Acoustics of Cubic Bubbles: Six Coupled Oscillators

We introduce cubic bubbles that are pinned to 3D printed millimetric frames immersed in water. Cubic bubbles are more stable over time and space than standard spherical bubbles, while still allowing large oscillations of their faces. We find that each face can be described as a harmonic oscillator coupled to the other ones. These resonators are coupled by the gas inside the cube but also by acoustic interactions in the liquid. We provide an analytical model and 3D numerical simulations predicting the resonance with very good agreement. Acoustically, cubic bubbles prove to be good monopole subwavelength emitters, with nonemissive secondary surface modes.

Figure 1

(a) Design of a cubic frame (the holding pedestal plate not represented). (b) Top view of the immersion of the cubic frame in water: a gas bubble is trapped, and interfaces are visible on each face. (c) Schematic drawing of a cross section of the cube, showing the amplitude of vibration ${\xi }_i$ of an interface labeled $i$, while other faces $j$ have an amplitude ${\xi }_j$.

Figure 2

(a) Signal emitted by a bubble ($2a=2.1\mathrm{mm}$) after a pulse excitation. (Inset) Frequency spectrum of the signal emitted, normalized by the signal without a bubble, showing the norm and the cosine of the phase. (b) Measured frequency when varying the inner space between pillars $2a$, keeping $e=0.5\mathrm{mm}$ fixed. Crosses, 3D simulation; line, theory for a cubic bubble with six coupled oscillators [Eq. (5)]; dashes, Minnaert frequency for a bubble of the same gas volume.

Figure 3

(a) Resonance frequency as a function of the number of open faces (orange stands for open faces). Lines express the full theory (continuous line) and Minnaert approximation (dashes), $2a=2.1\mathrm{mm}$. Crosses, 3D simulation. (b) Pressure scattered by a cubic bubble ($2a=1.4\mathrm{mm}$): here an instantaneous cross-section snapshot computed with a 3D finite-difference time-domain (FDTD) simulation in which the bubble oscillates at its natural resonance frequency after a wideband pulse excitation.

Figure 4

Surface modes at $f=760\mathrm{Hz}$, for a cube of inner size $2a=1.7\mathrm{mm}$. The red arrow shows the vibration of the reflection of the line of light. The image of a front measurement of the vibration amplitude performed on a single face is morphed and superimposed on the top face of the cube. It shows a standing wave with $3\times 3$ half wavelengths. Here the interfaces were slightly pushed inside when closing the watertight tank. See also the movie in the Supplemental Material showing the vibration mode [17].