Nonspherical modes nondegeneracy of a tethered bubble

When excited at sufficiently high acoustic pressures, a wall-attached bubble may exhibit asymmetric nonspherical modes. These vibration modes can be decomposed over the set of spherical harmonics $Y_{nm}(\theta ,\varphi )$ for a degree $n$ and order $m$. We experimentally capture the time-resolved dynamics of asymmetric bubble oscillations in a top-view configuration. A spatiotemporal modal analysis is performed and allowed recovering the set of zonal ($m=0$), tesseral ($0<m<n$), and sectoral ($m=n$) spherical harmonics that develop at the bubble interface. The analysis of the surface instability thresholds reveals that the frequencies of asymmetric modes differ from the standard Lamb spectrum. In addition, the nondegeneracy of asymmetric modes for a given degree $n$ is evidenced by noncompletely overlapping resonance bands. Finally, the coexistence between zonal and sectoral modes is analyzed through their modal interaction, amplitude interplay and relation of phase, as well as their geometric compatibility.

Figure 1

(a) Schematic representation of the experimental setup. (b) Side view of the microbubble tethered at the tank's bottom, making a contact angle $\alpha$ with the substrate in absence of ultrasound.

Figure 2

Snapshot series of an experimental (camera top-view) bubble, with its associated theoretical spherical harmonics (top and side views), oscillating on a sectoral mode $Y_{66}$ (a), on a zonal mode $Y_{60}$ (b) and on a tesseral mode $Y_{64}$ (c). The elapsed time between two consecutive snapshots is $14.9\mu \mathrm{s}$.

Figure 3

Instability thresholds of sectoral harmonics ($m=n$) as a function of the bubble equilibrium radius. Experimental wall-attached bubbles (geometric markers) are compared with free bubble theory (continuous lines), according to Refs. [10, 11]. Depending on the bubble size, the frame dimension varies following Table 1.

Figure 4

Change in spectral signature due to nodal lines parity. Spatiotemporal Fourier decomposition of the bubble contour $r(\varphi ,t)$ (normalized amplitude), processed for numerical (in red) and experimental (in black) cases of spherical harmonics $Y_{64}$ (a) and $Y_{74}$ (b). The discrepancy between numerical and experimental cases comes from the ombroscopic-induced artefact that affects the optical image but not the numerical projection. This is amplified in the $n-m$ odd situation due to the existence of a nodal line along the equatorial plane.

Figure 5

Modal decomposition of a top-view bubble contour including the $Y_{60}, Y_{64}$, and $Y_{66}$ modes triggered at time $(t_0,t_4,t_6)$ superimposed to a breathing mode $Y_{00}$. Comparison of the coefficients $c_{nm}^{*}\left(t\right)$ with the theoretical ones $a_{nm}\left(t\right)$ in case of (a) $(t_0,t_4,t_6)=(4,8,10)\mathrm{ms}$ and (b) $(t_0,t_4,t_6)=(8,10,4)\mathrm{ms}$. (c) Evidence of amplitude transfer between the subharmonic $c_{64}^{*}(\frac{f_0}{2},t)$ and fundamental $c_{64}^{*}(f_0,t)$ coefficients of the $Y_{64}$ tesseral mode.

Figure 6

Evolution of the modal coefficients $c_{nm}^{*}\left(t\right)$ resulting from a top-view contour in the case of a bubble (a) with equilibrium radius $R_0=133\mu \mathrm{m}$ oscillating on the set of $n=6$ spherical harmonics and (b) with equilibrium radius $R_0=83\mu \mathrm{m}$ oscillating on the set of $n=4$ spherical harmonics. Only the predominant nonspherical modes are displayed. Snapshot series illustrate significant changes in bubble shape.

Figure 7

Instability pressure thresholds of the first triggered nonspherical shape modes of degree $n=3$ (a), $n=4$ (b), and $n=7$ (c). Parabolic-like graphical fits (dashed lines) give an overview of the nondegenerate instability threshold curves. The continuous gray lines are the theoretical instability curves of free axisymmetric bubbles [10, 11], centered on the respective modal resonances.

Figure 8

Coexistence of sectoral and zonal modes is exposed relying on their modal interplay, phase relation and symmetry compatibility for the cases of modes of odd degree $n=3$ (a) and $n=5$ (b) and even degree $n=4$ (c) and $n=6$ (d). For each case (x), the information is structured as follows. (x1) Evolution of the coefficients $c_{nm}^{*}\left(t\right)$ for the radial, zonal and sectoral modes along a complete modulation period; (x2) evolution of the oscillatory behavior of the coefficients $c_{nm}\left(t\right)$ for the same modes, taken at particular times corresponding to coloured areas in (x1) and refolded over two acoustic periods. vertical purple lines correspond to the instants where occur the minima of radial oscillation; and (x3) side-view schematics of the extrema of bubble deformations for the selected shape modes.