⇦Chinese Physics Letters, 2017, Vol. 34, No. 11, Article code 114301⇨ Propagation Properties of Backward Lamb Waves in Plate Investigated by Dynamic Photoelastic Technique * Zhong-Tao Hu(胡中韬)1,2, Zhi-Wu An(安志武)1**, Guo-Xuan Lian(廉国选)1, Xiao-Min Wang(王小民)1 Affiliations 1State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190 2University of Chinese Academy of Sciences, Beijing 100049 Received 28 July 2017 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11374325 and 11427809.
^**Corresponding author. Email: anzhiwu@mail.ioa.ac.cn Citation Text: Hu Z T, An Z W, Lian G X and Wang X M 2017 Chin. Phys. Lett. 34 114301 Abstract The dynamic photoelastic technique is employed to visualize and quantify the propagation properties of backward Lamb waves in a plate. Higher energy leakage of second-order symmetric backward wave mode S$_{\rm 2b}$ in contrast to third-order anti-symmetric backward mode A$_{\rm 3b}$ is shown by the dispersion curve of a plate immersed in water, and then verified by experiments. To avoid the considerable high leakage, the plate is placed in air, both group and phase velocities of modes S$_{\rm 2b}$ and A$_{\rm 3b}$ in the glass plate are experimentally measured. In comparison with the theoretical values, less than 5% errors are found in experiments. DOI:10.1088/0256-307X/34/11/114301 PACS:43.35.-c, 46.40.Cd, 43.20.Bi © 2017 Chinese Physics Society Article Text Backward wave motion is broadly defined as the energy flux direction of wave motion, which is opposite to the direction of phase velocity or motion of individual wave front.[1-3] After Pendry proposed the idea that 'negative refraction makes a perfect lens', there have been discussions about control of acoustic energy using related phenomena.[4-7] By analyzing the characteristic of coupling modes in dispersion curves, it was proved that there are propagation modes near the cutoff frequency with negative group velocity or named 'backward wave motion'. For example, the backward wave branch S$_{\rm 2b}$ is connected to the S$_{2}$ mode by a purely imaginary branch, where S denotes the symmetric mode, and subscript b stands for backward wave.[1] Compared with phononic crystals or other methods, negative refraction is easier to realize in plates by backward Lamb waves.[8,9] In 1957, Tolstoy and Usdin found the existence of the backward mode in a homogeneous isotropic plate by numerical analysis of characteristic equation of a plate with a Poisson's ratio of 0.25.[10] Later, it was found that the backward waves exist over a certain range of Poisson's ratio or the bulk wave velocity ratio of the material.[11,12] In 1996, Negishi and Li visualized the third-order anti-symmetric backward mode (A$_{\rm 3b}$) excited by an ultrasonic pulse in water obliquely striking the glass plate using photoelastic technique, and claimed that the wavelength of S$_{\rm 2b}$ mode will become too large to be observed.[13] However, according to our numerical calculation and experimental results, it can be demonstrated that the leakage of acoustic energy for S$_{\rm 2b}$ mode is more rapid than that of A$_{\rm 3b}$ mode when the plate is immersed in water. As a result, S$_{\rm 2b}$ mode can only be detected during a very short time. In this study, the numerical calculation result of a plate immersed in water is given to show the energy leakage behavior of S$_{\rm 2b}$ mode in contrast to A$_{\rm 3b}$ mode. Experiments were carried out to verify the numerical predictions. Furthermore, to explore the backward propagation phenomenon more explicitly, we visualized as well as quantified the propagation properties of backward wave modes (namely, A$_{\rm 3b}$ and S$_{\rm 2b}$) in a plate placed in air. Both group and phase velocities of S$_{\rm 2b}$ and A$_{\rm 3b}$ modes, as well as their propagation directions, are measured using snapshots. Dynamic photoelasticity, with the mechanism of the temporary birefringent effect, is a powerful tool in experimental mechanics. In the ultrasound imaging system, optical glass usually serves as the wave-carrying medium. The glass is optically isotropic when it is free from stress. However, it will become anisotropic under stress and will display the characteristics similar to crystals. A pulsed YAG laser is used as the stroboscopic light source, and a synthesizer is utilized to adjust the time delay between the ultrasonic wave and laser to study the dynamic process. The laser emits a high intensity green light beam with very short duration (about 1 ns), which has the ability to seize 100 MHz ultrasound clearly. The sample is an optical glass plate with Poisson's ratio of 0.211 and a thickness of 3.02 mm. Its longitudinal wave velocity $c_{\rm L}$ and shear wave velocity $c_{\rm T}$ are measured to be 6000 m/s and 3631 m/s, respectively. The dispersion curves for group and phase velocities of S$_{1}$ and A$_{2}$ modes, obtained by the bisection technique, are shown in Fig. 1. The segments in S$_{1}$ and A$_{2}$ modes with negative group velocities are backward modes, i.e., S$_{\rm 2b}$ and A$_{\rm 3b}$, respectively. The marked points in each branch, as shown in Fig. 1, having the largest absolute value of group velocity, were selected to excite the two backward modes in the following experiments.

Fig. 1. The theoretical group velocity (a) and phase velocity (b) dispersion curves of S$_{1}$ and A$_{2}$ modes in the optical glass plate.

Fig. 2. The normalized attenuation (product of the attenuation constant and the wavelength ) for backward modes S$_{\rm 2b}$ (1, 1#) and A$_{\rm 3b}$ (2, 2#) of the sample placed in water (1, 2) and air (1#, 2#).

Let us clarify the confusion in Ref. [13] firstly. Figure 2 shows the normalized attenuation of S$_{\rm 2b}$ and A$_{\rm 3b}$ modes caused by the leakage when the plate is placed in water and air. It is quite obvious that the acoustic energy for either of the two modes leaks more rapidly when the plate is submerged in water. It is also found the normalized attenuation of S$_{\rm 2b}$ mode is several times larger than that of A$_{\rm 3b}$ mode. This feature of S$_{\rm 2b}$ was validated by the photoelastic experiment, as shown in Fig. 3. To generate S$_{\rm 2b}$ mode, a PZT transducer driven by a 968 kHz electrical tone burst of seven cycles radiated an ultrasonic wave into the plate at an incident angle of 4.21$^{\circ}$. As for A$_{\rm 3b}$ mode, the incident angle and frequency are 2.47$^{\circ}$ and 1786 kHz, respectively. The snapshots in Figs. 3 were captured by adjusting the time delay between the ultrasound and laser, where S$_{\rm 2b}$ and A$_{\rm 3b}$ modes are in frame intervals of 2 μs and 10 μs, respectively. It is clearly shown that S$_{\rm 2b}$ and A$_{\rm 3b}$ modes were generated with a negative (to the left) energy flux velocity in the plate. As the brightness of the figures is in proportion to acoustic energy, it becomes evident that S$_{\rm 2b}$ mode can propagate no more than 7.5 mm when the plate is placed in water, whereas A$_{\rm 3b}$ mode can propagate longer than 25.7 mm. It can be obtained that the main reason for the failure to visualize S$_{\rm 2b}$ mode in Ref. [13] is the energy leakage instead of large wavelength.

Fig. 3. The photoelastic images S$_{\rm 2b}$ mode (a) and A$_{\rm 3b}$ mode (b) in a plate immersed in water with a frame interval of 2 μs and 5 μs at $fd=2924$ kHz$\cdot$mm and $fd=5392$ kHz$\cdot$mm, respectively. The directions of energy flux velocity are indicated by red arrows.

To explore the propagation characteristics of backward waves in detail, the plate was placed in air to avoid energy leakage in the following experiments. The experimental setup including the sample, wedge and transducer is presented in Fig. 4(a). Both the symmetric (S$_{\rm 2b}$) and anti-symmetric (A$_{\rm 3b}$) backward Lamb modes in the glass plate were visualized using the dynamic photoelastic system. Then the group and phase velocities of backward modes are measured and compared with the theoretical predictions. In the experiment, distributions of shear stress component in the plate are visualized. To generate S$_{\rm 2b}$ mode, the point in the branch with $fd=2924$ kHz$\cdot$mm was selected. A 7-cycle sine electrical signal at frequency 968 kHz was delivered to drive a PZT transducer. Traveling through a polystyrene wedge, the ultrasound impinged upon the plate at an incident angle of 6.6$^{\circ}$. Figures 4(b) and 4(c) are the photoelastic images of S$_{\rm 2b}$ mode. In Fig. 4(b), the frame interval was set to be 0.1 μs. The phase velocity of S$_{\rm 2b}$ mode spreads to the right, which agrees with the direction of wave vector. The movement of wave package, which is much slower, can be observed over a longer time period. Figure 4(c) shows the snapshots of S$_{\rm 2b}$ mode with a frame interval of 10 μs, which gives a direct evidence of backward propagation. The actual length of each pixel in the figures can be calculated from Fig. 4(a). We repeated the measurement for 10 times and averaged the results. The group velocity is measured to be $-$780 m/s, and the theoretical value is $-$750 m/s, as indicated in Fig. 1(a). The phase velocity is measured to be 21400 m/s, and the theoretical value is 20410 m/s. The errors between the theoretical and experimental results, which mainly come from the jitter of the laser, are less than 5%. Similarly, to excite the A$_{\rm 3b}$ mode, the point in the branch with $fd=5392$ kHz$\cdot$mm was chosen. The incidence angle is calculated to be 3.92$^{\circ}$. The propagation direction of wave phase also spreads to the right as presented in Fig. 4(d), in which the frame interval is 0.05 μs. If the frame interval is extended to 10 μs, as shown in Fig. 4(e), the wave package is observed to go to the left. The group velocity of A$_{\rm 3b}$ mode is measured to be $-$520 m/s, which is comparable to the theoretical value $-$515 m/s. The measured phase velocity of A$_{\rm 3b}$ mode is 33400 m/s, and the theoretical value is 34580 m/s. The errors are less than 4%.

Fig. 4. (a) The experimental setup for visualizing backward wave mode S$_{\rm 2b}$. Photoelastic images of S$_{\rm 2b}$ mode ((b) and (c)) at $fd=2924$ kHz$\cdot$mm and A$_{\rm 3b}$ mode ((d) and (e)) at $fd=5392$ kHz$\cdot$mm in different frame intervals. The phase propagations of backward wave modes S$_{\rm 2b}$ (b) and A$_{\rm 3b}$ (d) were visualized in the frame interval of 0.1 μs and 0.05 μs, respectively. The energy flux propagations of S$_{\rm 2b}$ (c) and A$_{\rm 3b}$ (e) were in the frame interval of 10 μs. The propagation directions of phase and group velocity are indicated by red arrows.

In summary, the second-order symmetric backward wave mode (S$_{\rm 2b}$) in a plate has been visualized and studied by the photoelastic method for the first time. As discussed previously, S$_{\rm 2b}$ mode is hard to be visualized in a plate submerged in water due to its relatively high energy leakage. Our experimental results prove that S$_{\rm 2b}$ mode can only be detected during a very short time. To minimize this negative factor and to study the backward waves more explicitly, the glass plate is placed in air in our photoelastic experiments. An anti-symmetric (A$_{\rm 3b}$) backward wave in a plate is also visualized. The prediction of negative propagation is confirmed directly. The group and phase velocities are also measured in comparison with the theoretical values. Less than 5% errors are found, which demonstrate the applicability of measurement of backward waves by the photoelastic technique. We thank Dr. Hanyin Cui in the Institute of Acoustics, Chinese Academy of Sciences for useful discussion. References Investigation of Lamb waves having a negative group velocityNegative group velocity Lamb waves on plates and applications to the scattering of sound by shellsBackward waves with double zero-group-velocity points in a liquid-filled pipeSchlieren Visualization of Acoustic Propagation Characteristics in a One-Dimensional Phononic CrystalFocusing of Surface Acoustic Wave on a Piezoelectric CrystalAnalysis of Imperfect Acoustic Cloaking ResonancesElastic Wave Propagation in Two-Dimensional Ordered and Weakly Disordered Phononic CrystalsFocusing on Plates: Controlling Guided Waves using Negative RefractionManipulating Backward Propagation of Acoustic Waves by a Periodical StructureWave Propagation in Elastic Plates: Low and High Mode DispersionExistence of Negative Group Velocities in Lamb WavesLocal vibration of an elastic plate and zero-group velocity Lamb modesStrobo-Photoelastic Visualization of Lamb Waves with Negative Group Velocity Propagating on a Glass Plate

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