· Home · Table of Contents · Fundamental & Applied Research | Broadband dispersion behaviour of wedge wavesC. H. Yang and J. S. LiawDepartment of Mechanical Engineering, Chang Gung University 259 Wen-Hua 1^{st} Rd., Kwei-Shan, Taoyuan, Taiwan. Contact |
Wedge waves are guided acoustic waves propagating along wedge tips. Until now, no elasticity-based close-form solution is available for the wedge wave problems. By assuming the wedge as a thin plate with variable thickness, McKenna obtained a theoretical approximation for the prediction of dispersion relation of truncated wedges. However, the thin plate analysis can be expected to be valid only for small-angled wedges. Wideband dispersion relations of large-angled wedges cannot be estimated with confidence. This paper is focused on the investigation of the dispersion behaviors of wedge waves traveling along large-angled wedges with a broadband laser ultrasound measurement technique. It is found out that the thin plate theory is limited regarding to the bandwidth and apex angle.
Discovered by Lagasse [1,2] in early 1970's through a numerical study, wedge waves are guided acoustic waves propagating along the tip of a wedge, with energy tightly confined near the apex. Like Lamb waves, wedge waves with displacement field anti-symmetric about the mid-apex-plane are called anti-symmetric flexural (ASF) modes. ASF modes have attractive characteristics, including absence of dispersion, low phase velocity, and tight confinement of energy around the apex. These characteristics enable wedge waves for potential applications such as signal processing devices, under-water acoustics, sensors, and geophysics.
Up to date, no elasticity-based exact solution is available for the wedge wave problems. By assuming the wedge as a thin plate with variable thickness, McKenna obtained a theoretical approximation for the prediction of dispersion relation of truncated wedges [3]. Also, Krylov uses geometric acoustic approximation [4-7] to obtain phase velocity of ASF modes. Experimental works employing piezoelectric transducer, miniature non-contact EMAT transducers [8-10] or laser ultrasound techniques [11-13] have been conducted to investigate different aspects of ASF modes, including the influence of apex angle, apex truncation, water loading effects, and curvature effects of wedge apex [13].
A nominally sharp wedge may prove to be too dispersive for systems that require a large delay-bandwidth product. Therefore, it is an important task to predict and furthermore control the dispersion caused by wedge-tip truncation. For the prediction of dispersion relation of a truncated wedge, McKenna's analysis [3] based on the thin plate theory can be used for small-angle wedges. However, since the thin plate theory does not apply to large wedge angles, no estimate of the expected dispersion for other than small wedge angles can be made with any confidence. Therefore, dispersion due to truncation of the tip is a potential problem that should be investigated in the case of large-angled wedges.
In fact in 1992, Jia et. al. measured the dispersion behavior of duralumin wedges with apex angles of 30° and 60° using laser ultrasound technique [12]. However, in 1995, Chaumel [10] showed that Jia's measurements deviate from MeKenna's prediction substantially within the bandwidth of x_{0}k less than 1.0. At the same time, while Chamuel's measurement results and McKenna's prediction agree well while x_{0}k is within 0.5, the discrepancy begins to grow fast as x_{0}k increases.
Based on above observation, broadband dispersion relations of ASF modes traveling along large-angled wedges need further investigations. In this paper, laser ultrasound technique is applied to measure the broadband dispersion relation of ASF modes. On the other hand, the validity in using McKenna's thin plate approximation for the prediction of dispersion relations of ASF modes will be discussed.
Fig 1: A schematic showing the truncation of a straight wedge. |
Wedge waves propagating along brass wedge are investigated in the current research. The brass material has a longitudinal wave speed of 4440 m/sec and surface wave speed of 1950 m/sec. Three wedge specimens with different apex angles, 15°, 30°, and 60° are prepared. The wedge specimens are prepared to have relatively small truncations initially. After a dispersion relation has been measured, the wedges are further truncated and another measurement is taken. This process is repeated to obtain a sequence of measurements for different levels truncations on the same wedge. For each of the truncation stage, the truncation of the wedge specimen is measured with an optical microscope.
Fig 2: A schematic showing the laser generation/laser detection ultrasonic system. |
A laser-generation/laser-detection ultrasonic technique is used in the current measurement. As shown in Figure 2, the experiment configuration consists of a pulsed laser for ultrasonic wave generation and a laser-based acoustic wave detection system. The excitation source is a Nd:Yag laser with a 532 nm wavelength, and duration time of 6.6 ns. A laser optical knife-edge technique, with a broadband photoreceiver, is applied to detect the ASF modes propagating along the tip of wedges. Figure 3 shows the measured ASF modes propagating along a 30^{o} brass wedge. In the figure the fundamental ASF mode id denoted as A_{1}, while A_{2} denotes the second ASF mode.
Fig 3: ASF modes propagating along a 15° brass wedge measured with laser ultrasound technique. |
A B-scan scheme is used for the measurement of dispersion behavior of wedge waves. During the scanning, the optical detector is located at a fixed point, while the generation laser beam is scanned along the wedge tip. Along the wedge tip there are 150 scanning steps with a step size of 0.1 mm.
A two-dimensional fast Fourier transform (FFT), first taken with respect to time and second with respect to scanned position, is then used to obtain the c-f, phase velocity-frequency, dispersion relation.
Broadband dispersion measurement for the fundamental ASF mode of the 15^{o} brass wedge with the laser ultrasound technique is shown in Figure 5. The dispersion curves are plotted in a normalized dispersion space first used by McKenna in the thin plate theory [3]. The non-dimensional horizontal axis is x_{0}k, where x_{0} denotes the apex truncation as shown in Figure 1, k as the wavenumber of the ASF mode. The vertical axis V/V_{W} is phase velocity non-dimensionalized by the truncation-free phase velocity of the ASF wedge mode, V_{W}. Here V_{W} is determined by Lagasse's empirical formula [1]:
V_{W}=V_{R}sin(nq ), |
where V_{R} is the surface wave speed of the wedge material, q is the apex angle, and n=1 for the fundamental ASF mode. In fact, a horizontal line in Figure 5 labeled as V_{R} with a normalized phase velocity of 3.87 indicates the surface wave velocity of the brass material. Since the truncation-free phase velocity of the ASF wedge mode varies from wedge to wedge, the normalized phase velocity corresponding to the surface wave speed also varies for different wedges.
Fig 4: Measured dispersion relation of wedge waves for a straight 15^{o} brass wedge including different levels of truncations, compared with the thin plate theory by McKenna. |
As shown in Figure 4, there are 8 individual dispersion measurements on the same 15^{o} brass wedge specimen. Each of the dispersion curves corresponds to different levels of truncation: 39mm, 153 mm, 192 mm, 221 mm, 247 mm, 289 mm, 331 mm, and 371 mm. It is shown in Figure 5 that the dispersion curves have normalized phase velocity of about 1.0 at the low wavenumber limit. As the wavenumber increases, the normalized phase velocity also increases. However, after the wavenumber reaches the value of 17 or higher, the measurement results in Figure 4 show that the normalized phase velocity begins to saturate at a value very close to 3.87, which corresponds to the surface wave velocity of the brass material. This observation is very reasonable and can be explained as following. In the long wavelength regime, the truncation is so small compared with the wavelength of the ASF mode. Without any characteristic length in the wedge, the ASF mode takes the truncation-free ASF wave speed as its phase velocity, which corresponds to a normalized phase velocity of 1.0. As the wavelength of the ASF mode gradually decreases and finally becomes so small compared with the wedge truncation. In this case, the ASF mode is dominated by the surface wave characteristics and takes the surface wave speed as the saturated wave speed. Between the long wavelength and short wavelength limits, the dispersion curves of the fundamental ASF mode show gradual transition from the truncation-free ASF mode wave speed to the surface wave speed.
As shown in the Figure, except some scatter in the low x_{0}k regime, the 8 dispersion curves corresponding to different truncations cluster into a universal curve in the normalized dispersion space. Originally used by McKenna [3] in the thin plate theory, this universal curve is experimentally observed in the current experimental results. Universal curves are also observed for other 5 wedges belonging to different apex angles and different materials.
It is one of the objectives for the current research to explore the range of validity on applying the thin plate theory for the prediction of dispersion relations of ASF modes propagating along truncated wedges. Figure 4 indicates that for the 15^{o} brass wedge, the predicted dispersion relation based on McKenna's thin plate theory always over-estimate the phase velocity. For the low-wavenumber limit, no clear discrepancy can be observe between the prediction and measurements. However, the discrepancy grows fast as the wavenumber increases. In particular, one notices that the predicted normalized phase velocity based on the thin plate theory has no saturation behavior demonstrated by the laser ultrasound measurements, but increases almost linearly with respect to x_{0}k.
As pointed out by McKenna that the thin plate approximation is expected to be accurate only for low frequency ASF modes propagating along small-angled wedges. For the current case of 15^{o} brass wedge, the prediction accuracy is limited at the low frequency regime of x_{0}k<1.5. The over-estimate on phase velocity by the prediction can attribute to the assumptions of the thin plate theory. These assumptions include: middle-plane of the plate remain unstrained during bending, normals of the middle plane remain normals of the deformed plane after the deformation of the plate, and the neglecting of the shear stress components. These assumptions give extra deformation constraints on the displacement fields and result a stiffening effect when the thin plate theory is used for the prediction of ASF mode in the high frequency regime.
For brass wedge with larger apex angle, Figure 5 shows the measured dispersion relations for the 30^{o} brass wedge with 7 different levels of truncations. First of all, due to less scatter than that for the case of 15^{o} brass wedge, the 7 dispersion curves belonging to 7 truncations apparently cluster into universal curve in the normalized dispersion space. In the low-wavenumber limit, all the measured dispersion curves for different truncations begin with a normalized phase velocity very close to 1.0, corresponding to the truncation-free phase velocity. At the high-wavenumber limit, the measured dispersion curves indicate a saturated normalized phase velocity of 1.9, which again corresponds to the surface wave velocity of the brass material. Here one notices that the measured dispersion curves reach the saturated wave speed at a x_{0}k valued much earlier than that in the case of 15^{o} brass wedge case. The predicted dispersion relation based on the thin plate theory begins to deviate from the measurement earlier than that in the case of 15^{o} brass wedge. As shown in Figure 5, the predicted dispersion curve begins to deviate from the measurement from a x_{0}k value of about 0.7, less than that for the 15^{o} brass wedge of 1.5. Figure 6 shows the measured dispersion relation for 60^{o} brass wedge. The measurement results indicate that the 60^{o} brass wedge has an almost dispersion-free fundamental ASF mode. This plot also shows that the prediction is not really practical for the prediction of dispersion relation of such large-angled wedge.
Fig 5: Measured dispersion relation of wedge waves for a straight 30^{o} brass wedge including different levels of truncations, compared with the thin plate theory by McKenna. |
Fig 6: Measured dispersion relation of wedge waves for a straight 60^{o} brass wedge including different levels of truncations, compared with the thin plate theory by McKenna. |
Dispersion behaviors of fundamental ASF modes traveling along large-angled wedges are characterized with a broadband laser ultrasound measurement technique. Our measurement results on 6 wedges, covering 3 different apex angles and 2 materials, show that the dispersion curves measured form the same wedge with different levels of truncations cluster into a universal curve in the normalized dispersion space. And also, all the measured fundamental ASF modes saturate at a phase velocity of the material's surface wave velocity. Limited by its assumptions, the thin plate theory is proved to have prediction accuracy only within very limited low frequency regime. Beyond that, the phase velocities of the ASF modes are over-estimated by the thin plate theory. For these large-angled wedges with apex angle between 15^{o} and 60^{o}, the thin plate theory approximation developed by McKenna has good prediction accuracy in the low frequency regime from a x_{0}k value of 1.5 for the 15^{o} wedges to nearly 0 for 60^{o} wedges. The current study suggests that further theoretical models need to be developed for the prediction of dispersion behaviors of ASF modes propagating in large-angled wedges in wideband applications where practical applications are interested.
Support from National Science Council, Taiwan, through grant No. NSC89-2212-E182-003 is gratefully acknowledged.
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