Real-space topological singularities in structured flexural waves

Abstract

Real-space singularities underpin diverse wave phenomena yet remain largely unexplored in elastic wave systems. We report the observation of real-space topological singularities in structured flexural waves on finite-sized solids. These singularities are robust against perturbations and annihilate only through topological phase transitions. Moreover, they imprint dislocation lines on the radiated sound field, generating acoustic vortices in free space from an achiral source and structure. Our findings bridge continuum mechanics and topological physics, establishing elastic waves as a platform for exploring complex topological textures in real space and paving the way towards singular phononics.

Figures (12)

• Figure 1: (a) Solid shell hosting topological singularities of flexural fields. The shell has radius $r$ = 8 cm and thickness $t$ = 0.3 cm, and is excited by two point sources s$_1$ at ($\theta_1=90^{\circ}, \varphi_1=180^{\circ}$) and s$_2$ at ($\theta_2=90^{\circ}, \varphi_2=270^{\circ}$). (b) Normal (i.e., flexural, $u_{\perp}$) and tangent ($\mathbf{u}_{\|}$) components of the displacement field in the shell. The flexural field can excite the acoustic field in free space, while the tangent field cannot. (c) Displacement field and (d) pressure far field of the quadrupole eigenmode. (e) Excited flexural field $|u_{\perp}|$ and topological singularities. Two singularities “i” and “ii” appear on the front side, and the other singularities appear on the back side. The green and blue lines correspond to $u_{\perp}^{\Re}=0$ and $u_{\perp}^{\Im}=0$, respectively. (f) Singularities with charges $q=\pm 1$ in the complex plane and three simplest types of topological configuration. The black arrows indicate the traveling directions of phase $\Phi$ for a counterclockwise loop around the singularities in real space.
• Figure 2: (a) Experimental setup for mapping the displacement field of flexural waves on the shell, including the sample, scanning laser Doppler vibrometer (LDV), and wave function generator (WFG). The zoom-in shows the sample driven by two piezoelectric transducers s$_1$ and s$_2$ (moved to the front side of the sample for a better view). The green dot is the laser spot. The two black squares indicate the regions for mapping the displacement fields in the vicinity of singularities “i” and “ii”. (b) Simulated and measured displacement fields in the two regions in (a).
• Figure 3: (a) Phase dislocation lines of the pressure field generated by the shell. The red and blue arrows indicate the phase increasing directions. The shell color shows the phase of the flexural field $u_{\perp}$. (b) Pressure far field generated by the shell. (c) Amplitude and phase distributions in the areas enclosed by the squares in (a).
• Figure 4: (a) Topological phase transition involving the annihilation of two oppositely charged singularities “i” and “ii”, achieved by adjusting the location of source s$_2$ while source s$_1$ is fixed at $\theta_1=90^{\circ}, \varphi_1=180^{\circ}$. Star symbols denote the experimentally measured singularity positions for different values of $\varphi_2$ and fixed $\theta_2=90^{\circ}$. The colored lines show the simulated trajectories of the singularities. (b) Topological configurations before and after the topological phase transition. (c) Flexural wave singularities and pressure dislocation lines for $\varphi_2=280^{\circ}$, $300^{\circ}$, and $320^{\circ}$. (d) Simulated and experimentally measured displacement field $u_{\perp}$ in regions i and ii for $\varphi_2=280^{\circ}$, $300^{\circ}$, and $320^{\circ}$
• Figure 5: (a) Schematic for acoustic vortex generation by an ellipsoidal shell driven by a point source. The shell has the radii $r_y =1.05r_x$ with $r_x=r_z$ = 7 cm and the thickness $t=0.3$ cm. The point source s locates at ($\theta=80^{\circ}, \varphi=75^{\circ}$) with frequency $f$ = 2620 Hz. (b) Simulated and experimentally measured displacement fields near the four singularities "i-iv" in (a). (c) Multipole expansions of the acoustic fields radiated from the shell.