Broad-angle negative reflection and focusing of elastic waves from a plate edge

Guided elastic waves in plates, or Lamb waves, generally undergo reflection and mode conversion upon encountering a free edge. In the case where a backward-propagating Lamb wave is mode-converted to a forward-propagating wave or vice versa, the mode-converted wave is reflected on the same side of the surface normal as the incident wave. In this paper, we study such negative reflection and show that this effect can be achieved over a broad angular range at a simple plate edge. We demonstrate, through both numerical and experimental approaches, that a plate edge can act as a lens and focus a mode-converted Lamb wave field. Furthermore, we show that as the wave vectors of the incident and mode-converted Lamb waves approach each other, the mode-converted field nearly retraces the incident field. We propose that broad-angle negative reflection may find application in the nondestructive testing of structures supporting guided waves and in the development of new acoustic devices including resonators, lenses, and filters.

Figure 1

(a) Dispersion curve for symmetric Lamb waves in the vicinity of the ZGV point for a 1.0-mm-thick aluminum plate. (b) Mode conversion of an incident $S_{2b}$ mode at a free edge leads to negative reflection of the $S_0$ and $S_1$ modes. ${\theta }_I$ is the angle of incidence and ${\theta }_R$ is the angle of reflection. (c) The mode shapes of the incident $S_{2b}$ and the reflected $S_1$ modes near the ZGV point at a frequency of 2.85 MHz.

Figure 2

(a) Simulated out-of-plane surface displacements of the plate in the presence of a free edge at $x=25\mathrm{mm}$ (top, $y>0)$ and in the absence of a free edge (bottom, $y<0)$ for $f=2.859\mathrm{MHz}$ at $t=100\mu \mathrm{s}$. (b)–(d) Reflected wave fields from the free edge at (b) $f=2.889\mathrm{MHz}$ and $t=150\mu \mathrm{s}$, (c) $f=2.859\mathrm{MHz}$ and $t=150\mu \mathrm{s}$, and (d) $f=2.848\mathrm{MHz}$ and $t=250\mu \mathrm{s}$. The later time in (d) accounts for the slow group velocity in the close vicinity of the ZGV point.

Figure 3

Fourier domain representation of the wave fields at the frequencies of (a) 2.889 MHz, (b) 2.859 MHz, and (c) 2.848 MHz. The inner ring with the lowest spatial frequency corresponds to the excited $S_{2b}$ mode, and the outer circular segments represent the reflected $S_0$ and $S_1$ modes. The green and magenta lines show the maximum possible angles of reflection for the $S_0$ and $S_1$ modes, respectively. (d) Intensity plot of the reflected field for the lowest frequency $\left(f=2.848\mathrm{MHz}\right)$ showing the focusing of the acoustic energy just beyond the excitation point at $x=0\mathrm{mm}$.

Figure 4

(a) Schematic of the experimental setup used to excite and detect narrow-bandwidth Lamb waves and (b) the dimensions of the aluminum sample used for the measurements with the excitation point marked by the red dot.

Figure 5

(a) In-phase component of the raw experimental data at a frequency of 5.05 MHz. The wave field is dominated by the $S_{2b}$ mode with a wavelength of 3.6 mm. (b) Low-pass-filtered data (cutoff $k=3.4\mathrm{m}{\mathrm{m}}^{-1})$ isolating the $S_{2b}$ mode. (c) Band-pass-filtered data $\left(k=3.4-4.0\mathrm{m}{\mathrm{m}}^{-1}\right)$ to isolate the $S_1$ mode. Negative reflection and focusing of the field are apparent. (d) Band-pass-filtered data $\left(k=9.2-9.6\mathrm{m}{\mathrm{m}}^{-1}\right)$ to isolate the $S_0$ mode. Negative reflection leads to a concentration of the field along $y=0\mathrm{mm}$ for $x<-20\mathrm{mm}$.

Figure 6

Squared magnitude of the measured displacement fields at frequencies of (a) 5.050, (b) 5.025, and (c) 5.000 MHz. The data were band-pass filtered to isolate the $S_1$ mode with passbands of $k=3.4-4.0\mathrm{m}{\mathrm{m}}^{-1},k=3.3-4.0\mathrm{m}{\mathrm{m}}^{-1}$, and $k=3.1-4.0\mathrm{m}{\mathrm{m}}^{-1}$ for (a), (b), and (c), respectively. The $S_1$ focus recedes towards the excitation point $\left(x=0\right)$ as the frequency approaches the ZGV point (d) Ray diagram for negative reflection with mode conversion for 5.000 and 5.050 MHz.

Figure 7

Fourier domain representation of the wave fields from the experiment at frequencies of (a) 5.050, (b) 5.025, and (c) 5.000 MHz. As in Fig. 2, the inner ring corresponds to the excited $S_{2b}$ mode and the outer circular segments show the reflected $S_0$ and $S_1$ modes. The green and magenta lines show the maximum possible angles of reflection for the $S_0$ and $S_1$ modes, respectively, calculated from Snell's law using the experimentally obtained $k$ values.