p g q ¼
p¼1 q¼1
nS X
nS
X
cp cq exp½irðkp kq Þ ¼
p¼1 q¼1
nS X
nS
X
C pq expðihpq Þ
ð13Þ
p¼1 q¼1
where, hpq ¼ r½kp kq and C pq ¼ cp cq . Subsequently, the reconstructed input signal associated with the coupling of symmetric mode signals can be expressed as follows:
V R ¼ CKK V A ¼
nS X
nS
X
C pq expðihpq ÞKK V A
ð14Þ
p¼1 q¼1
Similar to Eq. (8), the reconstructed input signal in the time domain can be obtained by taking the inverse Fourier trans
form of Eq. (14) when an original input signal is a narrowband tone burst with a center frequency of x
V R ðtÞ ¼
Z 1
nS X
nS
X
1
V A exp½iðhpq þ xtÞdx
C pq j
2p 1
p¼1 q¼1
ð15Þ
Þcq ðx
Þ and j
ÞK ðx
Þ. Because hpq in Eq. (15) equals to zero when p = q, Eq. (15) can be rewritten as
¼ Kðx
where C pq ¼ cp ðx
follows:
V R ðtÞ ¼ V A ðT tÞ
nS
X
p¼1
þ
C pp j
Z 1
nS X
nS
X
1
V A exp½iðxt þ hpq Þdx
ð1 dpq Þ
C pq j
2
p
1
p¼1 q¼1
where dpq denotes the Kronecker delta in which dpq = 0 if p – q and dpq ¼ 1 if p = q.
ð16Þ
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
Note that there is no closed form solution for the integral term in Eq. (16). To get an approximate solution of this integral,
up to the first order term
hpq is expanded using a Taylor series near the driving frequency x
Þþ
hpq ¼ hpq ðx
dhpq dhpq Þ
ð
x
x
Þ
þ
H:O:T:
h
ð
x
Þ
þ
ðx x
pq
dx x¼x
dx x¼x
ð17Þ
, the wave number k, the group velocity w, and the phase velocBy using the relationships among the angular frequency x
ity v, Eq. (17) is expressed as follows:
s
pq þ xtpq
hpq x
ð18Þ
where
spq ¼
r
r
r
r
;
Þ wp ðx
Þ
Þ wq ðx
Þ
vp ðx
vq ðx
t pq ¼
r
r
Þ wq ðx
Þ
wp ðx
ð19Þ
and
dx ¼ wp ðxÞdkp ;
x ¼ vp ðxÞkp
ð20Þ
Using Eqs. (18)–(20), Eq. (16) is expressed as follows:
V R ðtÞ ¼
nS
X
V A ðT tÞ þ
C pp j
p¼1
nS X
nS
X
fexpðix
s
pq ÞV A ½T ðt þ t pq Þg
ð1 dpq ÞC pq j
ð21Þ
p¼1 q¼1
The first term on the right hand side of Eq. (21) indicates that both within-mode and multimode dispersions are compensated and Lamb wave modes converge to the main mode as long as the identical mode travels in both forward and backward
propagation directions. On the other hand, the second term reveals that, when Lamb waves travel at two different group
velocities in the forward and backward propagation directions, the corresponding modes in the reconstructed input signal
are shifted in the time domain by tpq , creating ‘‘sidebands” around the main mode where the most of the energy converges.
Note that the within-mode dispersion in the time domain is still fully compensated in the second term and the time shift
tpq depends only on the difference between the group velocities of the pth and qth modes. It is also noted that the amplis
pq Þ, which depends on the phase velocity difference between the pth and qth
tude of each sideband is proportional to expðix
s
pq ¼ s
qp and t pq ¼ t qp in
pq Þ can be interpreted as the phase shift of each sideband. Because s
modes. Here, the term expðix
Eq. (18), it can be shown that the second term of the reconstructed signal in Eq. (21) is symmetric with respect to the main
peak mode in the middle. Note that this symmetry of the reconstructed signal in Eq. (21) is valid regardless the symmetry of
the structure or the PZT transducer layout as long as the input signal is symmetric.
AA
AS
In a similar manner, V SA
R , V R , and V R terms in Eq. (9) can be calculated as follows:
V SA
R ðtÞ ¼
nA
nS X
X
SA
C SA
pq jfexpðixspq ÞV A ½T ðt þ t pq Þg
ð22Þ
AS
C AS
pq jfexpðixspq ÞV A ½T ðt þ t pq Þg
ð23Þ
p¼1 q¼1
V AS
R ðtÞ ¼
nA X
nS
X
p¼1 q¼1
V AA
R ðtÞ ¼
nA
X
p¼1
C AA
pp jV A ðT tÞ þ
nA X
nA
X
AA
ð1 dpq ÞC AA
pq jfexpðixspq ÞV A ½T ðt þ t pq Þg
ð24Þ
p¼1 q¼1
where superscripts S and A denote symmetric and anti-symmetric modes while nA denotes the total number of anti-symmetAS
. Eqs. (22) and (23) reveal that the V SA
ric modes at the excitation frequency x
R and V R terms in the reconstructed input signal
do not converge to the main mode but only create additional sidebands. The same can be said about V AA
R except that it converges on the main mode when p = q. However, it is noted that the symmetry of the reconstructed signal is still preserved
even in the presence of multiple symmetric and anti-symmetric modes because the TRP is a temporal correlation between
the multiple Lamb wave modes.
The effects of modal dispersion on the time reversibility of Lamb waves described in Eqs. (21)–(24) is validated through a
numerical simulation on an aluminum plate model with a pair of surface-bonded PZT transducers (PZTs A and B) in Fig. 6a.
The geometric configuration of the aluminum plate and the location of PZT transducers are properly arranged so that only
the effects of modal dispersion are included during the TRP without the interference of reflections. The parameters for the
PZT transducers were adopted from PSI-5A4E type of PZT sheets (thickness = 0.0508 cm) which is commercially available
[40]. The PZT transducers are assumed to be rigidly bonded on the host aluminum plate. The numerical simulation is conducted through ABAQUS 6.7-1 standard on IBM p595 in KISTI supercomputing center [41]. 1 mm 1 mm 8-node quadratic
plane strain solid elements are employed for the aluminum plate while 1 mm 0.507 mm 8-node quadratic plane strain piezoelectric elements are used for the pair of PZT transducers. For the accuracy of solution, the Newmark-b method is employed for numerical time integration with a time increment of 2.5 107 s (4 MHz).
The 100 kHz tone burst input signal in Fig. 7a is used to create the S0 and A0 modes as shown in Fig. 7b. Fig. 7b displays the
output response signal at PZT B when the created Lamb wave modes arrive at PZT B, 60 cm away from PZT A. The relative
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
PZT A
PZT B
1cm
6mm
Plate
60cm
100cm
100cm
(a) An aluminum plate model with a pair of surface-bonded PZT transducers
A'
Group velocity (m/msec)
6
S mode
5
0
4
3
A mode
2
A mode
0
1
1
A
0
0
100
200
300
Frequency (kHz)
(b) A dispersion curve of a 6 mm thick aluminum plate
Fig. 6. (a) An aluminum plate model with a pair of surface-bonded PZT transducers to validate the effects of modal dispersion on the time reversibility of
Lamb waves (the thickness of a PZT pair: 0.507 mm) and (b) a dispersion curve of a 6 mm thick aluminum plate as a function of the input driving frequency.
differences of the group velocities of the S0 and A0 modes estimated from Fig. 7b are less than 1% compared to those calculated in the theoretical dispersion curve in Fig. 6b. Note that the waveform of each mode in Fig. 7b deviates from that of the
original input signal in Fig. 7a due to the within-mode dispersion of Lamb waves.
Next, the effect of the time truncation point on the TRP is examined. For this investigation, the response signal measured
at PZT B is truncated at two different time points before reemitted to PZT B as shown in Fig. 7b. In Case 1, the response signal
is truncated at t = 0.19 ms so that only the S0 mode is included in the TRP. On the other hand, the response signal for Case 2 is
truncated at t = 0.30 ms so that both the S0 and A0 modes are used during the TRP. Fig. 7c presents the reconstructed input
signals at PZT A for Cases 1 and 2, respectively. For the brevity of description, the time point corresponding to the main peak
of the input signal is set to be zero in Fig. 7c. Using Eqs. (21)–(24), the reconstructed input signals for Cases 1 and 2 can be
represented as the following equations:
AS AS
AS
Case 1 : V R ðtÞ ¼ C SS
11 jV A ðT tÞ þ C 11 j expðixs11 ÞV A ½T ðt þ t 11 Þ
ð25Þ
SS þ C
AA Þj
SA j
AS SA
AS
V A ðT tÞ þ C
SA
AS
Case 2 : V R ðtÞ ¼ ðC
11
11
11 expðixs11 ÞV A ½T ðt þ t 11 Þ þ C 11 j expðixs11 ÞV A ½T ðt þ t 11 Þ
ð26Þ
The first term in Eq. (25) represents the main mode in the reconstructed input signal t = 0 at while the second term repAS
resents the sideband that is shifted from the main mode by t AS
11 in the time domain. From Eq. (19), the time shift, t 11 in Eq.
(25), is calculated as rð1=wA0 1=wS0 Þ ¼ 0:6 ð1=2:994 1=5:308Þ ¼ 0:0874 ms. From Fig. 7c, the time shift is estimated to be 0.0882 ms. The relative difference between these two time shift estimates is less than 1%. The first term in
Eq. (26) represents the main mode in the reconstructed signal at t = 0 while the second and third terms correspond to the
SA
SA
AS
sidebands that are shifted from the main mode by t SA
11 and t 11 , respectively. Note that t 11 and t 11 are the same time shift
SA
AS
with the opposite signs (t 11 ¼ t11 ), and this is clearly reflected on a symmetric pair of the side bands in Fig. 7c.
The reconstructed input signal for Case 2 in Fig. 7c is magnified in the vicinity of the main mode and compared to the
original input signal in Fig. 7d. For the better comparison of the signal’s shape, the reconstructed and original input signals
in Fig. 7d are normalized so that their maximum values are equal to 1.0 at t = 0. The within-mode dispersion of the forward
signal in Fig. 7b is compensated during the TRP, and the shape of the original input signal is recovered in the main mode of
the reconstructed input signal.
The effect of modal dispersion on the time reversibility is also experimentally validated using a pair of PZT transducers
(PZTs A and B) mounted on an aluminum plate shown in Fig. 8. The dimensions of the plate and each PZT transducer are
122 cm 122 cm 0.6 cm and 1 cm 1 cm 0.0508 cm, respectively. The properties of the PZT wafer transducers are identical to those used in the numerical simulation demonstrated above. The distance between the two PZTs is set to 40 cm in
order to avoid the effect of reflections, and a narrowband tone burst signal with a center frequency of 130 kHz is used as the
input signal.
H.W. Park et al. / Wave Motion 46 (2009) 451–467
100
Response signal at PZT B (V)
Original input signal at PZT A (V)
460
50
0
-50
-100
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Case 2
(0.3 msec)
Case 1
1.0
(0.19 msec)
0.0
S mode
0
-1.0
A mode
0
-2.0
0.00
0.05
0.10
0.15
0.20
Time (msec)
Time (msec)
(a) 100 kHz tone burst input signal exerted at PZT A
(b) Response signal at PZT B
0.25
0.30
0.02
0.03
1.5
1.0
0.0
-1.0
Case 1
Case 2
-2.0
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
Normalized amplitude
2.0
Normalized amplitude
2.0
0.8
0.0
-0.8
Reconstructed input signal
Original input signal
-1.5
-0.03
-0.02
Time (msec)
-0.01
0.00
0.01
Time (msec)
(c) Reconstructed input signal at PZT A through
(d) Comparison of the main modes in the original
the TRP (Case 1: only the S0 mode is truncated and
(dotted) and the reconstructed (solid) input
used for the TRP ; Case 2: Both S0 and A0 modes
signals
are used for the TRP)
Fig. 7. The effect of the multimode dispersion on the time reversibility of Lamb wave signals is numerically validated using the numerical setup in Fig. 6.
In Fig. 9, it is shown how selective truncation of specific mode(s) affects the reconstructed signal. In Case 1 shown in
Fig. 9a, the forward signal AB was truncated 0.105 ms after the input signal so that only the S0 mode could be reversed
and resent to the original location. In this case, only the S0 mode contributed to the main mode, and there was only one sideband (Fig. 9b). In Case 2 shown in Fig. 9a, the TRP was repeated by truncating the forward signal AB at 0.160 ms so that both
the S0 and A0 modes could be included. In this example, the main mode was composed of the contributions from both the S0
and A0 modes, and two sidebands were created as expected. Furthermore, the sidebands were symmetric along the main
AS and tSA ¼ tAS . Note that the signals corresponding to Cases 1
SA ¼ C
mode as illustrated in Case 2 in Fig. 9b because C
11
11
11
11
and 2 are scaled in Fig. 9b so that the maximum peak of the main mode is equal to one.
Finally, the time shift value between the main mode and the sidebands were measured experimentally from Fig. 9b and
compared with tSA in Eq. (19). The experimental group velocities of the S0 and A0 modes were 5.316 m/ms and 3.115 m/ms,
SA
SA
and the tSA
11 was 0.053 ms (t 11 = 0.4/5.316 0.4/3.115 = 0.053 ms). This t 11 value agreed well with the time gap between
the main mode and one of the sidebands observed from Fig. 9b (about 0.0531 ms). Therefore, it is successfully demonstrated
that Eq. (26) properly described the sidebands created by multimodes.
In conclusion, the main mode in the reconstructed signal is practically identical to the original input signal in spite of the
modal dispersion of Lamb waves. The symmetry of the side bands with respect to the main mode of the reconstructed input
signals is preserved as long as all symmetric and anti-symmetric modes of interest are included in the TRP.
3.4. Understanding the effect of reflections on the TRP
Similar to the multiple Lamb wave modes, the Lamb waves reflected from the boundaries of a structure create additional
sidebands in the reconstructed signal. The effect of the reflections on the TRP is illustrated in Fig. 10. For simplicity, it is
H.W. Park et al. / Wave Motion 46 (2009) 451–467
461
Fig. 8. An aluminum plate with PZT transducers used for experiment.
Fig. 9. Experimental verification of the multimodal effect by truncating the forward signals at different time point for the TRP: Case 1 truncated right after
the S0 mode and Case 2 after S0 and A0 modes.
assumed that a single mode travels unidirectionally and the structure has only one finite boundary where the Lamb wave is
reflected. When a single Lamb mode is generated at PZT A and travels to PZT B (Fig. 10a), the wave will take two different
paths to arrive at PZT B (Fig. 10b). In Fig. 10, P1 and P2 denote modes traveling along direct and reflection paths in a forward
propagation direction. P3 and P4 are defined in a similar fashion but in a backward propagation direction. When the P2 mode
is emitted back to PZT A due to the reflection (Fig. 10c), this mode generates two modes, P3/P2 and P4/P2, in the reconstructed signal due to the two different wave propagation paths in the backward propagation direction (Fig. 10d). Similarly,
when P1 is reemitted, it creates additional two modes, P3/P1 and P4/P1 (Fig. 10e). Finally, the reconstructed signal is composed of the main mode in the middle, which is the superposition of P3/P1 and P4/P2, and two symmetric sidebands due to
P3/P2 and P4/P1 (Fig. 10f). Note that the symmetry of the reconstructed signal is independent of the symmetry of the structure, the PZT layout and the boundary condition. In the example presented in Fig. 10, there is only one finite boundary where
waves can be reflected, but the reconstructed signal is still symmetric. The number of sidebands will increase if there are
additional wave reflections.
To examine the effect of the reflections in a more theoretical manner, the time reversal operator C in Eq. (4) can be
decomposed to those associated multiple wave propagation paths between the actuating and sensing PZT wafers. For brevity
of description, the discussion is first limited to a single Lamb mode with multiple wave propagation paths
C ¼ GG ¼
nR X
nR
X
g p g q
ð27Þ
p¼1 q¼1
where subscript p of a field variable denotes a specific wave propagation path, while nR, and g p represent the total number of
traveling paths and an individual transfer function associated with the pth traveling path, respectively.
Considering the amplitude and velocity dispersion of Lamb waves, g p can be simply expressed as follows:
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
Fig. 10. The effect of reflections on the TRP. (Note: P1 and P3 are waves propagating along the direct path between PZTs A and B, and P2 and P4 are waves
reflected at one end of the plate in forward and backward directions. P3/P2 denotes a signal arrived at PZT A through a direct path, when the reflected signal,
P2, is emitted back to PZT B after time reversal. P4/P2, P3/P1, and P4/P1 are similarly defined.)
g p ¼ c expðikrp Þ;
ð28Þ
where rp denotes a traveling distance from the actuating PZT wafer to the sensing PZT wafer associated with the pth traveling
path, respectively. Using Eq. (28), Eq. (27) is expressed as follows:
C¼
nR X
nR
X
g p g q ¼
p¼1 q¼1
nR X
nR
X
cc e½ikðrp rq Þ ¼
p¼1 q¼1
nR X
nR
X
C expðihpq Þ
ð29Þ
p¼1 q¼1
where hpq ¼ k½rp rq and C ¼ cc . From Eq. (29), the reconstructed input signal with multiple wave propagation paths can be
calculated as follows:
V R ¼ CKK V A ¼
nR X
nR
X
Ceðihpq ÞKK V A
ð30Þ
p¼1 q¼1
Similar to Eq. (6), the reconstructed input signal in the time domain can be obtained by taking the inverse Fourier trans
form of Eq. (30) if an original input signal is a narrowband tone burst with a center angular frequency x
V R ðtÞ ¼ V A ðT tÞ
nR
X
j
þ
C
p¼1
Z 1
nR X
nR
X
1
V A ðxÞ exp½iðxt þ hpq Þdx
ð1 dpq Þ
C j
2p 1
p¼1 q¼1
ð31Þ
Because the second term in Eq. (31) cannot be directly expressed in terms of VA(t), it is approximated by procedures similar to Eqs. (17)–(20)
V R ðtÞ ¼
nR
X
j
V A ðT tÞ þ
C
p¼1
nR X
nR
X
fexpðix
s
pq ÞV A ½T ðt þ tpq Þg
ð1 dpq ÞC j
ð32Þ
p¼1 q¼1
where
spq ¼ ðrp rq Þ
1
1
;
Þ wðx
Þ
vðx
t pq ¼ ðrp rq Þ
Þ
wðx
ð33Þ
Note that Eq. (32) is identical to Eq. (21) except that the summation is performed over the multiple reflection paths instead of the multiple symmetric modes. Therefore, the reflections create additional sidebands similar to the ones created by
the multiple modes, but do not change the symmetry of the reconstructed signal regardless of the symmetry of the structure’s boundary conditions. That is, the TRP can be interpreted as a temporal correlation among multiply reflected Lamb
wave modes at the boundary of the plate.
The effect of reflections on the time reversibility of Lamb waves described in Eqs. (32) and (33) is validated through a
numerical simulation on an aluminum plate model with four identical surface-bonded PZT transducers (PZTs A, B, C, and
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
PZT A
PZT B
1cm
6mm
PZT D
30cm
Plate
PZT C
80cm
20cm
Fig. 11. An aluminum plate model with two pairs of surface-bonded PZT transducers to validate the effect of reflections on the time reversibility of Lamb
waves (the thickness of a PZT pair: 0.127 mm).
Response signal at PZTs B and C (V)
Original input signal at PZT A (V)
D) in Fig. 11. The geometric configuration of the aluminum plate and the location of PZT transducers are properly arranged so that only the reflections from the left boundary of the plate can be included during the TRP without interference with modal dispersion. The polarization of PZTs A and D is configured so that only the A0 mode is generated when
PZTs A and D are excited simultaneously. The polarization of PZTs B and C is configured identical to that of PZTs A and
D. Reversing and re-emitting the response signal at PZTs B and C in the time domain, reconstructed input signals are
received at PZTs A and D. The PZT transducers were made from PSI-5A4E type of PZT sheets (thickness = 0.0127 cm).
The rest of the numerical setup here is identical to that in Section 3.3 unless otherwise mentioned.
Fig. 12a illustrates the 100 kHz tone burst input signal that is simultaneously exerted at PZTs A and D to create only the A0
mode in the forward propagation. The response signal at PZT B and C, 20 cm away from the excitation sources, is presented in
40
20
0
-20
-40
0.00
0.01
0.02
0.03
0.04
0.05
0.06
1.0
Case 2
Case 1
0.0
A mode
Reflected A mode
0
-1.0
0.00
0.05
0
0.10
0.15
Time (msec)
0.20
0.25
0.30
0.35
0.40
Time (msec)
(a) The 100 kHz tone burst input applied at PZTs A
(b) The response signal measured at PZT B
1.5
Normalized amplitude
1.0
Normalized amplitude
(0.40 msec)
(0.20 msec)
0.5
0.0
-0.5
Case 1
Case 2
-1.0
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.8
0.0
-0.8
Reconstructed input signal
Original input signal
-1.5
-0.03
-0.02
Time (msec)
-0.01
0.00
0.01
0.02
0.03
Time (msec)
(c) The Reconstructed input signal at PZT A
(d) Comparison of the main modes in the original a
through the TRP (Case 1: Only the direct A0 mode
(dotted) and the reconstructed (solid) input
is truncated and used for the TRP ; Case 2: Both
signals
the direct A0 and reflected A0 modes are included
for the TRP)
Fig. 12. The effect of reflection on the time reversibility of Lamb wave signals is numerically validated using the numerical setup in Fig. 11.
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
Fig. 12b. In Fig. 12b, the A0 mode propagating through the direct path arrives at PZT B first followed by the A0 mode reflected
from the left boundary of the plate. Note that the polarization of PZT C is arranged so that the response at PZT C is identical to
the response at PZT B when there are only anti-symmetric modes [42]. The estimated group velocity of the A0 mode in
Fig. 12b is approximately 2.959 m/ms while the theoretical one is 2.994 m/ms at 100 kHz in Fig. 6b. The relative difference
between the estimated and theoretical group velocities is less than 1%.
To investigate the effect of reflection on the time reversibility of Lamb waves, the response signal at PZT B is truncated at
two different time points as shown in Cases 1 and 2 of Fig. 12b, respectively. In Case 1, the response signal is truncated at
t = 0.20 ms so that only the A0 mode propagating along the direct path is included in the TRP. In Case 2, the response signal is
truncated at t = 0.40 ms to include both the direct and reflected A0 modes in the TRP. Note that the response signal measured
at PZT B is reversed and applied to PZTs B and C simultaneously as before to excite only the A0 mode.
Fig. 12c shows the reconstructed input signal measured at PZT A through the TRP for Cases 1 and 2, respectively. For the
simplicity of description, the time point corresponding to the main peak of the input signal is set to be zero. For Case 1 in
Fig. 12c, only a single side band is observed in the left hand side of the main mode. From Fig. 12c, the time shift of the side
band from t = 0 is estimated to be 0.206 ms. On the other hand, it is estimated to be t12 ¼ ðr 2 r 1 Þ=wA0 ¼
ð0:8 0:2Þ=2:994 ¼ 0:2 ms (0.2 0.8)/2.994 = 0.2 ms from Eq. (33), and the relative difference error between the theoretical and estimated time shifts is less than 3%.
For Case 2 in Fig. 12c, two identical side bands appear symmetrically with respect to t = 0. The time shift values of the left
and right hand sides are identical to that of Case 1. The reconstructed input signal for Case 2 is magnified near at t = 0 and
compared to the original input signal in Fig. 12d. In order to compare the shapes of the input and reconstructed signals, the
maximum peak value at t = 0 is normalized to be 1.0. The within-mode dispersion of the forward signals observed in Fig. 12b
is compensated during the TRP, and the shape of the original input signal is fully restored in the main mode of the reconstructed input signal.
Next, the appearance of sidebands due to reflections from boundaries and the symmetry of the resulting reconstructed
signal were experimentally demonstrated. The test was conducted using a pair of PZT patches attached to the aluminum
plate shown in Fig. 8. PZT A was mounted 10 cm away from the left side edge of the plate and PZT B was placed 32 cm away
from the right side boundary of the plate. In this configuration, multiple Lamb wave modes arrived at PZT B in the following
order as shown in Fig. 13a: (1) the S0 mode along the shortest (direct) wave path, (2) the S0 mode reflected from the left
boundary, and (3) the direct and reflected A0 modes. Because the direct and reflected A0 modes arrived later than the direct
and reflected S0 modes, it was possible to truncate the forward signal including only the S0 modes to study the effects of the
Fig. 13. Experimental investigation of the effect of reflections on the TRP by truncating the forward signals at different time points during the TRP: Case 1
includes only the direct S0 mode and Case 2 includes the reflected S0 mode as well in the TRP.
H.W. Park et al. / Wave Motion 46 (2009) 451–467
465
reflections. Note that the dynamic range of the data acquisition system is reduced here to improve the effective resolution of
the direct and reflected S0 modes in Fig. 13a. As a result, the A0 modes which had higher amplitudes than S0 modes were
partially saturated. The center frequency of the tone burst signal was set to 130 kHz.
To see the effect of the reflection, the truncation time point in the forward signal was varied. First, the forward signal in
Fig. 13a was truncated at 0.17 ms so that only the direct S0 mode was reversed and resent to the original location (Case 1).
Next, the truncation was done at 0.2125 ms to embrace both the direct and reflected S0 modes (Case 2). In Fig. 13b and c, the
sidebands created by the reflections are shown. When the TRP was conducted including only the direct S0 mode, a single
sideband appeared only on the left hand side of the main mode (Fig. 13b). On the other hand, two symmetric sidebands were
developed when both S0 modes were included during the TRP (Fig. 13c).
Next, it was investigated if the phase shift of these sidebands matched with the theoretical prediction. Based on the forward signal shown in Fig. 13a, the group velocity of the S0 mode was 5.239 m/ms. The arrival time of the right sideband in
Fig. 13c was estimated to be 0.0396 ms, and it was close to the t DR value obtained from Eq. (33)
Þ ¼ ð0:80 1:00Þ=5:239 ¼ 0:0382 msÞ. Furthermore, it must be noted that the shapes of the left
ðtDR ¼ ðrD r R Þ=wS0 ðx
and right sidebands were almost symmetrical along the main mode as illustrated in Fig. 13c.
In conclusion, the main mode in the reconstructed signal is practically identical to the original input signal in spite of the
modal dispersion of Lamb waves. The symmetry of side bands with respect to the main mode of the reconstructed input signals is preserved as long as all direct and reflected modes of interest are included in the TRP.
3.5. Summary of practical issues to be addressed for the TRP of Lamb waves on a plate
So far, various effects of Lamb waves on the TRP such as multimodes and reflections has been studied and validated by
numerical simulations as well as experiments in a laboratory environment. When the applicability of the TRP to real field
structures is sought, however, a number of additional practical issues need to be resolved in advance. For instance, it can
be very challenging to control the bonding condition of each PZT transducer. Also, the performance of the TRP under operational and environmental conditions should be investigated. Therefore, additional experiments have been designed and
conducted in laboratory and field environments to resolve the practical issues. Due to the page limitation, only the test results are briefly discussed in this section.
First, the effects of PZT size, orientation, shape and bonding condition on the TRP have been examined. When the bonding
conditions and areas of two PZT wafers were identical, the TRP could be successfully conducted regardless to their shapes
and orientations. When the TRP was tested using two PZTs with different sizes, the time shift of the main peak in the reconstructed signal was observed. This is because the impedance values of the two PZTs are different. Assuming two identical
PZTs are used for the TRP, the amount of time delay due to the impedance of the exciting PZT in the forward signal is identical to that of the backward signal, resulting in no time shift of the main peak. On the other hand, the time delay appears
when PZTs with different impedance values are used. However, in spite of the time delay, the shape of the main peak was
preserved in the TRP. Note that variation in the PZT bonding conditions has a similar effect as change in the PZT size, and the
time shift of the main peak in the reconstructed signal was observed when PZTs with different bonding conditions were used
for the TRP.
Next, the effects of environmental and operational variations on the TRP are investigated. In spite of temperature, boundary and surface condition changes, and ambient loading, the TRP could be successfully achieved. For instance, the time
reversibility of Lamb wave signals is tested by instrumenting a steel bridge girder under normal traffic load [43]. The temperature effect is also tested through temperature chamber experiments. It is noteworthy that Ribay et al. demonstrated the
same theoretical result in which the effect of temperature change on the time reversibility was negligible in the metallic
media such aluminum used in this study [44].
4. Summary
Guided waves such as Lamb waves have recently received much attention in NDT applications. NDT techniques using
guided waves are based on the premise that, when propagating waves encounter a defect, wave patterns are altered by
the defect. Therefore, the damage can be identified by comparing the baseline signal obtained from an intact condition of
the structure and the current signal obtained from an unknown condition of the structure. However, it has been reported
that this type of pattern comparison with the baseline signal can be susceptible to false alarms because other natural variations of the system can produce various changes in the signal. To relax this dependency on the baseline data, the authors
have been developing a damage detection technique that does not require a direct comparison of the test signal with the
baseline data [12]. The pivotal concept of this reference-free NDT technique is the TRP. While the TRP has been widely used
in modern acoustics, its use in Lamb waves has been limited due to the amplitude and velocity dispersion characteristics of
Lamb waves.
In this study, the applicability of the TRP to Lamb wave propagation is theoretically investigated. In particular, the primary interest in the TRP is to match the shape of the final response signal (the reconstructed signal after the TRP) with that
of the original input. Because the shape of the reconstructed signal deviates from the original input waveform when there is
a certain type of defect along the wave propagation path, this feature can be utilized for damage diagnosis. However, the
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H.W. Park et al. / Wave Motion 46 (2009) 451–467
interpretation of the TRP becomes complicated for Lamb wave propagation due to its dispersive characteristics. Based on the
theoretical, numerical and experimental studies presented in this paper, it has been shown that (1) within-mode dispersion
of a single mode is fully compensated during the TRP; (2) a narrowband input signal should be employed to minimize amplitude dispersion and to restore the shape of the input signal; (3) the existence of multiple Lamb wave modes and reflections
create additional sidebands around the main response mode in the reconstructed signal; and (4) the reconstructed signal is
symmetric along the main mode as long as a symmetric input signal is exerted regardless of the topology and boundary conditions of the structure. Finally, practical implementation issues relevant to field deployment are briefly discussed.
Acknowledgements
This work was supported by Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2008-331-D00590), in which main calculations were performed by using the supercomputing
resource of the Korea Institute of Science and Technology Information (KISTI), and the Radiation Technology Program under
Korea Science and Engineering Foundation (KOSEF) and the Ministry of Science and Technology (M20703000015-07N030001510). The second author would like to acknowledge the graduate fellowship program from Samsung Scholarship in Seoul,
Korea.
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