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All issues Volume 8 (2024) Acta Acust., 8 (2024) 66 Full HTML
Open Access
Review
Issue
Acta Acust.
Volume 8, 2024
Article Number 66
Number of page(s) 18
Section Ultrasonics
DOI https://doi.org/10.1051/aacus/2024069
Published online 06 December 2024

© The Author(s), Published by EDP Sciences, 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Plate-like metal structures are integral across various engineering fields, including shipbuilding, aerospace, automotive, energy production, and other critical industries. Often part of larger systems, these structures undergo diverse configurations and manufacturing processes to achieve their desired form and function. Given their critical role in ensuring system safety and integrity, maintaining the structural health of these materials is crucial to guarantee safe operation and minimize downtime [1–3]. Condition monitoring through non-destructive evaluation (NDE) techniques is essential throughout the lifespan of these materials. NDE methods detect features such as internal defects, such as cracks [4–6], porosity and inclusions [7, 8], and joint interface imperfections [9–11]. They also provide geometry information in areas inaccessible to visual inspection, like internal corrosion in pipeline-insulated sections [12–14].

Ultrasonics is a widely used NDE approach for inspecting metal plates. Using time-domain reflectometry with a hand-held transducer, the conventional A-scan technique is typical for detecting internal defects or material thickness loss [15, 16]. Integrating the pulse-echo method into an automated system allows for C-scanning, producing images that pinpoint defect locations or create 3D tomographic images [17]. However, C-scanning is slow and impractical for large structures like ship hulls, aircraft panels, pipelines, and storage tanks.

For large plate-like structures, long-range detection is preferred over exhaustive point measurements. Ultrasonic-guided waves offer a solution, leveraging their ability to travel long distances laterally in the plate. Specifically, Lamb or plate waves are highly effective for this purpose.

The Lamb waves technique for NDE has been extensively researched and applied to various materials and structural designs, demonstrating its practicality and versatility. It is used in condition monitoring of pipes [18], aerostructures [19, 20], and quality control of rolled steel products [21, 22]. Nevertheless, despite its potential, adapting the Lamb waves method to different materials can be challenging due to its complex behavior and the intricacies of long-distance propagation, especially in multimodal or dispersive wave excitations.

This paper highlights the significance of the Lamb waves method for NDE in metal plates, covering its theoretical foundations and practical applications. By presenting two case studies, the paper elucidates the method’s ability to detect structural features such as welded joints, stiffeners, and wall thickness loss while addressing the complexities of measurement and interpretation. Each case study includes detailed procedures, result analyses, and encountered challenges. These demonstrations provide readers with a comprehensive understanding of the method’s benefits and limitations, fostering further advancements and enhancing its robustness.

Furthermore, by presenting two case studies encompassing novel experiments and innovative approaches, this paper provides a comprehensive review designed to guide readers new to the field. The review offers practical insights and structured information, facilitating a deeper understanding and mastery of Lamb waves.

The first case study uses a B-scan image to locate weld joints and stiffener bars, which can be easily replicated with only a pair of PZT (Lead zirconate titanate) transducers and minimal signal processing. This contrasts with recent studies that use more transducers and advanced signal processing for higher accuracy [23–27]. The second case study employs a mode sensitivity approach, isolating a higher-order, dispersive mode post-measurement via frequency filtering rather than propagating a single target mode. This method, inspired by Alleyne, Cawley, and others [18, 28, 29], highlights the sensitivity of specific modes and uniquely identifies damage indicators through frequency filtering.

Although this paper primarily serves as a review, it is essential to highlight that the experiments presented within the two case studies are original and specifically conducted for this research. The experiments are designed and executed to provide fresh insights and validate the discussed methodologies.

2 Overview of Lamb waves

Lamb waves are guided waves in a solid plate. They occur due to the simultaneous propagation of longitudinal and shear waves in the plate, combined with geometrical constraints. Lamb waves induce parallel and orthogonal displacements to the plate’s principal plane. The foundational mathematical framework for Lamb waves was initially formulated by Horace Lamb [30], hence the name. An intricate property of Lamb waves that poses both benefits and challenges for its technical use is its multimodal propagation with dispersive behavior.

A Lamb wave may exist in a symmetric or antisymmetric mode, where symmetry is defined with respect to the midplane of the plate, as illustrated in Figure 1.

thumbnail Figure 1

Lamb modes: (a) symmetric and (b) antisymmetric.

Each mode of Lamb waves propagates at a velocity that depends on frequency, defined as the waves’ dispersion behavior. The frequency and velocity relations for symmetric and antisymmetric modes of Lamb waves are given by the Rayleigh-Lamb equations (1) and (2), respectively [31, 32].

 tan(qd)tan(pd)=-4k2pq(q2-k2)2, for symmetric modes$$ \begin{array}{cc}\frac{\enspace \mathrm{tan}\left({qd}\right)}{\mathrm{tan}\left({pd}\right)}=-\frac{4{k}^2{pq}}{{\left({q}^2-{k}^2\right)}^2},\enspace & \mathrm{for}\enspace \mathrm{symmetric}\enspace \mathrm{modes}\end{array} $$(1)

tan⁡(qd)tan⁡(pd)=-(q2-k2)24k2pq, for antisymmetric modes.$$ \begin{array}{cc}\frac{\mathrm{tan}({qd})}{\mathrm{tan}({pd})}=-\frac{{\left({q}^2-{k}^2\right)}^2}{4{k}^2{pq}},\enspace & \mathrm{for}\enspace \mathrm{antisymmetric}\enspace \mathrm{modes}\end{array}. $$(2)

Equations (1) and (2) assign the plate thickness as 2d (thus d is half-thickness), the other terms are defined as follows:

p2=ω2cL2-k2, q2=ω2cS2-k2, k=ωcp.$$ \begin{array}{ccc}{p}^2=\frac{{\omega }^2}{{{c}_{\mathrm{L}}}^2}-{k}^2,\enspace & {q}^2=\frac{{\omega }^2}{{{c}_{\mathrm{S}}}^2}-{k}^2,\enspace & k=\frac{\omega }{{c}_{\mathrm{p}}}.\end{array} $$(3)

where ω, cL, cS, k, cp are the angular frequency (2πf), velocities of longitudinal wave, velocity of shear wave, wavenumber, and phase velocity, respectively.

Dispersion curves are generated based on the relations in the preceding equations, plotted in a frequency axis against wavenumber (f-k) or phase velocity (f-cp). In addition, an important parameter that describes the dispersion of Lamb waves is group velocity (cg), obtained through equation (4).

cg=∂ω∂k=∂(kcp)∂k=cp+k∂cp∂k.$$ {c}_{\mathrm{g}}=\frac{{\partial \omega }}{{\partial k}}=\frac{\partial \left(k{c}_{\mathrm{p}}\right)}{{\partial k}}={c}_{\mathrm{p}}+k\frac{\partial {c}_{\mathrm{p}}}{{\partial k}}. $$(4)

Furthermore, since the relation of Lamb wave parameters is a function of the plate thickness, it is sometimes more convenient to express the plot in terms of the frequency-thickness product (fd) instead of the frequency only. That way, the plot is more generalized for any thickness of plates.

Pertaining to the discussion on dispersion curves, Figures 2a–2c exhibit the curves for typical aluminum plotted as relations of fd with wavenumber (k), phase velocity (vp), and group velocity (vg), respectively. These figures are generated numerically with Disperse®, by specifying the material density (2700 kg/m3), Poisson’s ratio (0.34), and Young’s modulus (70.76 GPa). The plots in Figure 2 show that there are 9 possible modes (5 Antisymmetric and 4 Symmetric) propagating in an aluminum plate within 0–9 MHz · mm range. The range is set as necessary by considering the computational time as it depends on the number of points involved in the numerical iteration.

thumbnail Figure 2

Dispersion curves of Aluminum, plotted as relations between frequency-thickness and (a) wavenumber, (b) phase velocity, and (c) group velocity.

Note that the horizontal axis is the fd product. Plotting in fd axis generalizes the chart, making it thickness independent. One can then conveniently convert the axis into frequency (f) when using the chart for a plate of known thickness, allowing for a more explicit spectrum analysis. Thus, for example, if the plate is of 6 mm thickness (d) as the one described later in Section 7, the frequency (f) range being observed in these plots equals fd range (0–9 MHz · mm) divided by 6 mm, hence 0–1.5 MHz.

The wavenumber and phase velocity dispersion curves, shown in Figures 2a and 2b, respectively, indicates the speed of the propagating modes as a function of frequency. Recall that wavenumber (k) and velocity (v) are related through k = f/v. This phase velocity information is important to determine the dispersiveness of the mode. Focusing on the phase velocity dispersion curves in Figure 2b, the non-dispersive regime is readily identified when the curve of a mode is substantially flat. Within this regime, the velocity of the wave does not significantly vary with frequency. For instance, the plot in Figure 2b suggests that exciting the fundamental symmetric mode (S0) at low fd product (up to 1.5 MHz · mm) generates non-dispersive waves. The waves, although having a range of frequencies, travel at a nearly uniform speed, preserving the apparent waveform even after long distances. This constant waveform shape simplifies the analysis of time-domain signals, making the non-dispersive mode at low frequencies widely preferred for material inspection applications.

On the other hand, operating in higher-order modes is less common for material inspection practices, primarily due to the challenges associated with capturing the non-dispersive regime of those modes. This target regime occurs at relatively high frequencies, as seen in the curves of A1, S1, and the next higher order modes. High-frequency waves experience elevated attenuation, reducing their propagation range. Another issue is the proximity of the velocity magnitudes of the non-dispersive regime (the flat part of the curves) among those higher-order modes. Referring to Figure 2b, at 7 MHz · mm where A1 is in the non-dispersive regime, other modes also exist at nearly around that velocity magnitude. As a result, these modes may merge or overlap in the waveform as they propagate. It is unlike the case of mode S0 at fd = 1.5 MHz · mm discussed earlier, where even though mode A0 can simultaneously exist at that fd, they have significantly distinct velocities. Thus, if both modes are propagating, they are still readily distinguishable because each one will show up at a well-distanced time delay to the other. Despite the challenges, it is possible and sometimes more effective to use higher order modes for inspecting a structure, by employing techniques such as those elaborated in Section 4 and demonstrated in the case studies (Sects. 6 and 7).

3 Generation and detection of Lamb waves

From the perspective of a practically oriented introduction to ultrasonic non-destructive evaluation of metals through Lamb wave inspections, it is relevant to first look at transducer types that have sufficiently found maturity in this field before proceeding further.

3.1 Piezoelectric transducer

The most used transducer type is built on piezoelectric material of lead zirconate titanate, Pb[ZrxTi1−x]O3 (0 ≤ x ≤ 1), commonly known as PZT. It stands out for its versatility and affordability. Its performance was extensively validated in various Lamb wave investigations for generation and detection purposes [33], evident from its widespread use across different materials and inspection scenarios. PZT transducers are also adaptable for oblique incident emission and detection, often paired with wedges as a standard industry practice [34–36]. Wedges for flat surfaces or curved ones in pipes or cylinders exist [37, 38]. In addition, by properly arranging of piezoelectric elements, phased array transducers can be designed.

Despite their simplicity, the size of PZT transducers often limits their use in setups where actuators or sensors are embedded in the structure, requiring careful consideration of transducer thickness. For those situations, piezoelectric wafer transducers (PWT), sometimes called piezoelectric wafer active sensors (PWAS), exist with micrometer-order thick elements readily integrable into the tested structure. Their utility was demonstrated in Lamb wave-based monitoring of large plate structures, such as in the aeronautics industry [39, 40].

Nevertheless, PWTs tend to be brittle and not suitable for curved structures. Comb [38, 41, 42] and interdigital [33, 43] transducers exist that overcome the shortcomings of PWTs. They are flexible composites containing a piezoelectric polyvinylidene fluoride polymer (PVDF).

Actuation and detection of Lamb waves using piezoelectric transducers often require contact with the structure. Conventional PZT transducers need consistent coupling for effective wave transmission, often affected by the coupling layer’s uniformity and surface roughness [44]. The adhesive bonding impacts performance for thin, embedded transducers due to impedance matching between the transducer, adhesive, and structure [45]. These factors are crucial in designing experiments and analyzing results.

3.2 Electromagnetic acoustic transducer (EMAT)

Unlike piezoelectric transducers, EMAT offers a non-contact solution for wave generation and detection, essential for high-temperature surfaces where direct contact or coupling is impossible. The transducer, comprising a magnet and coil, operates on the Lorentz force principle. An electric current through the coil induces an eddy current in the material, which, under the magnetic field, experiences Lorentz force, causing material vibration. Conversely, vibrations in the material induce an electric current in the coil, enabling signal detection by the EMAT [46, 47].

EMAT has been employed for Lamb wave excitation on metals such as aluminum [48], steel [49], and various metals [50, 51] in research and industrial settings. Nevertheless, its working principle limits its applications only to electrically conducting materials [52, 53]. Still, a technique exists to employ EMAT in nonconducting materials by depositing a thin metal layer on the surface [54]. However, despite its effectiveness, this solution involves additional preparation procedures and inevitably modifies the material surface. Moreover, EMAT transducers generally have lower signal-to-noise ratios than piezoelectric ones [55, 56].

3.3 Laser ultrasonic technique

Lasers are exploited for the non-contact generation of guided waves, including Lamb waves [57] or Rayleigh waves [58]. A high-power pulsed laser (Nd:YAG or CO2) creates a thermoelastic expansion pulse on the material at the point of laser incidence, generating propagating waves in the material. Pairing the laser-induced acoustic waves and laser Doppler vibrometry method provides a fully non-contact inspection setup.

Laser Doppler vibrometer is utilized for non-contact vibration detection. The device directs a laser beam toward the object while analyzing the Doppler shift of the reflected beam to capture the amplitude and frequency of vibrations. Through laser interferometry, the displacement or velocity of the targeted surface is determined [59].

Despite its non-contact benefits and the ability to operate at a considerable distance, laser Doppler vibrometry is restricted by high costs and delicate operation. Since the sensor detects the backscattered laser, a reflective surface is required to ensure a high signal-to-noise ratio. Thus, inspection on rough or non-reflective surfaces is challenging [60, 61]. A reflective tape attached onto the surface typically solves this problem. However, attaching reflective tape may be impractical due to surface conditions, and adding a new layer can alter wave propagation in thin plates.

4 Mode selection

Particular Lamb wave modes may be preferred based on criteria such as non-dispersiveness, minimum attenuation for long-range propagation, and sensitivity to specific defects [18, 29, 62]. Fundamental modes, A0 and S0, are famous in research and industry for their non-dispersiveness at low frequencies and maintain shape and amplitude over long distances. Despite these advantages, however, these fundamental modes are not necessarily sensitive to all defect types. Thus, in some cases, higher-order modes are preferred [63]. For instance, modes such as S1 and A1 provide higher sensitivity to near-surface defects in steel plates, while the fundamental modes are more suitable for in-depth detection [64]. The mode’s energy distribution causes the difference.

Higher-order modes have also proven their versatility for crack detection in multilayered structures of various materials [65–67]. Their sensitivity to defects in specific layers is more important than the dispersive behavior and inherently high attenuation of high-frequency waves, instilling confidence in their ability to detect specific defects.

Theoretically, Snell’s law provides a precise tool for mode selection by specifying the required angle of incidence, θ, of emission wave for a given frequency. For example, Snell’s law for a wedge with the sound speed cw, and the speed of the desired Lamb waves mode in the target material cmode, provides sin θ = cw/cmode, ensuring a secure understanding and application of this tool.

When angle beam excitation is not possible, such as when the desired wedge is unavailable, other excitation methods are necessary, such as normal incidence excitation. This excitation technique is often employed in Lamb wave inspection despite sacrificing particular mode selection and causing multiple Lamb wave generations. While signal processing and interpretation in multimodal Lamb waves are more complex than in single-mode waves, their practicality outweighs their drawbacks and, thence, their popularity [68–70].

5 Lamb waves-based non-destructive evaluation (NDE) methods

Employing Lamb waves as guided waves in NDE allows for long-range detection, contrasting with the conventional C-scan method using bulk waves for direct inspection. This capability makes the Lamb waves technique a more efficient measurement method.

Lamb waves NDE is extensively used for various materials, including isotropic ones like glass and metals, and anisotropic ones such as composites. Applications include defect inspection in rolled steel plates [21], pipelines [12, 71], railways [72], ship structures [73, 74], aerostructures [75–77], and other industrial uses. It is also employed in sustainable energy related fields for inspecting wind turbine blades [78, 79] and solar photovoltaics [80, 81]. With its broad applications, Lamb waves techniques have promising industrial prospects.

Lamb waves are closely linked to material properties and can be used for NDE of mechanical properties. Young’s modulus, indicating material stiffness, significantly affects wave propagation. In relation to the dispersive behavior of the waves, the difference in material stiffness manifests in the distinction of the dispersion curve pattern [82, 83]. Besides stiffness, Lamb waves can also be used to investigate hardness [84]. Therefore, Lamb waves are suitable for inspection of bulk properties and material uniformity in plates.

Lamb waves NDE is primarily used for inspecting discontinuities and geometry variations, such as cracks [85], porosity [86–89], inclusion [90], and thickness loss due to wear and corrosion [91–93]. Crack detection involves observing responses as waves travel through a structure. Encounters with cracks cause scattering, resulting in detectable changes in amplitude, phase, and frequency content, which indicate the presence and nature of cracks. While some defect indicators are evident in the time domain, they are often more apparent in the frequency domain [94, 95].

A pure fundamental mode, A0 or S0, is typically excited within the non-dispersive range, located at low frequency regime and less prone to attenuation. This simplifies defect identification, as the constant wave shape makes defect-induced alterations more noticeable. Alleyne and Cawley’s pitch-catch experiment on notched steel showed that response amplitudes are proportional to notch depth [34]. Similarly, Tua et al. demonstrated the detection and localization of cracks using Lamb waves in a time-of-flight analysis with PZT transducers [85]. Researchers have also utilized time of flight and amplitude analysis of Lamb waves in more complex structures to detect debonding in adhesive joints [96, 97].

Another method involves transmission and reflection coefficients for inspecting defects or features in the structure [27, 98]. However, the effectiveness of these techniques for flaw detection depends on the material, geometry, and the Lamb waves generated in the structure. In some instances, the phase shift is more noticeable than the decrease in amplitude when interacting with defects, as Chen et al. [99] noted in their study on fatigue damage. Advancements in data analysis and machine learning tend toward simultaneous analysis of multiple parameters for damage detection. He et al. [100] illustrate this approach by integrating correlation coefficient, amplitude change, and phase shift as co-indicators of fatigue crack detection.

In certain instances, analyzing signals in the frequency domain is preferable over evaluating them in the time domain. A subtle crack may not alter the amplitude or time of flight noticeably in the time domain but can affect the high-frequency component. High-frequency waves, more sensitive to defects, may reveal changes in the frequency spectrum. Defects can also induce nonlinear behavior, leading to higher harmonics in the spectrum, as shown in studies on cracked plates [101, 102].

Defects in the wave’s path can cause Lamb wave mode conversion, which is utilized for damage detection. The effect is more visible in the pure mode excitation as the change in waveform and spectrum can be readily identified. However, the approach also works in multimodal excitation [103].

6 Case study I: Detecting weld joint and plate stiffener with A0 Lamb mode

This section demonstrates the efficacy of Lamb waves in detecting welds and stiffeners on a steel plate, particularly relevant to ship construction. In the shipbuilding industry, steel plates are welded to form a continuous hull skin attached to stiffener frames, making weld integrity critical. Ultrasonic methods are standard for inspecting these features in marine applications [104]. This case study includes both numerical simulations and corresponding experiments, showing comparable results. However, some discrepancies exist due to differences between idealized conditions in simulations and practical conditions in experiments.

6.1 Numerical simulation

A commercial software, COMSOL Multiphysics®, simulates guided wave propagation on the modeled mock-up of steel plate with welds and stiffener. The fundamental antisymmetric Lamb wave A0 is chosen for its non-dispersive characteristics at low frequencies. The numerical study observed the interactions between the modes and the different features, i.e., welds and stiffeners. For all the numerical models, 2-D plane strain conditions are considered. The material used in the experiment is a Steel S235 plate without any treatment, only coated with an antifouling layer of 0.8 mm thickness. The steel plate is coated with an antifouling layer as in a real ship hull to protect the surface. In marine environments, this antifouling coating prevents the buildup of organisms on the hull panels. Specific to our plate specimen, the coating layer is part of the subject of investigation in another study on the effect of lift-off (distance between transducer to the material surface) on EMAT measurements, which is beyond the scope of this paper. Nevertheless, this thin coating layer does not affect the behavior of the propagating Lamb waves at a low-frequency regime.

In order to prove the negligible effect of the coating layer on a steel plate at low frequency, the Lamb waves dispersion curves of non-coated and coated plate, generated by Disperse®, are plotted on the same plot in Figure 3. The thickness of Steel S235 plate is known (8 mm). Hence the horizontal axis can be explicitly expressed as frequency (f) instead of fd product. As for the coated plate, a 0.8 mm epoxy coating layer is added on the top (Density, ρ = 1400 kg/m3; Young’s modulus, E = 3500 MPa; Poisson’s ratio, ν = 0.35).

thumbnail Figure 3

Dispersion curves of non-coated and coated steel plate (8 mm thick): Within the frequency band investigated in this study (80–250 kHz), mode A0 and S0 of non-coated and coated plates are identical.

In this study, the frequency regime in concern is within the range of 80–250 kHz, as stated and demonstrated later in this section’s numerical and experimental parts. By observing Figure 3, it is evident that within that frequency band, curves of the propagating modes (A0 and S0) in the non-coated and coated plate coincide, signifying identical behavior.

Since the presence of a thin coating layer does not change the behavior of Lamb waves in the frequency regime experimented in this work, it is safe to omit this thin layer from the model geometry, leaving a single layer of 8 mm Steel S235 plate in the simulated structure. This simplified model reduces computational cost in the simulation process. Accordingly, the numerical model is fed with the mechanical properties of the corresponding material, listed in Table 1.

Table 1

Mechanical properties of the Steel S235 (according to EN1993-1-1).

Three model cross-sections were realized to obtain three different geometrical configurations as follows. Specimen #1 is a free plate without any welds or stiffener and acts as a reference. The thin antifouling coating layer on all specimens is not considered in the numerical model since, as discussed before, it has a negligible effect on the propagation of low-frequency fundamental Lamb wave mode. Specimen #2 is a plate with a butt weld at the center of the model geometry. The weld is characterized by a single V-preparation joint of 10 mm in length with a clearance (trimming) and a heel of 2 mm. The weld material has the same properties as the plate material except for Young’s modulus, which has been reduced by 10% to consider the heat-affected zone (HAZ). Specimen #3 has a welded stiffener of 100 mm length and 10 mm width located at the center of the model geometry. The stiffener bar material is the same as the main plate material (Steel S235), with its properties listed in Table 1. These three configurations are chosen to represent the different cases of features encountered by the waves on the actual mock-up specimens experimented in this investigation. The geometric representations of the three numerical model specimens are illustrated in Figure 4.

thumbnail Figure 4

2-D models of the plate representing the realized mock-up. Specimen #1: a plain plate as the reference, Specimen #2: a plate with weld, Specimen #3: a plate with a stiffener bar.

An excitation of the A0 Lamb mode is simulated by using a theoretical periodic displacement in the direction perpendicular to the plate’s principal plane/surface (y-direction) with a 5-cycle sinusoidal tone burst enclosed in a Hanning window. This excitation is applied at the left boundary (x = 0 mm) at f = 180 kHz. The transient, time-dependent simulations are conducted with a time length of over 600 μs.

The simulation mimics a pitch-catch experiment with two transducers, where an emitter excites the A0 mode at a fixed point, and a receiver acquires the response at numerous positions away from the emitter. Here, the first acquisition position is 300 mm from the emitter. Signals are also captured from that point onwards up to the distance of 700 mm between transducers. In Specimen #2 and #3, the weld and stiffener positions are located in the middle of the signal acquisition range.

B-scan images can be constructed from the signals collected from all acquisition points of all simulated cases, as depicted in Figure 5. The color map represents the intensity (normalized value, arbitrary unit) in the compiled waveforms. In specimen #1, the reference specimen, the peaks move away in the time domain as the distance to the emitter increases, as shown in Figure 5a. Notice that the high-intensity peaks range has a consistent temporal length, forming a line of uniform width throughout the tested length. The consistency of the waveform shape is because the A0 mode is excited in its non-dispersive regime. Hence, there is no variation in velocity within the propagating mode’s frequency band. Moreover, low-frequency wave ensures minor attenuation, resulting in a constant intensity level of the peaks from the closest acquisition point (300 mm) to the farthest end (700 mm).

thumbnail Figure 5

B-Scan images of the A0 mode propagation along the three simulated models

While the explanation of the B-scan image of Specimen #1 is relatively trivial, more intricate phenomena arise in the presence of a welding joint or stiffener in the structure. Figure 5b depicts the B-scan from Specimen #2, i.e., a plate with a weld joint. From the x-positions of 300–500 mm, the waveform consists of a direct emitter-receiver and reflection signal from the weld. As one can infer from the image, the reflection signal is the weaker peaks that arrive later. As the acquisition point approaches the weld position (x = 500 mm), the time delay of the direct signal increases because the emitter-receiver distance extends.

In contrast, the reflection signal reduces because the position gets closer to the weld joint reflector. That explains the converging peaks from 300 mm to 500 mm in the B-scan plot. Beyond the weld, the waveform contains only the direct emitter-receiver signal, but the shape changes due to the interaction with the weld joint.

Additionally, the signal reflected by the weld and passed through it changes its form, which is readily discernible from the variations in width and colormap patterns. After interaction with the welded joint, either reflected by or transmitted through it, the signal is no longer a pure A0 mode, but part of the wave energy converts into the S0 mode. Therefore, the reflection signal before the weld point and the transmitted one beyond it are multimodal Lamb waves consisting of A0 and S0 modes, hence having a distorted shape compared to the incident wave.

The situation in Specimen #3 is similar to that of Specimen #2, where reflected signals are captured at locations leading to the position of the stiffener bar in the middle (x = 500 mm), and the response signal received behind the stiffener appears distorted. Likewise, mode conversion occurs in both reflected and transmitted signals, as the pure A0 incident wave is transformed into a multimodal A0 and S0 Lamb wave. The distortion effect is more prominent in Specimen #3, as exhibited in Figure 5c, since a prominent structural feature is involved in this stiffener case.

6.2 Experimental validation

Initially, experimental measurements are carried out in a non-defective area to validate the existence of the targeted Lamb mode, namely the A0. Then, the specimens are measured in transmission and reflection configurations to investigate the wave interaction with the butt weld and stiffener bar.

Piezoelectric transducers excite A0 Lamb mode, as commonly practiced elsewhere. In general, the mode S0 is simultaneously generated. Both wave modes are dispersive. It is important to note that, in such a configuration, the wave generation is omnidirectional. Trials are performed on the realized mock-up of 8 mm thick Steel S235 with a paint coating. It is found that the Vallen VS150-M transducers, working in the range of 80–450 kHz with the central frequency of 150 kHz, generate an energetic A0 mode. The excitation signal is a 5-cycle tone burst enclosed in a Hanning window. The measurements are conducted at different frequencies in the range of 80–250 kHz. Recall that within low frequency range, the coating layer has negligible effect on the behavior of Lamb waves in the steel plate, as discussed in the previous section (Fig. 3). Signal acquisition is facilitated by an NI USB 6356 module with a sampling rate of 1.25 MSample/s.

For transmission configuration, signal acquisition is made by recording the waveforms from 150 mm to 300 mm at 5 mm increments. In addition to detecting welds and stiffeners, this transmission experiment on the plain plate reveals the modes generated in the measured plate by constructing dispersion curves via two-dimensional Fourier transform (2D-FFT) method [105]. In reflection configuration, two transducers are placed side-by-side, forming a pseudo-pulse-echo experiment, with the main interest in the reflection echo from butt weld or stiffener. The arrangements of the transducers on the plate for experiments on butt-welded specimens are depicted in Figure 6. These setups are also employed to investigate the plate with a stiffener bar. As stated, the plain plate is only subjected to the transmission or pitch-catch experiment.

thumbnail Figure 6

Transducer arrangement in (a) transmission and (b) pseudo-pulse-echo reflection experiments on the specimen with butt weld (Specimen #2). The experiment for a plate with a stiffener bar (Specimen #3) employs identical arrangements. (E: Emitter, R: Receiver).

The signals collected from transmission experiments at evenly spaced points conducted on the plain plate (no weld and no stiffener) are transformed with the 2D-FFT procedure into dispersion curves in frequency-wavenumber space, shown in Figure 7. The dispersion curves reveal that our measurement setup excites A0 as the dominant mode, with S0 simultaneously propagating at a substantially lower intensity. Recall that the numerical simulation in the previous section assumes ideal pure A0 excitation. Nevertheless, such pure mode excitation is only possible with a correctly angled wedge in an oblique incident excitation method. The superiority of the A0 intensity over S0 in our experiment is exhibited in the dispersion curves in Figure 7.

thumbnail Figure 7

Dispersion curves, in frequency-wavenumber space, obtained from the transmission (pitch-catch) experiment on the plain plate (Specimen #1).

Figure 8 plots the results of measurements on all three experimented specimens as B-scan images. Those images are comparable to the ones in Figure 5, which result from numerical simulations. The frequency range that gives the best compromise of high amplitude signal and non-dispersive working point is 90–140 kHz. In that range, the velocity of A0 mode is around 2500–2700 m/s, which is in good agreement with the theoretical value.

thumbnail Figure 8

B-scan from experiments on (a) Specimen #1: plain plate; (b) Specimen #2: plate with butt weld; (c) Specimen #3: plate with stiffener bar at the backside.

When propagating through a butt weld or stiffener, the A0 mode experiences a decrease in signal amplitude. Figures 8b and 8c illustrate the location of these features, namely the butt weld and stiffener, by showing a decrease in intensity along the signal acquisition line. However, detecting A0 echoes from butt welds and stiffeners is challenging, unlike in numerical simulation B-scans where clear indicators are provided. With a pseudo-pulse-echo configuration, echo detection remains tricky even when appropriate measurement positions are carefully selected on the specimen. Generally, due to omnidirectional generation, the signal composition is complex, containing reflection components from all potential reflectors, including the free edges of the plate. It is important to note that echoes from discontinuities, such as stiffeners and butt welds, should have smaller amplitudes compared to those from free edges.

6.3 Remarks on the case study I

As demonstrated through numerical simulation and experimental measurements, Lamb wave inspection based on the excitation of a single, non-dispersive mode at a lower frequency provides a straightforward approach to detecting features in the plate. The concept is proven through numerical simulation, where a pure A0 mode is propagating in the plate, and its changes after interacting with the weld and stiffener indicate the presence and positions of those features. The mode conversion phenomenon is also readily identified from A0 to S0.

The ability to detect the presence and pinpoint the locations of features is demonstrated through experimentation, inferred from the fluctuations in intensity. However, analyzing the results obtained from experimental measurements proves to be more intricate. The waves generated by this simple normal incidence excitation setup are not inherently pure single mode. Moreover, free edges in the plate sample lead to a noisy wave field caused by scattering. These factors make identifying information-carrying signals from reflection and transmission more challenging. Despite these obstacles, Lamb wave methods still hold significant potential for non-destructive inspection of metal plate-like structures. Further signal processing techniques can refine the measurement results and aid the interpretation effort.

7 Case study II: Detecting thickness reduction with high-order Lamb mode in a multimodal excitation

This case study examines the nondestructive detection of thickness reductions in metal, a critical issue caused by corrosion or wear that lead to material loss [106, 107] and stress concentration. Such thinning is often hidden, particularly on inaccessible surfaces like the obstructed side of ship hull plates or the inner side of pipes, posing significant risks of material failure.

To simulate material loss, a 6 mm thick aluminum plate was pocket-milled to create three blind holes of 20 mm diameter at varying depths (h = 1 mm, h = 2 mm, and h = 3 mm). These defects are only visible from one side, simulating inaccessible damage, as shown in Figure 9.

thumbnail Figure 9

Specimen of Aluminum plate: (a) front side without any perceptible physical sign of pockets behind it, only showing traces of markers for experimental arrangement; (b) back view showing circular 20 mm diameter pockets of different depths, h.

Lamb wave inspection was performed using a pitch-catch experiment with two identical contact transducers mounted on a scanner arm for precise control of transducer motion and signal acquisition. This setup allows for consistent and accurate detection of the simulated defects. Subsequent sections detail the measurement techniques and findings, emphasizing the high-order Lamb modes’ effectiveness in identifying hidden thickness reductions.

7.1 Obtaining the characteristics of Lamb waves in the specimen

Understanding the characteristics of the propagating Lamb waves excited by the experimental setup is essential before arranging the inspection technique for defect detection. This is achieved by obtaining the dispersion curves of Lamb waves in the specimen. A numerical approach using dispersion curve calculator software like Disperse® [108], as used in this work, plots curves based on material properties and plate dimensions. Input parameters for the numerical simulation of the Aluminum plate are thickness (6 mm), density (2700 kg/m3), Poisson’s ratio (0.34), and Young’s modulus (70.76 GPa).

However, the calculated dispersion curves represent the theoretical full spectrum of all modes in the specimen plate. The actual propagating waves may differ in mode number and intensity distribution, depending on excitation conditions. Therefore, an experiment is conducted to obtain dispersion curves, which are then compared with numerical simulation results.

The dispersion curves are obtained experimentally by acquiring a dataset of waveforms recorded from pitch-catch measurements of varying distances between the emitter and receiver at a constant interval. This experiment is conducted in an area far from pockets and edges so that the specimen’s baseline, or undisturbed characteristics, is determined. Figure 10 illustrates the pitch-catch measurement performed in this work. In our experiment, a 5-burst pulse of 1 MHz sinusoidal signal from a function generator (Stanford Research System DS345) is sent to the emitting transducer (Panametrics V103) after a 10× gain amplification (Krohn-Hite 7500). Another identical transducer functions as the receiver records the response signal at numerous equally spaced points. In this measurement, the receiver is driven by a scanner arm (ITE/TransformNDT Polar C-Scan) to move in a straight line of 100 mm length and consistently record the waveform every 0.1 mm, thus registering the signals from 1000 points.

thumbnail Figure 10

Schematic of experiment to obtain dispersion curves: the emitter is stationary and receiver is moving with the scanner arm.

The measurement dataset is summarized as a two-dimensional matrix that contains temporal (a series of time stamps in the signal waveform) and spatial (a series of acquisition points) representations of the measurement. Then, a 2D-FFT procedure is applied to the dataset [105]. The procedure converts time-distance data into the frequency-wavenumber domain, producing the dispersion curves of Lamb waves in the specimen. Lamb wave characteristics depend on material thickness. Therefore, to account for different plate thicknesses, the horizontal axis is scaled as the product of frequency and thickness (fd) instead of frequency alone.

As stated before, a numerical simulation with Disperse® is made in conjunction with the experiment to generate the dispersion curves. Overlaying the curves from numerical simulation on the experimental-based curves, as shown in Figure 11, demonstrates that the results agree with each other regarding the shape and position of the curves. Nonetheless, not all of the possible modes are indeed propagating in our actual experiment. Four modes are prominent: S0, A1, S2, and A2. Four others appear in weaker intensities: A0, S1, A3, and A4, while S3 are virtually nonexistent.

thumbnail Figure 11

Dispersion curves (fd-k) in the pristine area.

The propagation of modes during the experiment, hence their appearance in the fd-k dispersion curves diagram, is affected by the excitation conditions of the waves. For instance, the resulting dispersion curves in Figure 10 are produced through an experiment with transducers oriented at normal incidence concerning the surface. Changing the angle of incidence would alter the distribution of mode intensities in the spectrum since specific angles excite particular modes differently [109, 110]. Normal incident excitation simplifies the experiment as it requires no wedge, only direct placement of the transducer on the plate surface with gel or water as coupling. Additionally, this setup provides omnidirectional excitation, allowing flexible receiver placement without alignment to the wedge’s line of sight.

Dispersion curves can also be plotted in a frequency-phase velocity (fd-vphase) space. Since it has been confirmed that the frequency-wavenumber (fd-k) relation of numerically generated dispersion curves matches the one from the experiment, using the corresponding f-vphase plot resulting from that numerical simulation is valid, as depicted earlier in Figure 2b. The velocity of a propagating mode can be determined by observing the curve of that mode in fd-vphase diagram in Figure 2b at the frequency range where it exhibits a high-intensity trace in the experimental fd-k diagram as shown in Figure 11. In that experimental-based dispersion curves, there are four prominent modes exhibited in: S0, A1, S2, and A2. Eventually, the fd-vphase diagram will be the reference for time domain analysis of the response signal, such as in comparing the apparent time of flight of the waves.

7.2 Lamb waves signal analysis for pocket feature detection

Despite its simplicity, normal incident excitation generates multiple modes, demonstrated in Figure 11, complicating time domain signal analysis due to each mode’s dispersive behavior. In contrast, oblique incident excitation allows efficient mode selection by adjusting the angle, determined by the sound speeds in the wedge and the target mode, and applying Snell’s law. However, this method is less practical.

A common Lamb wave inspection technique uses a non-dispersive mode, where wave velocity is constant across frequencies. Inspections using higher-order modes, which may be dispersive, have also proven effective [81, 111]. Our approach to isolate the target mode is through band pass filtering in the frequency domain of the response signal. The selection of the potential modes to be extracted and the frequency band to pass are based on the observation of dispersion curves in fd-k space (Fig. 11) and fd-vphase space (Fig. 2b). On top of the single mode approach, an analysis based on multimodal Lamb waves, i.e., the as-received response signal, is also exercised in this paper.

The schematic in Figure 12 is the setup of experiments to inspect the presence of damage with the specimen previously shown in Figure 9. Two contact-type transducers (emitter and receiver) are secured in a specially made, 3D-printed holder with a spacing of 6 mm. Other parameters on signal excitation (5-cycle sinusoidal pulse at 1 MHz) and the equipment involved are the same as the experiment outlined in Section 6.1. The transducer pair assembly is mounted on the scanner arm, providing linear motion. A line scan of 400 mm is performed with 0.2 mm resolution at the speed of 5 mm/s, thus acquiring waveforms from 2000 positions in 80 s. Note that the scanning resolution used in this measurement (0.2 mm, as indicated in Fig. 12) is coarser than that used for obtaining the dispersion curves (0.1 mm, see Fig. 10). This is because the experiment in the previous section (Fig. 10) prioritizes spatial precision to produce sharp and accurate dispersion curves with less concern for measurement time, hence smaller spacing between points is preferred. In contrast, this section’s feature detection scanning experiment (Fig. 12) considers practical aspects, balancing precision with efficiency-related factors like measurement time and data size. Therefore, a wider spacing is applied.

thumbnail Figure 12

Schematic of Lamb wave scanning to detect the pockets on the back side of plate: Emitter and Receiver are mounted on a common holder (custom made 3D printed part), hence simultaneously moving during the scan and measurement is conducted on the front side.

Measurements are made on the front side of the plate; hence, there is no visual indication of the pockets as they are on the backside. The motion path of each transducer is marked by the dashed lines in Figure 12, such that the pockets are in between the two transducers when the pair passes the pockets’ positions during the scanning process.

A typical waveform of the as-received signal, taken at a clean position, is presented in Figure 13a. This signal contains the whole spectrum by the band locations where high-intensity streaks appear in the dispersion curves in Figure 11. The first attempt in isolating a single mode is targeted on the low-frequency regime, namely the S0 mode at 0.9–1.5 MHz · mm. Recall that the horizontal axis in the dispersion curves is expressed in terms of frequency × thickness product (fd), hence the unit MHz · mm, and that the thickness of our specimen is 6 mm. Thus, this range is equivalent to 150–250 kHz in the frequency spectrum of the received signal. Although allowing single-mode extraction as needed, bandpass filtering within this frequency range captures the weak part of the spectrum. Consequently, the reconstructed signal (inverse Fourier transform) plotted in Figure 13b also has a low amplitude.

thumbnail Figure 13

Typical waveforms at a position without pocket: (a) no filtering; (b) band-pass filtered to isolate mode S0 (150–250 kHz); (c) band-pass filtered to isolate mode S2 (500-780 kHz).

A quick spectrum inspection suggests that the energy is concentrated at the mid-range of the Fourier space, as plotted in the inset image of Figure 13a. Referring to the dispersion curves in Figure 10, mode S2 exclusively appears with vigorous intensity in the 3–4.7 MHz · mm range. That range, divided by the plate thickness of 6 mm, equals 500–780 kHz in the spectrum of received Lamb waves signal. As expected, bandpass filtering in this region results in a high amplitude reconstructed signal exhibited in Figure 13c. Since the filter passes the regime that contains the most spectral energy, the reconstructed waveform closely resembles the original signal; only now does it have better-defined wave packets because the other contributing Lamb modes have been rejected.

The amplitude of the waveform series recorded with the setup in Figure 10 is visualized as B-scan images presented in Figure 14, where distinguishable characteristics emerge at the position of the pockets. Moreover, the characteristics are distinctive from one suspected position to the other, implying the physical variations among them, in this case, the depth of pockets. Note that the measurements are done on the clean side, without any perceptible sign of the pocket behind it. Thus, up to this point, it is evident that the Lamb waves technique can detect the presence of invisible pockets and provide insights into their depth.

thumbnail Figure 14

B-scans of different waveforms: (a) no filtering; (b) band-pass filtered to isolate single S0; (c) band-pass filtered to isolate single S2.

B-scan images in Figure 14 correspond to their typical waveform (A-scan) counterparts in Figure 13. The series of multimodal waveforms exemplified in Figure 13a produces a B-scan image shown in Figure 14a. Now, considering only the S0 mode component, as in Figure 13b, yields a B-scan image in Figure 14b that is less intelligible for feature detection. The high amplitude signals at the scanning positions of approximately 75–150 mm create ambiguity in interpreting the image because that region is, in fact, free of pocket. This inconsistency may arise due to the variation in signal excitation and reception conditions during measurement caused by inconstant pressure applied on the contact transducers, surface roughness, flatness, and the spread and thickness of the couplant. Scattering from surrounding edges is a less probable factor, although not being completely ruled out, because if it were, then similar phenomena would also appear in the region between 200 mm and 300 mm. Consequently, it is conclusive that with our setup, mode S0 is not an appropriate choice for sensing the pocket feature. However, an inspection by intentionally exciting a pure S0 mode utilizing a wedge at a correct angle is worth investigating, which is outside this paper’s scope. Thus, interested readers are suggested to find resources elsewhere, such as [64, 112].

Passing only mode S2, which lies in the high-intensity regime of the multimodal spectrum as described earlier in Figure 13c, produces a less noisy B-scan image, as exhibited in Figure 14c. With this mode, although the profiles of the regions in between pockets (60–190 mm and 210–340 mm) are non-identical, the image, in general, is easier to interpret. As the qualitative assessment settles on this S2 mode as the potential means of detection, investigation continues towards the effort to quantitatively describe the feature through analyzing the signal parameters.

It is evident from the scan result in Figure 14c that there is a considerable change in time of flight (TOF) at the positions where pockets exist. Thus, the TOF, or the speed of sound directly linked to it, can indicate the presence of pockets. However, by observing the appearance of the scan in Figure 14c, one may notice that the change of TOF has a different tendency for each pocket position. The TOF of the signal filtered for the S2 frequency band at each point along the scanning length, corresponding to the image in Figure 12, is plotted in Figure 15.

thumbnail Figure 15

Time of flight of signals along the scanning length. Abrupt changes indicate the presence of a pocket at the back side of the plate at 50 mm, 200 mm, and 350 mm.

While the TOF increases when encountering Pocket #1 and #3, signifying a slower wave velocity, the opposite occurs at the position of Pocket #2. A possible reason for this is related to the reflection of waves from plate edges, where Pocket #2 is in the middle of the plate and the others are near plate edges (see Fig. 9b).

As the Lamb waves interact with the pocket, which is a discontinuity in the thickness, the higher frequency components of the waves are suppressed. From the dispersion curves in Figure 11, the velocity at lower frequencies is higher within the range of mode S2 isolated in this analysis (3–4.7 MHz · mm). Consequently, it is understandable that the recorded TOF decreases when the wave passes through the pocket, as the low frequency, higher velocity components of the wave dominate the received waveform. For Pocket #2, this phenomenon occurs practically without being affected by the edge reflections since it is in the middle of the plate or away from the edges.

As they are near the edges, different circumstances apply for Pocket #1 and #3; hence, edge effects are expected. The waves that travel through the pocket are left out with the lower frequency, higher velocity components, as previously described in the case of Pocket #2. However, in those near-edge positions, waves reflect from the edges, which in turn interfere with the direct signals. Note that the intensity in the scanned image in Figure 14c is normalized since the primary interest is on the shape of the waveform while removing the variability from point to point due to the non-uniform contact conditions between transducers and the surface. Accordingly, the amplitude in the waveform is expressed in relative value, meaning that the TOF identification plotted in Figure 15 is based on the arrival of the dominant part.

7.3 Remarks on the case study II

This exemplary case demonstrates the efficacy and challenges of using a Lamb waves technique for detecting invisible or inaccessible thickness variation features in the plate. An unconventional approach is attempted in this case by extracting the frequency of interest from the multimodal response signal instead of emitting a single-mode wave in the first place. In addition, the approach does not depend on the non-dispersiveness of the propagating wave. This approach simplifies the measurement process since there is no need to target a particular mode, let alone the pure mode excitation.

By imposing a suitable band-pass filter, the changes in the signal attributed to the existence of an imperceptible reduction in thickness are accentuated. This approach offers insight into the location of the thickness reduction feature, represented here by a milled pocket, through observation of the arrival time in the band-pass filtered B-scan. Certain conditions may complicate the analysis, such as the interference of reflected waves during measurements conducted near the edges. Nevertheless, the primary information regarding the feature’s location can be easily obtained through uncomplicated interpretation.

8 Conclusion

This article provides a comprehensive review of Lamb wave-based NDE techniques for metal plates, serving as a contemporary and practically oriented introduction to this research field. Through two detailed case studies, we illustrate the practical applications and advancements in Lamb wave generation and detection, highlighting their importance in contemporary industrial contexts.

The first case study demonstrates the efficacy of the A0 Lamb wave mode in detecting weld joints and stiffeners on steel plates, particularly relevant to the shipbuilding industry where weld integrity is critical. By combining numerical simulations with experimental validations, this study not only showcases the potential of Lamb waves to identify these features, but also their ability to overcome practical challenges, such as discrepancies between idealized conditions and real-world scenarios.

The second case study focuses on detecting thickness reductions in aluminum plates, simulating conditions like corrosion or wear that can lead to material thinning and potential failure. Using high-order Lamb modes in a multimodal excitation setup, this study highlights the precision of Lamb waves in identifying hidden defects, providing valuable insights into the technique’s applicability in scenarios where damage is not easily accessible.

Both case studies underscore the method’s practical relevance and effectiveness in real-world applications, demonstrating how Lamb wave techniques can address contemporary challenges in inspecting metal structures. The detailed exploration of transducer technologies, mode selection, and the intricacies of signal interpretation further enriches the paper. By presenting two exemplary case studies, we emphasize these techniques’ practical orientation and contemporary relevance, showcasing their potential to ensure the structural integrity and safety of metal plates in various industrial applications. The findings advocate for continued research and development in this domain, particularly in advanced signal processing and analysis, to enhance the accuracy and efficiency of Lamb wave-based inspections.

Acknowledgments

The authors thank M. Yves Guillermit, from Weez-U Welding, for helping to model the representative mock-up and for his knowledge of the welding process.

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 871260.

Conflicts of interest

The authors declare that they have no conflict of interest in relation to this article.

Data availability statement

The data are available from the corresponding author on request.

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Cite this article as: Silitonga D. Declercq NF. Walaszek H. Vu QA. Saidoun A, et al. 2024. A comprehensive study of nondestructive localization of structural features in metal plates using single and multimodal Lamb wave excitations. Acta Acustica, 8, 66. https://doi.org/10.1051/aacus/2024069.

All Tables

Table 1

Mechanical properties of the Steel S235 (according to EN1993-1-1).

In the text

All Figures

thumbnail Figure 1

Lamb modes: (a) symmetric and (b) antisymmetric.
In the text
thumbnail Figure 2

Dispersion curves of Aluminum, plotted as relations between frequency-thickness and (a) wavenumber, (b) phase velocity, and (c) group velocity.
In the text
thumbnail Figure 3

Dispersion curves of non-coated and coated steel plate (8 mm thick): Within the frequency band investigated in this study (80–250 kHz), mode A0 and S0 of non-coated and coated plates are identical.
In the text
thumbnail Figure 4

2-D models of the plate representing the realized mock-up. Specimen #1: a plain plate as the reference, Specimen #2: a plate with weld, Specimen #3: a plate with a stiffener bar.
In the text
thumbnail Figure 5

B-Scan images of the A0 mode propagation along the three simulated models
In the text
thumbnail Figure 6

Transducer arrangement in (a) transmission and (b) pseudo-pulse-echo reflection experiments on the specimen with butt weld (Specimen #2). The experiment for a plate with a stiffener bar (Specimen #3) employs identical arrangements. (E: Emitter, R: Receiver).
In the text
thumbnail Figure 7

Dispersion curves, in frequency-wavenumber space, obtained from the transmission (pitch-catch) experiment on the plain plate (Specimen #1).
In the text
thumbnail Figure 8

B-scan from experiments on (a) Specimen #1: plain plate; (b) Specimen #2: plate with butt weld; (c) Specimen #3: plate with stiffener bar at the backside.
In the text
thumbnail Figure 9

Specimen of Aluminum plate: (a) front side without any perceptible physical sign of pockets behind it, only showing traces of markers for experimental arrangement; (b) back view showing circular 20 mm diameter pockets of different depths, h.
In the text
thumbnail Figure 10

Schematic of experiment to obtain dispersion curves: the emitter is stationary and receiver is moving with the scanner arm.
In the text
thumbnail Figure 11

Dispersion curves (fd-k) in the pristine area.
In the text
thumbnail Figure 12

Schematic of Lamb wave scanning to detect the pockets on the back side of plate: Emitter and Receiver are mounted on a common holder (custom made 3D printed part), hence simultaneously moving during the scan and measurement is conducted on the front side.
In the text
thumbnail Figure 13

Typical waveforms at a position without pocket: (a) no filtering; (b) band-pass filtered to isolate mode S0 (150–250 kHz); (c) band-pass filtered to isolate mode S2 (500-780 kHz).
In the text
thumbnail Figure 14

B-scans of different waveforms: (a) no filtering; (b) band-pass filtered to isolate single S0; (c) band-pass filtered to isolate single S2.
In the text
thumbnail Figure 15

Time of flight of signals along the scanning length. Abrupt changes indicate the presence of a pocket at the back side of the plate at 50 mm, 200 mm, and 350 mm.
In the text

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