Ultrasonics 142 (2024) 107362
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Ultrasonics
journal homepage: www.elsevier.com/locate/ultras
Short communication
Experimental validation of zero-group-velocity feature guided waves in a
welded joint utilizing the pitch-catch measurement technique with
air-coupled ultrasonic transducers
Xiangdi Meng a, Mingxi Deng a, *, Weibin Li b, *
a
b
College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
School of Aerospace Engineering, Xiamen University, Xiamen 361005, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Resonance frequency
Zero-group-velocity (ZGV)
Feature guided waves (FGWs)
Welded joint
Air-coupled ultrasonic transducers
Zero-Group-Velocity (ZGV) Lamb waves in elastic plates had been conducted extensive theoretical and experi­
mental researches in the field of ultrasonic nondestructive testing. The ZGV modes in complex structures had
been studied theoretically, but less attention had been paid to their experimental investigation. This paper re­
ports the experimental observation of Zero-Group-Velocity Feature Guided Waves (ZGV-FGWs) in a welded joint
using the pitch-catch measurement technique with air-coupled ultrasonic transducers. Firstly, for the elastic
plate, it is verified that the received time-domain signal using the pitch-catch measurement method with aircoupled ultrasonic transducers is indeed ZGV Lamb waves. Subsequently, we applied the same pitch-catch
measurement method with air-coupled ultrasonic transducers to receive time-domain signals at different exci­
tation frequencies in the welded joint. It is observed that the received time-domain signals in the welded joint
oscillate for extended periods of time. By performing short-time Fourier transforms on the received time-domain
signals, we analyze the frequency content of the received time-domain signals at different excitation carrier
frequencies. By analyzing the spectral amplitude variations of these signals at different excitation carrier fre­
quencies, it can be demonstrated that the spectral amplitude corresponding to the resonance frequency is the
largest. These findings collectively affirm that the received time-domain signals in the welded joint exhibit ZGV
characteristics, identified as ZGV-FGWs. Consequently, from an experimental perspective, the presence of ZGVFGWs in the welded joint is verified. Moreover, the experimentally determined resonance frequency of ZGVFGWs concurs with the results obtained through simulation. This study confirms the feasibility of using the
pitch-catch measurement method with air-coupled ultrasonic transducers to excite ZGV-FGWs in a welded joint
and provides a reference for future experimental investigations of ZGV-FGWs in complex structures.
1. Introduction
Ultrasonic testing is a versatile non-destructive testing (NDT)
method, finding applications across a wide spectrum of industries.
Notably, advancements in transducer efficiency and signal processing
have paved the way for non-contact air-coupled ultrasonic testing to
emerge as a viable alternative to traditional immersion or contact testing
techniques [1–3].
Furthermore, ultrasonic guided waves have gained widespread
recognition and utilization in non-destructive evaluation (NDE) [4–6]
and structural health monitoring (SHM) [7,8]. The prominent advantage
of ultrasonic guided wave technology lies in its capacity to enable highly
accurate inspections, demonstrating remarkable sensitivity to structural
damages and defects. Lamb waves in a plate loaded with droplets were
successfully excited and detected using air-coupled ultrasonic trans­
ducers [9], demonstrating that the air-coupled ultrasonic testing tech­
nique had been successfully applied to ultrasonic guided waves.
Among the various types of Lamb waves utilized for inspection
purposes in elastic plates, Zero-Group-Velocity (ZGV) Lamb waves stand
out as a unique category. ZGV Lamb waves are characterized by a nonzero wavenumber while exhibiting the intriguing property of vanishing
group velocity. Significantly, the energy associated with ZGV Lamb
waves remains locally resonant within the excitation region, refraining
from propagating along the elastic plate. Tolstoy and Usdin were the
* Corresponding authors.
E-mail addresses: mxdeng@cqu.edu.cn (M. Deng), liweibin@xmu.edu.cn (W. Li).
https://doi.org/10.1016/j.ultras.2024.107362
Received 31 March 2024; Received in revised form 16 May 2024; Accepted 29 May 2024
Available online 6 June 2024
0041-624X/© 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
X. Meng et al.
Ultrasonics 142 (2024) 107362
first to predict the presence of ZGV Lamb waves in elastic plates [10].
Subsequent research efforts had further confirmed the resonance phe­
nomenon of ZGV Lamb waves in elastic plates [11–13]. Prada et al.
introduced the potential application of ZGV resonance in precision
thickness measurements for micron-scale plates and membranes [14].
Zhang et al. explored the use of the electromechanical impedance
technique to extract and enhance ZGV and cutoff frequency resonances
in a waveguide structure [15]. Additionally, investigations into ZGV
Lamb waves have extended to various structures, including elastic and
viscous thin bonded Aluminum/Adhesive/Aluminum plates [16],
multilayered plates [17], and high-performance anisotropic Cu–Al–Ni
alloy plates [18]. In another study, combining air-coupled ultrasonic
testing technique with ZGV Lamb waves, Holland and Chimenti
employed a broadband, focusing, air-coupled transducer to excite the
ZGV S1 mode within plates [19]. Nevertheless, to date, there has been an
absence of experimental investigations into Zero-Group-Velocity
Feature Guided Waves (ZGV-FGWs) within complex structural config­
urations utilizing air-coupled ultrasonic transducers.
In this paper, we present our findings derived from measurements
involving the resonance phenomenon of ZGV-FGWs within a welded
joint. These measurements were conducted using the pitch-catch mea­
surement technique with air-coupled ultrasonic transducers. The timedomain signals acquired through pitch-catch measurements using aircoupled ultrasonic transducers have been confirmed to represent ZGV
Lamb waves in the elastic plate. By applying the same pitch-catch
measurement technique with air-coupled ultrasonic transducers, we
obtained time-domain signals from the welded joint. Through a
comprehensive analysis of these received time-domain signals, coupled
with corresponding Short-Time Fourier Transforms (STFT), we have
definitively established that the received time-domain signals within the
welded joint indeed exhibit ZGV characteristics, identified as ZGVFGWs. This validation unequivocally demonstrates the feasibility of
utilizing the pitch-catch measurement method with air-coupled ultra­
sonic transducers to effectively generate and detect ZGV-FGWs within
welded joints.
layers were added around the air domain to absorb the reflected wave at
the interface. The material properties of the PML layer were consistent
with those of air. We applied low-reflecting boundary conditions on both
sides of the elastic plate to minimize signal reflection, while the sound
soft boundaries were imposed on the boundaries enclosing the air
domain. The coupling between the air and the elastic plate domains was
realized through the acoustic structure interface. The material chosen
for the elastic plate was aluminum alloy, with material parameters
including the mass density (ρ) of 2750 kg/m3, Young’s modulus (E) of 74
GPa, and Lame’s constants (λ and μ) of 51 GPa and 26 GPa, respectively.
Fig. 2 presents acoustic field distributions of u1 and u2 in the elastic
plate in the x1 and x2 directions. Displacement field distribution of u1 is
symmetric and displacement field distribution of u2 is antisymmetric
along the x1 axis, and the effect of the air medium on displacement field
distributions in the elastic plate is negligible. The frequency corre­
sponding to acoustic field distributions can be determined as the reso­
nance frequency of ZGV Lamb wave in the elastic plate. For the selected
geometric and material parameters of the elastic plate, the resonance
frequency of ZGV Lamb wave is 479.4 kHz.
In the case of the welded joint, we present the geometric parameters
of the 2D FE model, illustrated in Fig. 3. The model settings of the
welded joint were consistent with those of the elastic plate described
above. Acoustic field distributions of u1 and u2 in the welded joint in
Fig. 4 are obtained, and referring to the analysis of acoustic field dis­
tributions of ZGV-FGW in our previous work [20], the frequency at
which the resonance phenomenon occurs in the welded joint is 399.8
kHz. Displacement field distribution of u1 is symmetric along the x1 axis,
while displacement field distribution of u2 is antisymmetric along the x1
axis. The influence of sound pressure on displacement field distributions
on both sides of the welded joint can be disregarded. This allowed us to
determine the resonance frequencies of ZGV Lamb wave in the elastic
plate and ZGV-FGW in the welded joint.
3. Experimental validation of ZGV-FGWs in a welded joint
3.1. Experimental preparation
2. Determination of resonance frequency of ZGV guided wave
The experimental setup using the pitch-catch measurement method
with air-coupled ultrasonic transducers is shown in Fig. 5(a). A
computer-controlled transmitter–receiver (Ritec 5000 SNAP system)
was used to generate RF tone burst voltages for exciting the air-coupled
ultrasonic transducer Tx (center frequency 400 kHz, diameter 40 mm,
400 K-20 N-R50-T) and to receive and process the detected signals with
the air-coupled ultrasonic transducer Rx (center frequency 400 kHz,
diameter 40 mm, 400 K-20 N-R50-R). The amplified RF tone burst
voltage was passed through a specific attenuator to suppress the tran­
sient behavior of the RF amplifier. When a N-cycle sinusoidal tone burst
voltage with a carrier frequency of f was applied to Tx, an ultrasonic
wave tone burst will be excited and propagate through the detected
To determine the resonance frequencies of ZGV Lamb wave in the
elastic plate and ZGV-FGW in the welded joint, we employed a twodimensional (2D) finite element (FE) method to analyze acoustic field
distributions at resonance frequencies.
The geometric parameters of the 2D FE model of the elastic plate in
the air medium are depicted in Fig. 1. The entire domain was discretized
with a free triangular node. Excitation (prescribed pressure of 10 Pa)
was applied at bounded excitation source to simulate the transducer Tx
excitation situation, the receiver source was used to simulate the
reception of the time-domain signal with the transducer Rx. The width
of the bounded excitation source was set at 40 mm. The 6 mm thick PML
Fig. 1. Schematic of the 2D model for the elastic plate in the air medium.
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Ultrasonics 142 (2024) 107362
Fig. 2. Acoustic field distributions of u1 (a) and u2 (b) in the elastic plate in the x1 and x2 directions at different scales (color coding visualizes the variance of
displacements or sound pressure over the cross section).
Fig. 3. Schematic of the 2D model for the welded joint in the air medium.
specimen (the elastic plate or the welded joint). The pulse-echo response
of the excited ultrasonic wave tone burst was received by Tx through the
sampler, while the ultrasonic signal propagating through the detected
specimen was received by Rx, which was then amplified by the pre­
amplifier. Fig. 5(c) illustrates the position of air-coupled ultrasonic
transducers in relation to the detected specimen, and there is no contact
between air-coupled ultrasonic transducers and the detected specimen,
which provides one of the prerequisites for the formation of ZGV guided
wave mode.
We machined and customized the elastic plate and the welded joint
based on the dimensions of the 2D FE model in Section 2. The di­
mensions of the elastic plate and welded joint are 500 mm × 500 mm ×
6 mm, and the dimensions of the x1x2 cross-section of feature region of
the welded joint refer to Fig. 3.
3.2. Experimental observation of ZGV Lamb wave in the elastic plate
We selected the 10-cycle Hanning-windowed sinusoidal tone burst
voltage with the carrier frequency of 480 kHz and 520 kHz to excite
ultrasonic guided wave in the elastic plate. Fig. 6(a) and 6(b) present the
received time-domain signals at excitation carrier frequencies of 480
kHz and 520 kHz in the elastic plate, respectively. It can be straight­
forward to observe that the received time-domain signals in the elastic
plate oscillate during a long time. To analyze the frequency content in
the received time-domain signal, we employed STFT technique for more
detailed analysis. Time-frequency spectra of the time-domain signals in
Fig. 6(a) and 6(b) enhances our comprehension of the formation process
of the received time-domain signals in the elastic plate over time, as
illustrated in Fig. 6(c) and 6(d). It can be observed that the resonance
phenomenon still persists after approximately 500 μs and the maximum
value of the spectral amplitude at 800.8 μs corresponds to the frequency
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Ultrasonics 142 (2024) 107362
Fig. 4. Acoustic field distributions of u1 (a) and u2 (b) in the welded joint in the x1 and x2 directions at different scales (color coding visualizes the variance of
displacements or sound pressure over the cross section).
Fig. 5. (a) Experimental setup using the pitch-catch measurement method with air-coupled ultrasonic transducers; (b) enlarged view of Ritec 5000 SNAP system; (c)
the position of air-coupled ultrasonic transducers in relation to the detected specimen (the case of the welded joint).
of 477.2 kHz in the elastic plate. The experimentally determined reso­
nance frequencies in the elastic plate closely concurs with the results
obtained through simulation in Fig. 2, and the error in the resonance
frequency is only 0.46 %.
Additionally, carrier frequencies in the range of 430 kHz to 530 kHz
were selected for excitation in the elastic plate, with intervals of 10 kHz.
This allows us to receive time-domain signals under different excitation
carrier frequencies. The time-domain signals received at different
excitation carrier frequencies in the elastic plate oscillate for extended
periods of time. Resonance phenomena are observed in the time-domain
signals received at different excitation carrier frequencies, with a
consistent resonance frequency of 477.2 kHz. The spectral amplitude at
different excitation carrier frequencies is further extracted, and the re­
lationships between the normalized spectral amplitude at 800.8 μs and
excitation carrier frequency in the elastic plate are given in Fig. 7. It can
be concluded that the normalized spectral amplitude is the largest when
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X. Meng et al.
Ultrasonics 142 (2024) 107362
(a)
(b)
0.08
0.08
480 kHz
520 kHz
0.04
Volt (V)
Volt (V)
0.04
0.00
-0.04
-0.08
0.00
-0.04
0
200
400
600
800
-0.08
1000
Time (µs)
0
200
400
600
800
1000
Time (µs)
Fig. 6. The received time-domain signals at excitation carrier frequencies of 480 kHz (a) and 520 kHz (b) in the elastic plate; time vs frequency spectra of the
received time-domain signals at excitation carrier frequencies of 480 kHz (c) and 520 kHz (d).
wave in the elastic plate, we conducted experiment in the welded joint
using a similar experimental protocol of pitch-catch measurement
technique with air-coupled ultrasonic transducers. We selected the 10cycle Hanning-windowed sinusoidal tone burst voltage with the car­
rier frequency of 400 kHz and 440 kHz to excite ultrasonic guided wave
in the welded joint.
Fig. 8(a) and 8(b) present the received time-domain signals at exci­
tation carrier frequencies of 400 kHz and 440 kHz in the welded joint,
respectively. The time-domain signals received in the welded joint can
oscillate for extended periods of time. The corresponding time­
–frequency spectra are presented in Fig. 8(c) and 8(d). When the exci­
tation carrier frequency deviates from the resonance frequency,
resonance is still formed at the resonance frequency. Multiple frequency
components are present until 400 μs. As the time increases, the fre­
quency components are gradually single and the resonance persists. The
received time-domain signal exhibits ZGV characteristics based on the
time–frequency spectra. The frequency corresponding to the maximum
of the spectral amplitude at 800.8 μs is 397.9 kHz. The experimentally
determined resonance frequency in the welded joint differs from that in
Fig. 4 by only 1.9 kHz.
Furthermore, carrier frequencies in the range of 350 kHz to 450 kHz
are selected for excitation in the welded joint, with intervals of 10 kHz.
The time-domain signals received at different excitation carrier fre­
quencies in the welded joint oscillate during a long time at a consistent
resonance frequency of 397.9 kHz. Fig. 9 presents the relationship be­
tween the normalized spectral amplitude at 800.8 μs and excitation
carrier frequency in the welded joint. The spectral amplitude increases
and then decreases with increasing excitation carrier frequency, and the
spectral amplitude at the resonance frequency is the largest. The above
analysis determines that the received time-domain signals exhibit ZGV
characteristics. Conclusively, the presence of ZGV-FGWs is successfully
verified at the experimental stage using the pitch-catch measurement
method with air-coupled ultrasonic transducers.
Normalized spectral amplitude
1.2
1.0
0.8
0.6
0.4
0.2
430
455
480
505
530
Excitation carrier frequency (kHz)
Fig. 7. Relationship between normalized spectral amplitude and excitation
carrier frequency in the elastic plate.
the excitation carrier frequency is the resonance frequency in the elastic
plate. These findings collectively affirm that the received time-domain
signals in the elastic plate are characterized by ZGV Lamb waves. The
above analysis proves the feasibility of this experimental method of
pitch-catch measurement technique with air-coupled ultrasonic trans­
ducers in detecting ZGV Lamb waves in the elastic plate.
3.3. Experimental observation of ZGV-FGW in the welded joint
Analogous to the experimental observation process of ZGV Lamb
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X. Meng et al.
Ultrasonics 142 (2024) 107362
(a)
(b)
0.12
400 kHz
0.00
0.00
-0.06
-0.06
-0.12
440 kHz
0.06
Volt (V)
Volt (V)
0.06
0.12
0
500
1000
1500
-0.12
2000
0
500
1000
1500
2000
Time (µs)
Time (µs)
Fig. 8. The received time-domain signals at excitation carrier frequencies of 400 kHz (a) and 440 kHz (b) in the welded joint; time vs frequency spectra of the
received time-domain signals at excitation carrier frequencies of 400 kHz (c) and 440 kHz (d).
welded joint oscillate for extended periods of time. Using STFT on the
received time-domain signals at different excitation carrier frequencies
allows us to analyze the frequency content of these signals. Our findings
revealed that the spectral amplitude is the largest when the excitation
carrier frequency corresponds to resonance frequency. The received
time-domain signals at different excitation carrier frequencies exhibit
ZGV characteristics, identified as ZGV-FGWs. Importantly, the experi­
mentally determined resonance frequency aligns with the results ob­
tained through simulation. Ultimately, we experimentally confirm the
presence of ZGV-FGWs in the welded joint.
This investigation substantiates the viability of using the pitch-catch
measurement method with air-coupled ultrasonic transducers to excite
ZGV-FGWs in the welded joint. Furthermore, there is an anticipation
that ZGV-FGWs can be applied to evaluate the localized damage of the
material and measure the thickness of complex feature structures, it
serves as a valuable reference for the prospective utilization of ZGVFGWs in NDE and SHM of complex feature structures.
Normalized spectral amplitude
1.2
0.9
0.6
0.3
0.0
350
375
400
425
450
Excitation carrier frequency (kHz)
CRediT authorship contribution statement
Fig. 9. Relationship between normalized spectral amplitude and excitation
carrier frequency in the welded joint.
Xiangdi Meng: Writing – original draft, Validation, Software,
Methodology, Investigation, Data curation. Mingxi Deng: Writing –
review & editing, Supervision, Methodology, Investigation, Funding
acquisition, Formal analysis, Conceptualization. Weibin Li: Writing –
review & editing, Validation, Investigation, Funding acquisition,
Conceptualization.
4. Conclusions
Our investigation was rooted in the experimental observation of
ZGV-FGWs in the welded joint, facilitated by the pitch-catch measure­
ment method with air-coupled ultrasonic transducers. We first verified
the feasibility of the pitch-catch measurement method with air-coupled
ultrasonic transducers by observing ZGV Lamb waves in the elastic plate.
Utilizing the same pitch-catch measurement method with air-coupled
ultrasonic transducers, we observed the received time-domain signals
in the welded joint. Notably, the received time-domain signals in the
Declaration of competing interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
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X. Meng et al.
Ultrasonics 142 (2024) 107362
Data availability
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Data will be made available on request.
Acknowledgments
This project was supported by the National Natural Science Foun­
dation of China, Grant Nos. 12134002 and 12074050.
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