Attenuation of a Slow Subsonic A0 Mode Ultrasonic Guided Wave in Thin Plastic Films

The ultrasonic testing technique using Lamb waves is widely used for the non-destructive testing and evaluation of various structures. For air-coupled excitation and the reception of A0 mode Lamb waves, leaky guided waves are usually exploited. However, at low frequencies (<100 kHz), the velocity of this mode in plastic and composite materials can become slower than the ultrasound velocity in air, and its propagation in films is accompanied only by an evanescent wave in air. To date, the information about the attenuation of the slow A0 mode is very contradictory. Therefore, the objective of this investigation was the measurement of the attenuation of the slow A0 mode in thin plastic films. The measurement of the attenuation of normal displacements of the film caused by a propagating slow A0 mode is discussed. The normal displacements of the film at different distances from the source were measured by a laser interferometer. In order to reduce diffraction errors, the measurement method based on the excitation of cylindrical but not plane waves was proposed. The slow A0 mode was excited in the polyvinylchloride film by a dry contact type ultrasonic transducer made of high-efficiency PMN-32%PT strip-like piezoelectric crystal. It was found that that the attenuation of the slow A0 mode in PVC film at the frequency of 44 kHz is 2 dB/cm. The obtained results can be useful for the development of quality control methods for plastic films.

alone reaches 5 million tons annually, and the non-destructive evaluation of the quality of such films is very important [29].
The paper is organized as follows. In Section 2, the measurement method of the attenuation coefficient is proposed and theoretically analysed. In Section 3, the experimental set-up and measurement procedures are described. In Section 4, the measurement results are presented. In Section 5, the conclusions drawn from the theoretical and experimental results are presented.

Theoretical Background for the Measurement of the Attenuation
The phase and group velocities of the A0 mode guided wave are frequency-dependent. Dependencies of those velocities versus frequency are called dispersion curves. The dispersion curves of the A0 Lamb wave mode phase velocity in PVC films of different thicknesses were calculated using the semi analytical finite element (SAFE) method and are presented in Figure 1. The calculations were performed using the following parameters of the PVC film [30]: the Young's modulus E = 2.937 GPa; the Poisson coefficient ν = 0.42; the density ρ = 1400 kg/m 3 ; and the thickness of the plate d = 50-300 μm. The film was split into a finite number of thin layers, each of which is described in the one axis direction by three nodes, and in the second axis direction, it is assumed that the plate is infinite [6]. From the results presented in Figure 1, it follows that in a quite wide range of frequencies (<900 kHz), the phase velocity of A0 mode in PVC films is lower than the ultrasound velocity in air cair = 342 m/s. The low-frequency range in which the slow Lamb wave A0 mode was generated is shown in Figure 1 by the rectangle. At those frequencies, the propagating A0 mode is called a slow or sub-sonic wave [13]. In this case, the wave is trapped inside the film and accompanied not by a leaky but an evanescent wave in air. This means that the attenuation of such a slow A0 mode should be only due to losses in the plastic material.
Usually, in the case of the air-coupled ultrasonic non-destructive evaluation, the reception of guided waves leaky waves in air is exploited. This is created only by normal displacements of a film under a test. Therefore, we shall further discuss the measurement of the attenuation of normal displacements ) , ( t x j  caused by the propagating slow A0 mode. Here, xj is the spatial coordinate and t is the time. From the results presented in Figure 1, it follows that in a quite wide range of frequencies (<900 kHz), the phase velocity of A 0 mode in PVC films is lower than the ultrasound velocity in air c air = 342 m/s. The low-frequency range in which the slow Lamb wave A 0 mode was generated is shown in Figure 1 by the rectangle. At those frequencies, the propagating A 0 mode is called a slow or sub-sonic wave [13]. In this case, the wave is trapped inside the film and accompanied not by a leaky but an evanescent wave in air. This means that the attenuation of such a slow A 0 mode should be only due to losses in the plastic material.
Usually, in the case of the air-coupled ultrasonic non-destructive evaluation, the reception of guided waves leaky waves in air is exploited. This is created only by normal displacements of a film under a test. Therefore, we shall further discuss the measurement of the attenuation of normal displacements ξ(x j , t) caused by the propagating slow A 0 mode. Here, x j is the spatial coordinate and t is the time.
Usually, the attenuation of an ultrasonic wave is obtained from the amplitudes of a plane ultrasonic wave measured at two different distances. Application of the plane wave allows us to avoid measurement errors due to diffraction. However, in our case, at frequencies lower than 50 kHz, to excite a plane A 0 mode wave in the film is unrealistic.
Therefore, for attenuation measurement, instead of a plane wave, we proposed the use of a cylindrical wave excited by a point type source. As such a source, a transducer creating displacements normal to the film with lateral dimensions smaller than wavelength of A 0 mode can be used. In this case, the normal displacement ξ(x j , t) of the film at the distance x j from the source is given by [13] where t is the time, ξ 0 is the normal displacement at the centre of the point type source, k is the wave number, and α(f ) is the frequency f -dependent attenuation coefficient. The normal displacements of the film at different distances from the source can be measured by a laser interferometer. Such measurements allow the avoidance of the influence of the ultrasonic wave propagating from the excitation source in air.
The spectrum of the normal displacement at the distance x j is given by where FFT is the fast Fourier transform. From spectra obtained at different distances at selected frequencies, it is possible to obtain the attenuation coefficient α(f ): where S m (x m , f ) and S n (x n , f ) are the spectra of the normal displacements at the distances x m and x n , respectively. Another way to obtain the attenuation coefficient α (f ) is based on approximation of the measured normal displacement dependence versus distanceξ(x j ) by the theoretical dependence given by Equation (1): During approximation, the attenuation coefficient α(f ) in Equation (4) is modified until the best fit to the experimental results is obtained. Such a procedure can be performed by the Nelder-Mead simplex algorithm [31], which enables us to find the minimum of the unconstrained multivariable function using a derivative-free method: where J is the total number of the measurement points. The value of the attenuation coefficientα( f ) at which the minimum of the function in Equation (5) is obtained corresponds to the measured attenuation coefficient of the A 0 mode in the film. This approach should be more accurate than the method based on measurement at two different distances because, in the latter case, the measurement result is obtained from many amplitude measurements at different distances x j from the source.
The accuracy of such a method first of all depends on how close the used source of the A 0 mode is to a point type source. If this requirement is not fulfilled, then the decay of the normal displacement amplitude along the distance from the source cannot be described by Equations (1) and (2), which will introduce additional diffraction errors into the results of measurements. From a practical point of a view, any kind of source possesses finite dimensions. For the excitation of the A 0 mode, a contact-type transducer with a high efficiency PMN-32%PT strip-like piezoelectric crystal operating in a longitudinal-extension mode has been used [32]. This transducer is described in more detail in Section 3. The exciting aperture is a rectangular tip of the transducer with dimensions of 1 × 5 mm 2 . In order to check how close the aperture is to a point type source at the operation frequency of 44 kHz, finite element modelling of the excitation and propagation of the guided wave in a PVC film was performed.

Finite Element Modelling
The numerical investigation of the propagation of a pulsed ultrasonic A 0 Lamb wave in the 135-µm thick 3D PVC film was carried out using the Abaqus 6.16 finite element software. To solve the transient wave equation, an explicit algorithm was used. In order to reduce the computational time, only a quarter of the plate with symmetry boundary conditions was modelled. A graphical representation of the model is presented in Figure 2. The accuracy of such a method first of all depends on how close the used source of the A0 mode is to a point type source. If this requirement is not fulfilled, then the decay of the normal displacement amplitude along the distance from the source cannot be described by Equations (1) and (2), which will introduce additional diffraction errors into the results of measurements. From a practical point of a view, any kind of source possesses finite dimensions. For the excitation of the A0 mode, a contact-type transducer with a high efficiency PMN-32%PT strip-like piezoelectric crystal operating in a longitudinal-extension mode has been used [32]. This transducer is described in more detail in Section 3. The exciting aperture is a rectangular tip of the transducer with dimensions of 1 × 5 mm 2 . In order to check how close the aperture is to a point type source at the operation frequency of 44 kHz, finite element modelling of the excitation and propagation of the guided wave in a PVC film was performed.

Finite Element Modelling
The numerical investigation of the propagation of a pulsed ultrasonic A0 Lamb wave in the 135-µ m thick 3D PVC film was carried out using the Abaqus 6.16 finite element software. To solve the transient wave equation, an explicit algorithm was used. In order to reduce the computational time, only a quarter of the plate with symmetry boundary conditions was modelled. A graphical representation of the model is presented in Figure 2. The modelled film is placed in a vacuum. The attenuation coefficient of the A0 mode in the material was assumed to be negligible.
In order to excite the asymmetric A0 guided wave in the PVC film, the normal transient excitation force of 1 N was applied to the selected nodes on the ultrasonic transducer zone. The modelling was performed using two different dimensions of the excitation zone: a 1 × 1 mm 2 zone that corresponds to a point type source, and a 1 × 5 mm 2 zone which corresponds to the dimensions of the radiating aperture used in the experimental investigation. The excitation force is uniformly distributed on the excitation surface. The excitation force was of 44 kHz 5 periods burst with the Gaussian envelope. The frequency was chosen to be the same as the operation frequency of the high-efficiency ultrasonic transducer used in the experimental investigations. The time diagram and the spectrum of a transient excitation force are presented in Figure 3a,b.
The modelling of wave propagation was performed by solving the following dynamic equation: The 3D geometry of the PVC film was meshed using 8 nodes and linear brick elements of C3D8R with a hourglass control. The hourglass control prevents the propagation of zero energy modes through the mesh, which may lead to inaccurate solutions. The size of the finite elements was The modelled film is placed in a vacuum. The attenuation coefficient of the A 0 mode in the material was assumed to be negligible.
In order to excite the asymmetric A 0 guided wave in the PVC film, the normal transient excitation force of 1 N was applied to the selected nodes on the ultrasonic transducer zone. The modelling was performed using two different dimensions of the excitation zone: a 1 × 1 mm 2 zone that corresponds to a point type source, and a 1 × 5 mm 2 zone which corresponds to the dimensions of the radiating aperture used in the experimental investigation. The excitation force is uniformly distributed on the excitation surface. The excitation force was of 44 kHz 5 periods burst with the Gaussian envelope. The frequency was chosen to be the same as the operation frequency of the high-efficiency ultrasonic transducer used in the experimental investigations. The time diagram and the spectrum of a transient excitation force are presented in Figure 3a,b.
The modelling of wave propagation was performed by solving the following dynamic equation: wave at the frequency of 44 kHz. The central difference integration method is conditionally stable, and the most critical variable using this method is the time step size Δt. The time step Δt must be smaller than the stability limit of the central difference method. If the time step size is not small enough, then the solution becomes unstable. In our case, the stable solutions are obtained with a time step duration of 10 ns. In the case of the point type excitation, and when the attenuation in the material is neglected, the amplitude of the propagating guided wave should decrease only due to the diffraction effect according to the 1/ r law, where r is the distance from the excitation source. In order to check this, the calculated normal displacements at each node along the x axis were recorded, the spectra of all signals were calculated and the spectra values at the frequency of 44 kHz were extracted. The result is presented in Figure 4 by the solid line. The normal displacement amplitudes versus distance were approximated according to Equations (4) and (5) by the above-described Nelder-Mead simplex algorithm. The obtained result is shown in Figure 4 by the dashed line. It was found that, in this case, 1/ r curve-fitting revealed the presence of an additional 0.75 dB/cm attenuation ( Figure 4). This phenomenon can be explained by a numerical attenuation introduced by the finite element software, and therefore should be neglected as a numerical artefact. In the Abaqus explicit software, a small amount of numerical damping is introduced by default in a form of bulk viscosity to control high-frequency oscillations [33]. The 3D geometry of the PVC film was meshed using 8 nodes and linear brick elements of C3D8R with a hourglass control. The hourglass control prevents the propagation of zero energy modes through the mesh, which may lead to inaccurate solutions. The size of the finite elements was 65 µm. This size corresponds to 1/45 th of the wavelength of the slowest A 0 mode ultrasonic Lamb wave at the frequency of 44 kHz. The central difference integration method is conditionally stable, and the most critical variable using this method is the time step size ∆t. The time step ∆t must be smaller than the stability limit of the central difference method. If the time step size is not small enough, then the solution becomes unstable. In our case, the stable solutions are obtained with a time step duration of 10 ns.
In the case of the point type excitation, and when the attenuation in the material is neglected, the amplitude of the propagating guided wave should decrease only due to the diffraction effect according to the 1/ √ r law, where r is the distance from the excitation source. In order to check this, the calculated normal displacements at each node along the x axis were recorded, the spectra of all signals were calculated and the spectra values at the frequency of 44 kHz were extracted. The result is presented in Figure 4 by the solid line. The normal displacement amplitudes versus distance were approximated according to Equations (4) and (5) by the above-described Nelder-Mead simplex algorithm. The obtained result is shown in Figure 4 by the dashed line. It was found that, in this case, 1/ √ r curve-fitting revealed the presence of an additional 0.75 dB/cm attenuation ( Figure 4). This phenomenon can be explained by a numerical attenuation introduced by the finite element software, and therefore should be neglected as a numerical artefact. In the Abaqus explicit software, a small amount of numerical damping is introduced by default in a form of bulk viscosity to control high-frequency oscillations [33].
In the case of the 1 × 5 mm 2 size excitation zone, a modulus of the particle velocity field at two different time instants is shown in Figure 5. Spatial distributions of the particle velocity field were analysed along x and y symmetry axes (see Figure 2). A-scans at different distances from the excitation zone along the x axis are presented in Figure 6. The signal amplitudes were normalized according to the amplitude maximum of the signal at the excitation zone. An approximation of the calculated dependences was performed in the same way as in the case of the point type source. An analysis of the signal spectra along x and y axes revealed that amplitude decay law is very close to the results obtained for a point type excitation (Figure 7a,b). In the case of the signals obtained along the x axis, the additional numerical attenuation is 0.52 dB/cm, and in the case of the signals along the y axis, the additional numerical attenuation is 0.7 dB/cm. Both those numerical attenuations may be neglected as before because no material attenuation was taken into account during finite element modelling. The obtained results allow us to make the assumption that a 1 × 5 mm 2 size ultrasonic transducer at 44 kHz can be analysed in the same way as a point type transducer, and for the A 0 mode attenuation measurement, it is possible to use the method presented above (Equations (4) and (5)).  In the case of the 1 × 5 mm 2 size excitation zone, a modulus of the particle velocity field at two different time instants is shown in Figure 5. Spatial distributions of the particle velocity field were analysed along x and y symmetry axes (see Figure 2). A-scans at different distances from the excitation zone along the x axis are presented in Figure 6. The signal amplitudes were normalized according to the amplitude maximum of the signal at the excitation zone. An approximation of the calculated dependences was performed in the same way as in the case of the point type source. An analysis of the signal spectra along x and y axes revealed that amplitude decay law is very close to the results obtained for a point type excitation (Figure 7a,b). In the case of the signals obtained along the x axis, the additional numerical attenuation is 0.52 dB/cm, and in the case of the signals along the y axis, the additional numerical attenuation is 0.7 dB/cm. Both those numerical attenuations may be neglected as before because no material attenuation was taken into account during finite element modelling. The obtained results allow us to make the assumption that a 1 × 5 mm 2 size ultrasonic transducer at 44 kHz can be analysed in the same way as a point type transducer, and for the A0 mode attenuation measurement, it is possible to use the method presented above (Equations (4) and (5)).  In the case of the 1 × 5 mm 2 size excitation zone, a modulus of the particle velocity field at two different time instants is shown in Figure 5. Spatial distributions of the particle velocity field were analysed along x and y symmetry axes (see Figure 2). A-scans at different distances from the excitation zone along the x axis are presented in Figure 6. The signal amplitudes were normalized according to the amplitude maximum of the signal at the excitation zone. An approximation of the calculated dependences was performed in the same way as in the case of the point type source. An analysis of the signal spectra along x and y axes revealed that amplitude decay law is very close to the results obtained for a point type excitation (Figure 7a,b). In the case of the signals obtained along the x axis, the additional numerical attenuation is 0.52 dB/cm, and in the case of the signals along the y axis, the additional numerical attenuation is 0.7 dB/cm. Both those numerical attenuations may be neglected as before because no material attenuation was taken into account during finite element modelling. The obtained results allow us to make the assumption that a 1 × 5 mm 2 size ultrasonic transducer at 44 kHz can be analysed in the same way as a point type transducer, and for the A0 mode attenuation measurement, it is possible to use the method presented above (Equations (4) and (5)).

Experimental Set-Up
For measurements of the attenuation of a sub-sonic A0 Lamb wave mode, a thin polyvinyl chloride (PVC, London, UK) film was chosen. The film with lateral dimensions 210 × 297 mm 2 and 0.135 mm thickness was fixed in a specially made rectangular frame (Figure 8). For the excitation of the A0 Lamb wave mode, a rectangular strip-like piezoelectric element with dimensions of 15 × 5 × 1 mm 3 made of the PMN-32%PT piezoelectric crystal (HC Materials Corporation, Bolingbrook, IL, USA) was exploited. The piezoelectric element was excited in the transverse-extension mode at the lowest resonance frequency f0 = 44 kHz. Regarding radiation, the tip of the piezoelectric strip with a rectangular aperture of 5 × 1 mm 2 was used. In this case, the sides of the radiating aperture a and b are close to or less than the wavelength A0 of the A0 mode in the film; e.g., a/A0 = 1.58 and b/A0 = 0.32. This means that such a source of ultrasonic waves may be considered as close to a point type source, and correspondingly, attenuation values measured along directions x and y in the case of an isotropic material should be close to each other.

Experimental Set-Up
For measurements of the attenuation of a sub-sonic A 0 Lamb wave mode, a thin polyvinyl chloride (PVC, London, UK) film was chosen. The film with lateral dimensions 210 × 297 mm 2 and 0.135 mm thickness was fixed in a specially made rectangular frame (Figure 8). For the excitation of the A 0 Lamb wave mode, a rectangular strip-like piezoelectric element with dimensions of 15 × 5 × 1 mm 3 made of the PMN-32%PT piezoelectric crystal (HC Materials Corporation, Bolingbrook, IL, USA) was exploited. The piezoelectric element was excited in the transverse-extension mode at the lowest resonance frequency f 0 = 44 kHz. Regarding radiation, the tip of the piezoelectric strip with a rectangular aperture of 5 × 1 mm 2 was used. In this case, the sides of the radiating aperture a and b are close to or less than the wavelength λ A 0 of the A 0 mode in the film; e.g., a/λ A 0 = 1.58 and b/λ A 0 = 0.32. This means that such a source of ultrasonic waves may be considered as close to a point type source, and correspondingly, attenuation values measured along directions x and y in the case of an isotropic material should be close to each other. The used strip-like PMN-32%PT [34] piezoelectric elements in a transverse-extension mode possess a very high electromechanical coupling coefficient k32 (0.84-0.90), which makes them attractive for low-frequency applications [31]. To improve the bandwidth and efficiency of radiation, a quarter wavelength matching strip made of AIREX T90.210 [35] type polystyrene foam (AIREX AG, Sins, Switzerland) was bonded to the tip of the piezoelectric strip. The excitation of A0 Lamb wave was performed via a dry contact between the matching strip and the film.
For attenuation measurements, normal displacements of the film versus distances along x and y directions perpendicular to the long and short sides of the rectangular aperture 5 × 1 mm 2 were measured by the Polytec laser interferometer OFV-5000 (Polytec GmbH, Waldbronn, Germany) [36] ( Figure 8). For this purpose, the supporting frame and the PVC film were scanned together with the dry-contact ultrasonic transducer with respect to the laser beam. This allowed us to achieve a stable acoustic contact during measurements between the strip-like ultrasonic transducer and the film. Two-dimensional scanning along x and y directions was performed by the 2D scanner 8MTF-75LS05 [37] (Standa, Vilnius, Lithuania) ( Figure 8).
The ultrasonic transducer was excited by AFG-3051 [38] (GW INSTEC, Taiwan) generator using the 70-cycle duration electric pulse with a 0.5 V amplitude and with a 0.5 s repetition period. The used ultrasonic transducer is relatively narrowband; therefore, in order to achieve the steady state amplitude of vibrations and to increase the accuracy of measurements, 70-cycle excitation is required. The normal displacement waveforms of the film registered by the Polytec laser interferometer were digitized and recorded by the analog-digital converter ADQ214 [39] (Teledyne SP Devices, Vaxholm, Sweden) with a sampling frequency of 50 MHz. In order to improve the signal to noise ratio at each measurement point, 5-10 signals were averaged. The synchronization of the whole measurement system and saving of recorded signals is performed by a master computer. The view of the experimental set-up is shown in Figure 9. An example of the normal displacement signal of a thin film measured above the centre of the ultrasonic transducer aperture is shown in Figure 10. The used strip-like PMN-32%PT [34] piezoelectric elements in a transverse-extension mode possess a very high electromechanical coupling coefficient k 32 (0.84-0.90), which makes them attractive for low-frequency applications [31]. To improve the bandwidth and efficiency of radiation, a quarter wavelength matching strip made of AIREX T90.210 [35] type polystyrene foam (AIREX AG, Sins, Switzerland) was bonded to the tip of the piezoelectric strip. The excitation of A 0 Lamb wave was performed via a dry contact between the matching strip and the film.
For attenuation measurements, normal displacements of the film versus distances along x and y directions perpendicular to the long and short sides of the rectangular aperture 5 × 1 mm 2 were measured by the Polytec laser interferometer OFV-5000 (Polytec GmbH, Waldbronn, Germany) [36] ( Figure 8). For this purpose, the supporting frame and the PVC film were scanned together with the dry-contact ultrasonic transducer with respect to the laser beam. This allowed us to achieve a stable acoustic contact during measurements between the strip-like ultrasonic transducer and the film. Two-dimensional scanning along x and y directions was performed by the 2D scanner 8MTF-75LS05 [37] (Standa, Vilnius, Lithuania) ( Figure 8).
The ultrasonic transducer was excited by AFG-3051 [38] (GW INSTEC, Taiwan) generator using the 70-cycle duration electric pulse with a 0.5 V amplitude and with a 0.5 s repetition period. The used ultrasonic transducer is relatively narrowband; therefore, in order to achieve the steady state amplitude of vibrations and to increase the accuracy of measurements, 70-cycle excitation is required. The normal displacement waveforms of the film registered by the Polytec laser interferometer were digitized and recorded by the analog-digital converter ADQ214 [39] (Teledyne SP Devices, Vaxholm, Sweden) with a sampling frequency of 50 MHz. In order to improve the signal to noise ratio at each measurement point, 5-10 signals were averaged. The synchronization of the whole measurement system and saving of recorded signals is performed by a master computer.
The view of the experimental set-up is shown in Figure 9. An example of the normal displacement signal of a thin film measured above the centre of the ultrasonic transducer aperture is shown in Figure 10.

Experimental Results
The measurement of the attenuation of the A0 mode Lamb wave was based on the collection of waveforms of normal displacement signals ) , , ( t y x i j  at different distances from the exciting ultrasonic transducer and comparing them with calculated amplitude versus distance dependences made under the assumption that the wave is excited by a point type source. Such measurements were performed at two orthogonal directions, x and y, which are perpendicular to the longer and shorter sides of the radiating aperture.
The scanning of the frame with the attached ultrasonic transducer was performed from 10 mm to 80 mm between the incident laser beam and the ultrasonic transducer. The 0 mm distance corresponds to the centre of the radiating aperture.

Experimental Results
The measurement of the attenuation of the A 0 mode Lamb wave was based on the collection of waveforms of normal displacement signals ξ(x j , y i , t) at different distances from the exciting ultrasonic transducer and comparing them with calculated amplitude versus distance dependences made under the assumption that the wave is excited by a point type source. Such measurements were performed at two orthogonal directions, x and y, which are perpendicular to the longer and shorter sides of the radiating aperture.
The scanning of the frame with the attached ultrasonic transducer was performed from 10 mm to 80 mm between the incident laser beam and the ultrasonic transducer. The 0 mm distance corresponds to the centre of the radiating aperture.
The displacement signals were registered by the laser interferometer with initial laser beam positions of x 1 = 10 mm or y 1 = 10 mm, depending on the measurement direction. The scanning step was 0.5 mm. All collected A-scans were normalized according to the maximum amplitudes of the A-scans at the distances x = 10 mm or y = 10 mm. The B-scan of the measured normal displacements of the PVC film is presented in Figure 11. The amplitudes of the normal displacement presented in Figure 11 by different colours were normalized with respect to the maximal amplitude obtained at the distance x = 10 mm. According to this B-scan, the phase velocity of the A 0 mode in the investigated PVC film was 139 m/s; e.g., much slower than the ultrasound velocity in air. distances xj and yi from the excitation aperture were found. Their dependencies versus distance are shown in Figure 13a,b by black solid lines. The amplitudes presented in Figure 13 were normalized with respect to the amplitude value at the position 10 mm. The results are presented for two orthogonal directions, x and y, corresponding to the wide and the narrow edge of the radiating aperture.  The collected raw A-scans at the distances x j = 20 mm and x j = 60 mm are shown in Figure 12a,b. From the presented signal, it follows that they are affected by noise. Therefore, in order to increase the accuracy of measurements, they were filtered by a bandpass Gaussian filter with the following parameters: a central frequency of 44 kHz, and a filter bandwidth of 0.9 kHz. The filtered signals are presented in Figure 12c,d.
In those equations, x K 1 and y K 1 are the normalization coefficients, and x K 2 and y K 2 represent the measured attenuation coefficients   f  of the A0 mode Lamb wave in x and y directions. The results obtained from those approximations are shown in Figure 13 by red solid lines. There is a very good correspondence between the measurement data and the approximations. This means that the used ultrasonic transducer with a rectangular aperture can be considered as a point like source and the attenuation coefficient can be found from Equations (8) and (9). The attenuation coefficients calculated from those curves are the following: along the x direction, 1.99 dB/cm, and along the y direction, 2.03 dB/cm; e.g. the difference between the attenuation coefficients in x and y directions is only 0.04 dB/cm. Taking into account the fact that this difference is very small, it is possible to make the conclusion that the attenuation of the slow A0 mode in a PVC film at the frequency of 44 kHz is 2 dB/cm.
The measurement uncertainty mainly depends on the uncertainty of normal displacement measurements performed by the laser interferometer and the uncertainties of spatial coordinates xj and yi. The latter uncertainty is due to the 2D scanner performing scanning. The spatial resolution of it is very high; the minimal scanning step is 10 μm. This means that uncertainty due to the scanner may be neglected.  The attenuation of the A 0 mode is frequency-dependent; therefore, it was evaluated at a fixed selected frequency. For this purpose, the spectra of the filtered waveforms ξ(x j , y i , t) were calculated using a fast Fourier transform: In each spectrum, at the frequency of 44 kHz, the amplitude values A S (x j , y i ) at different distances x j and y i from the excitation aperture were found. Their dependencies versus distance are shown in Figure 13a,b by black solid lines. The amplitudes presented in Figure 13 were normalized with respect to the amplitude value at the position 10 mm. The results are presented for two orthogonal directions, x and y, corresponding to the wide and the narrow edge of the radiating aperture. Therefore, the uncertainty of the measured normal displacement amplitudes is found to be the standard deviation of the peak amplitude of the spectrum at the selected frequency in the whole measurement range from 10 mm to 90 mm: where J is the total number of the measurement points. The total number of measurements from which the uncertainty was estimated was 1500. The scattering of the measured values  

Conclusions
For the measurement of the attenuation of the slow A0 mode in films, the method based on the application of a point type source of a guided wave was proposed. Such an approach allows the avoidance of diffraction errors that would be impossible to eliminate in the case of measurements with quasi-planar waves at low frequencies. For the excitation of the A0 mode, an ultrasonic transducer made of a high-efficiency PMN-32%PT strip-like piezoelectric crystal operating in a longitudinal-extension mode was used. For radiation, the tip of the piezoelectric strip with a rectangular aperture of 5 × 1 mm 2 was used. In this case, the sides of the radiating aperture are close or less than the wavelength A0 of the A0 mode in the film at the frequency of 44 kHz, which means that such a source may be considered as close to a point type source. This was also confirmed by a finite element modelling and by measurements performed in two orthogonal directions. Measurements of the attenuation of the sub-sonic A0 Lamb wave mode were performed in a polyvinyl chloride (PVC) film with lateral dimensions of 210 × 297 mm 2 and a thickness of 0.135 mm. The measured attenuation coefficient for this mode in PVC film at the frequency of 44 kHz is 2 dB/cm. The measurements at different frequencies will be correct if the assumption of the point type Such measurements allow us to evaluate how the used source with an aperture of 5 × 1 mm 2 is close to the point type source. The maximum amplitudes of the spectra A S (x j ) of signals versus x and y directions corresponding to different orientations of the ultrasonic transducer were approximated by expressions which are valid for a point type source: In those equations, K x 1 and K y 1 are the normalization coefficients, and K x 2 and K y 2 represent the measured attenuation coefficients α( f ) of the A 0 mode Lamb wave in x and y directions. The results obtained from those approximations are shown in Figure 13 by red solid lines. There is a very good correspondence between the measurement data and the approximations. This means that the used ultrasonic transducer with a rectangular aperture can be considered as a point like source and the attenuation coefficient can be found from Equations (8) and (9). The attenuation coefficients calculated from those curves are the following: along the x direction, 1.99 dB/cm, and along the y direction, 2.03 dB/cm; e.g., the difference between the attenuation coefficients in x and y directions is only 0.04 dB/cm.
Taking into account the fact that this difference is very small, it is possible to make the conclusion that the attenuation of the slow A 0 mode in a PVC film at the frequency of 44 kHz is 2 dB/cm.
The measurement uncertainty mainly depends on the uncertainty of normal displacement measurements performed by the laser interferometer and the uncertainties of spatial coordinates x j and y i . The latter uncertainty is due to the 2D scanner performing scanning. The spatial resolution of it is very high; the minimal scanning step is 10 µm. This means that uncertainty due to the scanner may be neglected.
Therefore, the uncertainty of the measured normal displacement amplitudes is found to be the standard deviation of the peak amplitude of the spectrum at the selected frequency in the whole measurement range from 10 mm to 90 mm: where J is the total number of the measurement points. The total number of measurements from which the uncertainty was estimated was 1500. The scattering of the measured values A S x j was evaluated with respect to the approximation curve A Sapp x j and the maximum possible deviation of the attenuation coefficient α( f ) was obtained. The uncertainties of the attenuation coefficients evaluated in such a way are the following: along the x direction, ±0.21 dB/cm, and along the y direction, ±0.16 dB/cm.

Conclusions
For the measurement of the attenuation of the slow A 0 mode in films, the method based on the application of a point type source of a guided wave was proposed. Such an approach allows the avoidance of diffraction errors that would be impossible to eliminate in the case of measurements with quasi-planar waves at low frequencies. For the excitation of the A 0 mode, an ultrasonic transducer made of a high-efficiency PMN-32%PT strip-like piezoelectric crystal operating in a longitudinal-extension mode was used. For radiation, the tip of the piezoelectric strip with a rectangular aperture of 5 × 1 mm 2 was used. In this case, the sides of the radiating aperture are close or less than the wavelength λ A 0 of the A 0 mode in the film at the frequency of 44 kHz, which means that such a source may be considered as close to a point type source. This was also confirmed by a finite element modelling and by measurements performed in two orthogonal directions. Measurements of the attenuation of the sub-sonic A 0 Lamb wave mode were performed in a polyvinyl chloride (PVC) film with lateral dimensions of 210 × 297 mm 2 and a thickness of 0.135 mm. The measured attenuation coefficient for this mode in PVC film at the frequency of 44 kHz is 2 dB/cm. The measurements at different frequencies will be correct if the assumption of the point type source for those frequencies is still valid.
It is necessary to point out that, during the manufacturing of PVC films, various defects such as wrinkles, holes, a rough surface and thickness variations arise [40]. Some of these are detected by optical methods, but such defects as holes and especially thickness variations could be found by ultrasonic methods using the discussed A 0 mode guided waves. However, the application of such a technique is possible when the attenuation of such waves is not too high. The determined attenuation values are suitable for ultrasonic testing methods, and the obtained results can be useful for the development of quality control methods of plastic films.