# Enhancement of Guided Wave Detection and Measurement in Buried Layers of Multilayered Structures Using a New Design of V(z) Acoustic Transducers

^{*}

## Abstract

**:**

_{0}and S

_{0}modes corresponding to the steel layer inside the three-layer structure. In this study, we also develop a specific tracking method for particular guided waves possessing large phase velocity variations over the considered frequency range, as is the case for the S

_{0}mode of the steel sub-layer.

## 1. Introduction

_{1}Lamb mode of a bronze plate, for a frequency-thickness product roughly between 2.35 and 2.65 MHz·mm, is not observed with the use of a classical V(z) transducer.

## 2. Theoretical Considerations and Methods

#### 2.1. Principles of V(z) Signature with Single and Multi-Element Focused Transducers

_{c}and half aperture angle ${\theta}_{ap}$) and the segmented V(z) transducer (the same ones and the length $\mathrm{\u2206}l$) are given. The z axis origin is chosen at the focus of the transducers. Hence, when the transducer focuses inside the sample, the (z) coordinate of its top surface is negative. The numbering of the piezoelectric elements in Figure 1b is as follows: the central piezoelectric element around the (z) axis is numbered (0), the next one to its right is numbered (1) and symmetrically (−1) for the next one to its left, and so on for all subsequent elements. This numbering will be used in the following theoretical section. These piezoelectric elements have a curvilinear abscissa ${l}_{i}$ as indicated in Figure 1b.

#### 2.2. V(z) Curve Modelling for Single and Multi-Element Focused Transducers

- ${k}_{x}$ is the horizontal component of the wave vector ${k}_{0}$ in the coupling fluid;
- ${U}_{f}^{s}\left({k}_{x}\right)$ is the angular spectrum of the incident field of the piezoelectric source (s) defined at the focal plane (f);
- ${U}_{s}^{f}\left({k}_{x}\right)$ is the angular spectrum of the transducer response when a plane wave of unit amplitude is emitted from the focal plane (f) to the piezoelectric source (s);
- $R\left({k}_{x}\right)$ is the reflection coefficient of the fluid-loaded sample, as a function of the incident wave number component ${k}_{x}$;
- $\mathrm{exp}\left(-2i{k}_{z}z\right)$ is the phase shift applied to the spectrum ${U}_{s}^{f}\left({k}_{x}\right)$ when the surface sample is located at a distance (−z) from the focal plane during defocusing process, where ${k}_{z}$ is the vertical component of the wave vector ${k}_{0}$ in the coupling fluid, thus given by ${k}_{z}=\sqrt{{k}_{0}^{2}-{k}_{x}^{2}}$.

#### 2.3. V(z) Processing and Associated Spectral Representation

_{s}. Such a spectrum is represented in Figure 2c. Thus, each wave number value ξ

_{p}associated with a peak in the spectrum corresponds to a leaky guided mode, and is linked to the periodicity $\mathrm{\u2206}z$ by:

_{p}in Equation (9). Hence, the choice was made to adopt another representation of the V(z) spectrum. Thus, replacing ξ

_{p}by ξ from Equation (9) into Equation (8), and the result into Equation (7), allows the spectrum of the V(z) curves to be expressed as a function of an equivalent angle of incidence ${\theta}_{0}$:

_{1}, S

_{0}, and A

_{0}are identified through the dispersion curves of the aluminum plate that will be studied hereafter in Section 3.2. It can also be noted that in regard to Equation (10) and Figure 2c,d, low wave numbers ξ correspond to low equivalent incident angles ${\theta}_{0}$, and conversely.

## 3. Results and Comparison

#### 3.1. Transducer and Material Properties

#### 3.2. Detection and Measurement of Guided Waves on the Three-Layer Structure

_{0}and M

_{1}modes that do not possess a cut-off frequency. It should be noted that the angular axis is limited to 50° for clarity, but the M

_{1}mode tends toward 90° when frequency tends toward 0. In Figure 3a the label of certain modes was repeated near the final frequency values, when the corresponding resonant modes of the reflection coefficient seem to partially disappear, such as M

_{0}, or seem to intersect, such as M

_{2}and M

_{3}, which is not the case when looking at Figure 3b. It was also verified that M

_{8}and M

_{9}modes do not intersect near 42.8 MHz. The minima of the reflection coefficient modulus are found to be in excellent agreement with the positions of the free guided modes.

_{3}mode approximately corresponds to the S

_{0}mode of the aluminum layer, in the frequency range around 18 to 31 MHz. In the same manner, the M

_{1}mode approximately corresponds to the A

_{0}mode of the steel layer, in the frequency range around 20 to 40 MHz. Modes of the three-layer structure that do not match one of those of a single layer correspond to “coupling modes” that propagate inside the whole of the structure. It was verified that no guided modes correspond to Lamb modes of the epoxy layer in the frequency range from 0 to 50 MHz.

_{0}, M

_{3}, M

_{4}, M

_{5}, M

_{6}, and M

_{7}are fairly well detected and determined, since their measured values are well superimposed on the corresponding guided modes of the dispersion curves. Figure 5a,b show that, except for M

_{7}mode, all the modes appearing for incident angles lower than about 7°, which correspond to phase velocities higher than approximately 12,308 m/s, cannot be detected. This is due to the fact that their low frequency spectral components are superimposed or “drowned” in the low frequency spectral components of the V(z) curves. This is not the case for M

_{7}mode, which is detectable from about 4°, since it possesses a corresponding deep peak in the modulus of the reflection coefficient, as it can be observed when looking at its black trace in Figure 5a.

_{0}mode is almost unobservable in the minima of the modulus of reflection coefficient in Figure 5a, whereas its velocity values are well detected and determined with the classical V(z) transducer over all the frequency range 20–40 MHz, as indicated in Figure 5b. An explanation for this is that, although its corresponding minimum in the reflection coefficient modulus is very small, and can even disappear, as illustrated at 25 MHz in Figure 6a, conversely, the last one continues to exhibit a large associated phase shift, as it can be seen in Figure 6b. This can be generalized to the entire frequency range from 20 to 40 MHz, as can be seen on the 3D views of the reflection coefficient’s modulus and phase in Figure 6c,d, respectively. In these figures, for clarity, only the modes that we discuss are named. The theoretical positions of the M

_{0}mode peak are marked with white dots on the modulus of the reflection coefficient in Figure 6c to better compare them with the associated large phase shifts seen in Figure 6d. Concerning the M

_{1}mode, Figure 5a,b show that it is completely undetectable with the classical V(z) transducer. This is due to a combination of two effects. Firstly, the neighboring M

_{0}mode is excited with more energy (due to its corresponding large phase shift in the reflection coefficient), and since it is very close to the M

_{1}mode, its spectral components in the V(z) signal processing predominate and “drown” those of the M

_{1}mode. Secondly, the M

_{1}mode tends by itself to be less and less detectable with frequency, since its corresponding peak and phase shift in the reflection coefficient tend to decrease and become very small, as can be observed at 25 MHz in Figure 6a,b, respectively. Globally, the 3D view of Figure 6c shows that the peaks of the refection coefficient corresponding to the M

_{1}mode still exist between approximate frequency ranges 0–25 MHz and 35–40 MHz, but the associated phase shifts almost totally disappear in the frequency range 10–40 MHz, as illustrated in Figure 6d with white dots.

_{2}mode is detectable for the first four frequency values (20 to 23 MHz), by tracking its corresponding spectral peak that starts around 17.3° at 20 MHz. However, its spectral peak rapidly decreases with frequency and becomes undetectable for frequencies higher than about 23 MHz, due to its corresponding peak and phase shift in the reflection coefficient that rapidly decrease and even disappear with frequency, as seen in Figure 6c,d, respectively. For M

_{3}mode, its corresponding peak in the modulus of reflection coefficient tends to disappear roughly between 30 MHz and 50 MHz, as illustrated in Figure 6c where the corresponding white dots show its theoretical position. However, its velocity values are well detected and determined with the classical V(z) transducer over all the frequency range 20–40 MHz, as shown in Figure 5b. As for M

_{0}mode, this is due to its corresponding phase shift that is very large over all this frequency range, as illustrated in Figure 6d. It was verified that these results can be generalized: for a given mode, its corresponding phase shift has a predominant effect on its detection, compared to the associated peak of the modulus of the reflection coefficient. Thus, this conclusion can be applied to M

_{4}mode between roughly 37 and 38 MHz, and the totality of M

_{8}mode in the considered frequency range, which are not detected in these frequency ranges, as shown in Figure 5b. Indeed, Figure 6d shows that they possess a phase shift that vanishes in the aforementioned frequency ranges. White dots corresponding to M

_{8}mode curve are added in the frequency range 36–43 MHz, to better locate its theoretical position.

_{2}mode is detectable only for the first four frequency values, while M

_{1}mode is completely undetectable. However, detection of M

_{1}mode, and a better detection of M

_{2}mode in the frequency range from 20 to 40 MHz would be of particular interest since they approximately correspond to modes in the steel layer, as represented in Figure 4. Indeed, assuming that this three-layer structure is mounted or clamped into a casing, V(z) measurements will only be possible with the aluminum layer at the top surface. Thus, it would be interesting to detect these modes in order to characterize the elastic properties of the steel layer, which would correspond to a “buried” layer in such a configuration.

_{1}mode with a segmented V(z) transducer, one can note that for the considered frequency range, it possesses an angular variation lower than about 2°, around an average value of 32°. Thus, it is possible to choose a particular central angular aperture and most importantly a Rayleigh angular aperture that will be constant, since diffraction effects will allow to generate, and thus detect, this mode over its angular variation range. Trials have shown that a very good detection of its spectral peak is obtained for a central angular aperture of 2°, and a Rayleigh angular aperture of 3°, ranging between approximately 30° and 33°. Figure 7a,b show the angular spectra of modes detected at 20 MHz, for the classical V(z) transducer and segmented one, respectively. Attention must be paid to the fact that results for a classical V(z) transducer are here given first, to facilitate further explanations. Indeed, knowing in advance the theoretical position of the M

_{1}mode with the help of the dispersion curves of Figure 5b, it is possible to indicate its position on the spectrum. Hence, one can observe in Figure 7a that it corresponds to a very low peak amplitude value, for example lower than the peaks of numerical artefacts appearing between 20 and 30 MHz.

_{0}mode to be detected, but with a lower precision than for the case of the classical V(z) transducer. This is due to the angular proximity of M

_{0}and M

_{1}modes, and the relatively large peak of mode M

_{1}whose spectral components disturbs the repartition of the M

_{0}spectral ones. It is then possible to follow the peak of the M

_{1}mode over the frequency range from 20 up to 30 MHz with a step of 1 MHz, using the segmented V(z) transducer. Its spectral peak becomes undetectable in the frequency range from about 30 to 33 MHz, due to its too small corresponding peak and phase shift in the reflection coefficient, and then becomes again detectable from 34 MHz upwards, since its corresponding phase shift reappears, as observed in Figure 6d.

_{0}mode, and more importantly of M

_{1}mode, to be detected, as represented in Figure 8a,b with orange crosses for M

_{0}and M

_{1}. These figures represent the superimposition of this detected mode with the modulus of the reflection coefficient and dispersion curves, respectively. It thus can be observed in Figure 8b that M

_{1}mode is clearly detected and superimposes well on the corresponding dispersion curve. Results concerning the generation and detection of the M

_{2}mode with a segmented transducer are also added on Figure 8a,b, and will be discussed below.

_{1}mode, the M

_{2}mode varies with a relatively large angular range of about 12°. Thus, even when taking into account diffraction effects, it is impossible to detect and track it with a fixed Rayleigh aperture angle all along the frequency range from 20 to 40 MHz. The method we have developed here consists of searching at start the best Rayleigh aperture values at the first frequency value of 20 MHz, corresponding to a peak of the M

_{2}mode that predominates in the angular spectrum. It was found that a central aperture angle of 2° and a Rayleigh aperture comprised between 18° and 20° allow a predominant peak of the M

_{2}mode at 20 MHz to be obtained. This leads to the spectrum represented in Figure 9a, while Figure 7a is repeated as Figure 9b for comparison with the spectrum obtained with the classical V(z) transducer. Then, the idea consists of gradually increasing the frequency, in small steps, and, when necessary, increasing the angular aperture position (while keeping a 2° aperture value) in order to continuously track the position of M

_{2}peak.

_{2}mode starts at 20 MHz with a small slope, the initial segmented transducer configuration allows to detect it from 20 up to 24 MHz, as represented by the five green crosses in Figure 8a,b. It can be noted that the first four values are measured with a better accuracy than the ones measured with a classical V(z) transducer. Then, it was found that an angular aperture position shifted by 2 degrees every 2 MHz was a good criterion to track the position of the M

_{2}peak. For instance, in the frequency range from 25 to 26 MHz, the angular aperture position is shifted between 20° and 22°, corresponding to the two red crosses in Figure 8a,b. In the frequency range from 27 to 28 MHz, it corresponds to an angular aperture position shifted between 22° and 24°, corresponding to the two grey crosses, and so on. This method allows the M

_{2}mode to be detected up to 30 MHz, while only up to 24 MHz with the classical V(z) transducer. Between roughly 31 and 37 MHz, M

_{2}mode becomes undetectable even with the segmented V(z) transducer, since it corresponds to peaks (lack of black trace) in the modulus of the reflection coefficient and phase shifts that completely vanish, as observed in Figure 6c,d, respectively. Although this mode theoretically exists in the dispersion curves and for this frequency range, it corresponds in practice to a mode that requires a lot of energy to be generated, and/or that leaks too little energy in water to be detected. As represented in Figure 8a,b by the last three brown crosses, M

_{2}mode becomes detectable again for frequencies from about 38 to 40 MHz (three orange crosses), but measured with low precision due to the presence of the neighboring M

_{1}mode that disturbs the spectral components of its peak.

_{1}and M

_{2}modes correspond to relatively high incident angles, i.e., to low phase velocities (see Equation (7)), which corresponds to small periods of oscillations $\mathrm{\u2206}z$ (see Equation (8)) in the V(z) curves. Thus, even if the radius of curvature is decreased, leading to a lower extent of defocus values in the V(z) curves, the number of periods is still high enough to be correctly estimated in the corresponding spectra. The main difference that was observed, compared to the results with the initial radius of curvature of 25 mm, is the fact that some modes are detected in a slightly smaller frequency range. For instance, it was verified that the M

_{1}mode remains undetectable in a frequency range from about 28 up to 35 MHz for a radius of curvature ${R}_{c}$ of 15 mm, and from about 26 up to 36 MHz for a radius of curvature of 10 mm.

_{1}and M

_{2}modes with the same geometry of segmented V(z) transducer, due to the increase in diffraction effects at low frequencies. For these modes, the acoustical energy that is emitted by the segmented V(z) transducer in the direction of the incident angles of the M

_{1}and M

_{2}modes is lower than previously (for the frequency range from 20 to 40 MHz) since a part of it is spread in other directions. Since these modes are excited with less energy, their peaks do not sufficiently stand out in the spectrum to be accurately measured or even detected. Concerning more specifically M

_{1}mode in the frequency range 0–5 MHz, diffraction effects also imply that more energy is leaked and absorbed by its neighboring M

_{0}mode, whose spectral peak components predominate again and tend to “drown” those of M

_{1}.

## 4. Conclusions

_{1}and M

_{2}—corresponding to modes in the “buried” steel layer in the frequency range between 20 and 40 MHz—are not (or poorly) detected in the V(z) spectra, due to too small corresponding minimum and/or phase shift in the associated reflection coefficient. Contrary to this situation, the segmented V(z) transducer allows these guided modes to be detected and correctly measured in this frequency range. As M

_{1}mode possesses a small angular variation for frequencies up to 20 MHz, it is possible to set the range of Rayleigh angles once and for all for the entire frequency range, since diffraction effects allow to spread acoustical energy in an incident angle range large enough to cover all the corresponding incident angle values. Unlike M

_{1}mode, M

_{2}mode possesses a large angular variation in the frequency range from 20 to 40 MHz. Thus, when searching to complete its detection with the segmented V(z) transducer, it is impossible to maintain a constant range of Rayleigh angles over this frequency domain. In this case, an original technique was developed. It consists of searching the best initial configuration of the segmented transducer for the first frequency value, in order to obtain the largest corresponding peak in the spectrum of V(z) curve. Then, the Rayleigh angles range is kept constant as long as the corresponding peak of M

_{2}mode is predominant in the spectrum. When it becomes lower than another spectral peak, the Rayleigh angles range is progressively increased until the peak of M

_{2}becomes again the predominant one. This method allows the M

_{2}mode to be detected and tracked up to 30 MHz, while only 24 MHz can be reached with the classical V(z) transducer.

_{1}of the three-layer structure studied here and possessing a small angular variation for a given high enough frequency range, they allow to maintain the Rayleigh angles range constant over this frequency range for modes detection. However, when working at much lower frequencies (roughly ten times lower), the diffraction effects increase and become an inconvenient. Indeed, they spread the radiated energy, and thus decrease the energy absorbed by the mode that is desired to be generated and detected. This leads to inaccurate results in terms of detection and measurements of modes that are excited with too little energy, as it was verified to be the case for the M

_{1}and M

_{2}modes when using a frequency range from 0 to 5 MHz.

_{1}and M

_{2}in our example) correspond to a specific layer that needs to be characterized, but not directly accessible, such as a “buried” layer, or a bottom layer in a structure that cannot be reversed for practical reasons.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Rokhlin, S.I.; Huang, W. Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: Exact and asymptotic-boundary-condition methods. J. Acoust. Soc. Am.
**1992**, 92, 1729–1742. [Google Scholar] [CrossRef] - Ismaili, N.A.; Chenouni, D.; Lakhliai, Z.; El-Kettani, M.E.C.; Morvan, B.; Izbicki, J.L. Determination of epoxy film parameters in a three-layer metal/adhesive/metal structure. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2009**, 56, 1955–1959. [Google Scholar] [CrossRef] [PubMed] - Mustapha, S.; Ye, L. Propagation behaviour of guided waves in tapered sandwich structures and debonding identification using time reversal. Wave Motion
**2015**, 57, 154–170. [Google Scholar] [CrossRef] - Balvantín, A.J.; Diosdado-De-la-Peña, J.A.; Limon-Leyva, P.A.; Hernández-Rodríguez, E. Study of guided wave propagation on a plate between two solid bodies with imperfect contact conditions. Ultrasonics
**2018**, 83, 137–145. [Google Scholar] [CrossRef] [PubMed] - Guo, Z.; Achenbach, J.D.; Madan, A.; Martin, K.; Graham, M.E. Modeling and acoustic microscopy measurements for evaluation of the adhesion between a film and a substrate. Thin Solid Films
**2001**, 394, 188–200. [Google Scholar] [CrossRef] - Baltazar, A.; Wang, L.; Xie, B.; Rokhlin, S.I. Inverse ultrasonic determination of imperfect interfaces and bulk properties of a layer between two solids. J. Acoust. Soc. Am.
**2003**, 114, 1424–1434. [Google Scholar] [CrossRef] - Boström, A.; Golub, M. Elastic SH wave propagation in a layered anisotropic plate with interface damage modelled by spring boundary conditions. Q. J. Mech. Appl. Math.
**2009**, 62, 39–52. [Google Scholar] [CrossRef] [Green Version] - Golub, M. Propagation of elastic waves in layered composites with microdefect concentration zones and their simulation with spring boundary conditions. Acoust. Phys.
**2010**, 56, 848–855. [Google Scholar] [CrossRef] - Leiderman, R.; Figueroa, J.C.; Braga, A.M.B.; Rochinha, F.A. Scattering of ultrasonic guided waves by heterogeneous interfaces in elastic multi-layered structures. Wave Motion
**2016**, 63, 68–82. [Google Scholar] [CrossRef] - Leiderman, R.; Braga, A.M.B. Scattering of guided waves by defective adhesive bonds in multilayer anisotropic plates. Wave Motion
**2017**, 74, 93–104. [Google Scholar] [CrossRef] - Siryabe, E.; Rénier, M.; Meziane, A.; Galy, J.; Castaings, M. Apparent anisotropy of adhesive bonds with weak adhesion and non-destructive evaluation of interfacial properties. Ultrasonics
**2017**, 79, 34–51. [Google Scholar] [CrossRef] [PubMed] - Fraisse, P.; Schmit, F.; Zarembowitch, A. Ultrasonic inspection of very thin adhesive layers. J. Appl. Phys.
**1992**, 72, 3264–3271. [Google Scholar] [CrossRef] - Xu, P.C.; Lindenschmidt, K.E.; Meguid, S.A. A new high-frequency analysis of coatings using leaky lamb waves. J. Acoust. Soc. Am.
**1993**, 94, 2954–2962. [Google Scholar] [CrossRef] - Rogers, J.A.; Dhar, L.; Nelson, K.A. Noncontact determination of transverse isotropic elastic moduli in polyimide thin films using a laser based ultrasonic method. Appl. Phys. Lett.
**1994**, 65, 312–314. [Google Scholar] [CrossRef] - Rokhlin, S.I.; Ganor, M.; Degtyar, A.D. Ultrasonic characterization of plasma spray coating. In Review of Progress in Quantitative Nondestructive Evaluation; Springer: New York, NY, USA, 1997; pp. 1585–1591. [Google Scholar] [CrossRef] [Green Version]
- Van de Rostyne, K.; Glorieux, C.; Gao, W.; Gusev, V.; Nesladek, M.; Lauriks, W.; Thoen, J. Investigation of elastic properties of CVD-diamond films using the lowest order flexural leaky lamb wave. Phys. Stat. Sol. A
**1999**, 172, 105–111. [Google Scholar] [CrossRef] - Alleyne, D.; Cawley, P. A 2-dimensional Fourier transform method for the measurement of propagating multimode signals. J. Acoust. Soc. Am.
**1991**, 89, 1159–1168. [Google Scholar] [CrossRef] - Abbate, A.; Koay, J.; Frankel, J.; Schroeder, S.C.; Das, P. Application of wavelet transform signal processor to ultrasound. Proc. IEEE Ultrason. Symp.
**1994**, 2, 1147–1152. [Google Scholar] [CrossRef] - Titov, S.A.; Maev, R.G.; Bogachenkov, A. Measurements of velocity and attenuation of leaky waves using an ultrasonic array. Ultrasonics
**2006**, 44, 182–187. [Google Scholar] [CrossRef] - Titov, S.A.; Maev, R.G.; Bogachenkov, A. Lens multielement acoustic microscope in the mode for measuring the parameters of layered objects. Acoust. Phys.
**2017**, 63, 583–589. [Google Scholar] [CrossRef] - Titov, S.A.; Maev, R.G. An Ultrasonic Array Technique for Material Characterization of Plate Samples. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2013**, 60, 1435–1445. [Google Scholar] [CrossRef] - Lemons, R.A.; Quate, C.F. Acoustic microscope. Phys. Acoust.
**1979**, XIV, 1. [Google Scholar] [CrossRef] - Kushibiki, J.I.; Chubachi, N. Material characterization by line-focus beam acoustic microscopy. IEEE Trans. Son. Ultrason.
**1985**, 32, 189–212. [Google Scholar] [CrossRef] - Nayfeh, H.; Chimenti, D.E. Propagation of guided waves in fluid-coupled plates of fiber-reinforced composite. J. Acoust. Soc. Am.
**1988**, 83, 1736–1743. [Google Scholar] [CrossRef] - Nagy, P.B.; Adler, L. Adhesive joint characterization by leaky guided interface waves. In Review of Progress in Quantitative Nondestructive Evaluation; Springer: New York, NY, USA, 1989; pp. 1417–1424. [Google Scholar] [CrossRef] [Green Version]
- Philibert, M.; Yao; Gresil, M.; Soutis, C. Lamb waves-based technologies for structural health monitoring of composite structures for aircraft applications. Eur. J. Mater.
**2022**, 2, 436–474. [Google Scholar] [CrossRef] - Gorgin, R.; Luo, Y.; Wu, Z. Environmental and operational conditions effects on Lamb wave based structural health monitoring systems: A review. Ultrasonics
**2020**, 105, 106114. [Google Scholar] [CrossRef] - Liu, G.R.; Achenbach, J.D.; Kim, J.O.; Li, Z.I. A combined finite element method/boundary element method technique for V(z) curves of anisotropic-layer/substrate configurations. J. Acoust. Soc. Am.
**1992**, 92, 2734–2740. [Google Scholar] [CrossRef] - Lee, Y.C.; Kim, J.O.; Achenbach, J.D. V(z) curves of layered anisotropic materials for the line-focus acoustic microscope. J. Acoust. Soc. Am.
**1993**, 94, 923–930. [Google Scholar] [CrossRef] - Achenbach, J.D.; Kim, J.O.; Li, W. Measuring thin-film elastic constants by line-focus acoustic microscopy. Adv. Acoust. Micros.
**1995**, 129, 153–208. [Google Scholar] [CrossRef] - Guo, Z.; Achenbach, J.D.; Madan, A.; Martin, K.; Graham, M.E. Integration of modeling and acoustic microscopy measurements for thin films. J. Acoust. Soc. Am.
**2000**, 107, 2462–2471. [Google Scholar] [CrossRef] - Lee, Y.C.; Cheng, S.W. Measuring Lamb wave dispersion curves of a bi-layered plate and its application on material characterization of coating. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2001**, 48, 830–837. [Google Scholar] [CrossRef] - Bourse, G.; Xu, W.J.; Mouftiez, A.; Vandevoorde, L.; Ourak, M. Interfacial adhesion characterization of plasma coatings by V(z) inversion technique and comparison to interfacial indentation. NDT E Int.
**2012**, 45, 22–31. [Google Scholar] [CrossRef] - Lematre, M.; Benmehrez, Y.; Bourse, G.; Xu, W.J.; Ourak, M. Acoustic microscopy measurement of elastic constants by using an optimization method on measured and calculated SAW velocities: Effect of initial C
_{ij}values on the calculation convergence. NDT E Int.**2002**, 35, 279–286. [Google Scholar] [CrossRef] - Loukkal, A.; Lematre, M.; Bavencoffe, M.; Lethiecq, M. Modeling and numerical study of the influence of imperfect interface properties on the reflectance function for isotropic multilayered structures. Ultrasonics
**2020**, 103, 106099. [Google Scholar] [CrossRef] [PubMed] - Vijaya Kumar, R.I.; Bhat, M.R.; Murthy, C.R.I. Some studies on evaluation of degradation in composite adhesive joints using ultrasonic techniques. Ultrasonics
**2013**, 53, 1150–1162. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Principle of interference with a classical V(z) transducer; (

**b**) corresponding scheme with a segmented V(z) transducer.

**Figure 2.**(

**a**) Example of V(z) curve of an aluminum plate; (

**b**) corresponding filtered V(z) curve; (

**c**) spectral representation in the wave number space; (

**d**) corresponding spectral representation in the incident angle space.

**Figure 3.**(

**a**) Modulus of the reflection coefficient in the aluminum/epoxy/steel three-layer structure; (

**b**) corresponding dispersion curves in the incident angle space.

**Figure 4.**Superimposition of the dispersion curves of the three-layer structure (blue color) with the Lamb wave curves of the aluminum layer (green color) and steel layer (red color), considered separately.

**Figure 5.**(

**a**) Superimposition of the detected mode velocities (red circles), obtained with the classical V(z) transducer, with the modulus of the reflection coefficient; (

**b**) corresponding superimposition with the dispersion curves.

**Figure 6.**(

**a**) Modulus of the reflection coefficient on the three-layer structure at 25 MHz; (

**b**) corresponding phase of the reflection coefficient; (

**c**,

**d**) corresponding 3D global views of the modulus and phase of the reflection coefficient, respectively.

**Figure 7.**(

**a**) Spectral representation of the modes detected at 20 MHz with the classical V(z) transducer; (

**b**) corresponding spectra obtained with the segmented V(z) transducer.

**Figure 8.**(

**a**) Superimposition of all the mode velocities, generated and detected with the classical V(z) transducer (circles) and the segmented one (crosses), with the modulus of the reflection coefficient; (

**b**) corresponding superimposition with the dispersion curves.

**Figure 9.**(

**a**) Spectral representation of modes detected at 20 MHz with the segmented V(z) transducer; (

**b**) corresponding spectra obtained with the classical V(z) transducer.

Material | Mass Density (kg/m^{3}) | Longitudinal Wave Velocity (m/s) | Transversal Wave Velocity (m/s) | Longitudinal Attenuation (Np/m) | Transverse Attenuation (Np/m) |
---|---|---|---|---|---|

Aluminum | 2740 | 6190 | 3128 | - | - |

Epoxy | 1548 | 2380 | 1400 | 20 | 53 |

Steel | 7850 | 5940 | 3240 | - | - |

Water | 1000 | 1500 | - | - | - |

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**MDPI and ACS Style**

Lematre, M.; Lethiecq, M.
Enhancement of Guided Wave Detection and Measurement in Buried Layers of Multilayered Structures Using a New Design of *V*(*z*) Acoustic Transducers. *Acoustics* **2022**, *4*, 996-1012.
https://doi.org/10.3390/acoustics4040061

**AMA Style**

Lematre M, Lethiecq M.
Enhancement of Guided Wave Detection and Measurement in Buried Layers of Multilayered Structures Using a New Design of *V*(*z*) Acoustic Transducers. *Acoustics*. 2022; 4(4):996-1012.
https://doi.org/10.3390/acoustics4040061

**Chicago/Turabian Style**

Lematre, Michaël, and Marc Lethiecq.
2022. "Enhancement of Guided Wave Detection and Measurement in Buried Layers of Multilayered Structures Using a New Design of *V*(*z*) Acoustic Transducers" *Acoustics* 4, no. 4: 996-1012.
https://doi.org/10.3390/acoustics4040061