Sound emission from oscillating bubbles trapped by the collapse of drop-impact craters

TL;DR

This work resolves the time-resolved mechanism of underwater sound produced by rain-analog drop impacts by tracking the pinch-off of the crater bottom dimple and the ensuing oscillations of the entrapped bubble with ultra-high-speed imaging up to $5\times10^6$ fps, synchronized to a submerged hydrophone. Across multiple dimple shapes and impact conditions, the acoustic amplitude scales with the maximum compression of the pinched-off bubble (up to ~50%), and smaller bubbles produce higher resonance frequencies that closely follow the Minnaert relation $f=\frac{1}{2\pi a_0}\sqrt{\frac{3\gamma P_0}{\rho_0}}$ for the entrapped gas, though non-spherical geometry and coupling with a second tiny bubble modulate the spectrum. In singular jetting cases where no bubble pinches off, the signal is weaker and broader, and the dominant frequencies arise from dimple retraction rather than a resonant bubble, with a forcing frequency related to the retraction time scale $T_R\sim L/V$. The tiny initial bubble acts as a phase-delayed, forced oscillator that can create double crests in the acoustic signal, highlighting a complex interplay between macro- and micro-bubble dynamics. Overall, the results illuminate the physical mechanisms behind rain-induced underwater acoustics and clarify how crater geometry, pinch-off dynamics, and bubble coupling shape the emitted sound.

Abstract

When a drop impacts a deep pool, it forms a crater which subsequently rebounds. Under certain conditions, a dimple forms at the crater bottom, which pinches off to entrap a small bubble. The oscillation of this entrapped bubble is the primary source of the underwater sound produced by rain. We use simultaneous ultra-high-speed video imaging and synchronized acoustic recording, with an immersed hydrophone, to investigate the details of the sound formation, over a range of impact Weber numbers and different dimple shapes. With frame-rates as high as 5 million fps, we can track the shape evolution of the pinched off dimple-bubble, which experiences large volumetric compression, by as much as 50%. The subsequent volume oscillations are consistent with the observed $\simeq 125$ Pa acoustic pressure amplitude, for the strongest compression. For our configuration the sound amplitude increases for smaller bubbles pinched off from the dimple. The acoustic forcing mechanism is therefore the inertial focusing of the momentum of the liquid outside the dimple, as it pinches off. The acoustic frequency agrees well with the Minnaert theory for freely oscillating spherical bubbles, of the same size. The finer details of the acoustic signal reveal an interplay between the larger dimple bubble and the tiny bubble entrapped during the initial contact between the drop and pool. For the singular dimple where no bubble is pinched off, the sound generation has a broader range of frequencies, with the tiny bubble oscillating at $\sim 100$ kHz, after being deformed by the rapid vertical retraction of the dimple below the singular jet.

Figures (12)

• Figure 1: (a) Sketch of the experimental set-up, with two high-speed video cameras viewing the dynamics from perpendicular directions. One camera captures the dimple dynamics inside the pool (within red square), while the other views the drop above the pool level to measure the diameter and impact velocity onto its surface. (b) The full long-duration acoustic signature (blue curve) of a drop impact on liquid pool (corresponding to Figure \ref{['Fig5_2047_5.0Mfps']}), which includes the initial low-amplitude impact pulse, at $t=-25$ ms, caused by the first contact of the drop with the pool surface. It also shows the larger-amplitude oscillatory signal from the entrapped bubble. The black baseline represents Kirana high-speed camera synchronization signal (Voltage on the right axis). (c) Zoomed-in details of the 200 ns framing pulses from the Kirana acquisition at 5 Mfps.
• Figure 2: Experimental range in the Fr-We parameter space, for water/glycerin drops impacting a pool of the same liquid. The two solid curves are the bounds of the regular bubble entrapment regime, where the bottom dimple is pinched off from the rebounding crater, as measured by Pumphrey1990 and fitted by Oguz1990. The snapshots of Mesler entrainment and regular bubble entrapment are reproduced from Thoroddsen2012Thoroddsen2018. The symbols correspond to four typical dimple shapes, with the same names as used in Thoroddsen2018. Specifically these are: triangular dimple pinch-off ($\triangle$), teardrop pinch-off ($\triangle$), small bubble pinch-off ($*$) and singular jetting ($\ovoid$). The specific shapes for each pinched-off bubble are shown in the next figure.
• Figure 3: Typical acoustic signals (middle column) and time-frequency wavelet transforms (right side) generated by four different dimple collapse shapes, resulting from different impact conditions, for the same liquid in the drop and pool. These four cases correspond to the main dimple shapes observed in Thoroddsen2018 over a narrow range of $We$: Triangular dimple pinch-off ($\triangle$) $D=3.63$ mm, $U=1.41$ m/s, $Fr=56$, $We= 122$; Teardrop pinch-off ($\triangle$) $D=3.63$ mm, $U=1.34$ m/s, $Fr=51$, $We= 109$; Small bubble pinched off ($*$) $D=3.55$ mm, $U=1.36$ m/s, $Fr=53$, $We= 110$ and singular jet: ($\ovoid$) $D=3.54$ mm, $U=1.38$ m/s, $Fr=55$, $We= 113$. The scale bars are 100 $\mu$m long. The small bubble visible below the dimples come from the central air-disk entrapped under the drop at its first contact with the pool, see Thoroddsen2003.
• Figure 4: Comparison of high-speed images and acoustical data for the triangular dimple pinch-off ($\triangle$), for $U$ = 1.41 m/s and $D$ = 3.63 mm, giving $Re$= 802, $Fr$= 56, $We$ = 122. (a) Typical frames showing the triangular dimple pinch-off shape and subsequent bubble shape oscillations, taken from a video sequence recorded at 1 Mfps. The circled numbers marked with correspond to the acoustic signal is (b): ① shows the dimple pinch-off time; ② The conical neck pulls back and the bubble compresses and reach the minimum volume; ③ bubble expanded to reach maximum volume; The bubble finishes another compression-expansion cycle, then repeats it in (④⑤⑥). (b) The acoustical signal (left axis) following the dimple pinch-off, including the measured bubble volumes (right axis) from the high-speed video with 1 $\mu$s interframe time. The normalized volume of the bubble pinched off from the dimple marked with a red line and the tiny bubble from the initial drop impact marked with the dark line. (c) The evolution of the bubble shape edge used for entrapped bubble volume calculation with revolution. (d) Time-frequency wavelet transform of the sound from dimple pinch-off ($\triangle$). (e) The diameter comparison between horizontal and vertical radius, effective radius $r$, $2r= \sqrt[3]{a^{2}b}$, volume radius $R$, and Minnaert prediction. (f) The damped acoustic signal evolution for extended 2 ms duration.
• Figure 5: Combined high-speed photography and acoustical data showing the process of teardrop dimple pinch off, for $U$ = 1.34 m/s and $D$ = 3.63 mm, giving $Re$= 760, $Fr$= 51, $We$ = 109. (a) Typical frames of dimple pinch off and bubble oscillation taken from a video sequence recorded at 5M fps, corresponding to the circled numbers marked with in the acoustic signal in (b); The scale bar is 50 $\mu$m long. (b) The acoustical signal (left axis) during the dimple pinch off, with measured bubble volumes (right axis) from high speed camera video. The normalized volume of bubbles, entrapped bubble pinched off from the dimple marked with red line and tiny bubble from the initial drop impact marked with dark line, are tracked from the dimple pinch off. The camera capture regime is also noted with bubble compression and expansion area at the top axis with 200 ns frame time. (c) Time-frequency wavelet transform of the acoustic signal from teardrop pinch-off ($\triangle$). (d) The diameters comparison with Minnaert prediction. (e) The acoustic signal evolution for 2 ms duration.