Sound of effervescence

Capillary bubbles burst at a free surface following a rapid sequence of events occurring at different length scales and timescales: hole nucleation, fast retraction of the micron-thick liquid film in a few microseconds preluding the much slower overall collapse of the millimeter-sized bubble in a matter of milliseconds. Each of these steps is associated with unsteady fluid forces and accelerations, and therefore with sound radiation. In this experimental study we focus on the airborne sound generated during bubble bursting. Investigating the physical mechanism at the root of sound emission with the help of synchronized fast imaging and sound recordings, we quantitatively link the film retraction dynamics with the frequency content of the acoustic signal. We demonstrate that, contrary to a Minnaert resonance scenario, the frequency here drifts and increases, consistent with a Helmholtz-type resonance of the cavity being more and more opened as the thin film retracts. We propose as an extension a simple model based on a collection of drifting Helmholtz resonators capturing the main features of the fizzing sound of an effervescing beverage.

Figure 1

Simultaneous acquisition of the airborne sound emitted by champagne in a glass and of a movie of the liquid surface viewed from the top. (a) Picture of a glass of champagne showing the microphone placed a few centimeters above the glass edge. (b) Histogram of the bubble radii extracted from a picture of the liquid surface at time $t=960$ ms (average radius $<R>=327\mu \mathrm{m}$). (c) 1 s duration time history of the acoustic pressure high-pass filtered with a $50\mathrm{Hz}$ cutoff frequency and a sampling frequency of 500 kHz. (d) 5 ms duration part of the signal centered on a single emission event occurring at $t=961$ ms. (e–h) Top view pictures of the liquid surface acquired at 200 fps using a high-speed camera synchronized with the acoustic signal acquisition. Pictures (e, f) and the enlargement of their framed part (g, h) evidence the disappearance of one bubble inside the red frame between $t=960$ ms and $t=965$ ms. (i) Acoustic spectra of the 1 s duration acoustic recording (c) (black curve) and of its 5 ms duration enlargement (d) (green curve). The blue horizontal dashed line indicates the noise level, and the vertical dashed line indicates the limit between the audible domain and the ultrasound domain.

Figure 2

Sketch of a floating bubble at the initial stage of its collapse corresponding to the retraction of the liquid film: (a) profile view and (b) top view. (c) Sequence of pictures of a floating air bubble with $R=1.9$ mm equivalent radius viewed from the side above the free water surface, showing the retraction of the water film separating the bubble from the atmosphere. The black arrow indicates the position on the rim of the retracting film. (d) Sequence of pairs of pictures of a $R=1.1$ mm collapsing air bubble viewed from the side above and below the free water surface using two synchronized high-speed cameras. (e) Synchronized airborne acoustic pressure signal emitted during the collapse and detected using a microphone located 20 mm right above the bubble. (f) Fourier spectrum of pressure signal (dashed line: noise level) (normalized by the maximum of the spectrum).

Figure 3

(a) Pair of pictures from the side below (left) and above (right) the surface of a $R=1.9$ mm air bubble floating at a water free surface. ${\mathrm{\Sigma }}_1$: liquid film separating the inner air from the atmosphere. ${\mathrm{\Sigma }}_2$: water-air interface below the free surface. ${\mathrm{\Sigma }}_3$: free liquid surface. $a$ (respectively $b$): semiminor (respectively semimajor) axis of the elliptic best fit of the bottom of the bubble profile. Red (respectively blue, green) dashed curve: shape of ${\mathrm{\Sigma }}_1$ (respectively ${\mathrm{\Sigma }}_2$, ${\mathrm{\Sigma }}_3$) solution of the Young-Laplace problem (see text). (b) Measured $r_c$ versus their theoretical prediction. The solid line corresponds to the perfect agreement between experiment and theory. Inset: theoretically predicted variation of $r_c$ versus $R$ for pure water (black solid curve) and SDS solution (green solid curve) together with Eq. (5) for pure water (black dashed curve) and SDS solution (green dashed curve).

Figure 4

(a) Measured film opening duration $t_b$ and measured duration ${\tau }_b$ of the frequency chirp of the acoustic emission versus $R$. Black line: theoretical prediction (8) for $t_b$ corresponding to the average value $\langle h\rangle =2.7\mu \text{m}$ of the retracting liquid films. Inset: measured duration ${\tau }_b$ of the frequency chirp of the acoustic emission versus measured film opening duration $t_b$. (b) Time-dependent distance $x$ traveled since film breaking by the rim (up to $t_b$), then by the ${\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_3$ junction (from $t_b$) for three different values of $R$. Lines are best linear fits. Inset: thickness $h$ of the retracting liquid films for the water bubbles deduced from Eq. (6) versus $R$. Dashed line: average value of $h$ for water bubbles.

Figure 5

(a) Airborne acoustic pressure signal emitted during the collapse of a $R=1.1$ mm air bubble floating at a water free surface. Red curve: absolute value of the Hilbert transform of the signal. (b) Sketch of the opening bubble. (c) Spectrogram of the signal shown in (a). Dots: maxima of the instantaneous spectra. The vertical dashed line indicates the time $t_b$ defined by Eq. (8) at which the film retraction is completed. Black solid curve: prediction of $f_H$ based on Eqs. (11), (13), (12), and (8).

Figure 6

(a) Time-dependent peak frequency $f_0\left(t\right)$ of the instantaneous spectra of the airborne acoustic emission of collapsing bubbles for three different values of the bubble equivalent radius $R$. (b) Measured duration $\delta t$ of the acoustic signal and duration of the bubble collapse $\mathrm{\Delta }t$. Solid line: scaling law for $\mathrm{\Delta }t$ from [9]. (c) Prefactor of the scaling law $f_H={\beta }_{\text{exp}}{t}^{1/2}$ as a function of its theoretical prediction ${\beta }_{\text{theo}}$ using by Eq. (12). The solid line corresponds to the perfect agreement between experiment and theory. Inset: Variation of the section $S^{\prime }$, defined in Eq. (10), over the variation of area $S$ defined for a circular hole. (d) Variation versus $R$ of the peak frequency $f_0\left(t_b\right)$ of the instantaneous spectra of the airborne acoustic emission of collapsing bubbles at the instant $t_b$ at which the liquid film opening is ending. Black line (respectively dashed line): theoretical prediction (13) with $\varepsilon =8/\left(3\pi \right)$ [respectively $\varepsilon =16/\left(3\pi \right)]$. Inset: acoustic wavelength $\lambda$ versus $R$. Solid line: law $\lambda =10R$.

Figure 7

(a) Measured quality factor $Q$ versus $R$. Solid line (respectively dashed line): theoretical prediction Eq. (16) computed using $\delta =8/\left(3\pi \right)$ (respectively $\delta =16/\left(3\pi \right)$. Inset: theoretical ratio of the characteristic thermal length to hole radius as function of bubble radius. (b) Dimensionless radiated acoustic energy $E/\left(\sigma R^2\right)$ versus $R$.

Figure 8

Black solid curve: acoustic spectrum of the 1 s duration airborne acoustic recording above a glass of champagne. Green bars: histogram of the resonant frequency of acoustic emission of the floating bubbles whose histogram of radii is shown. Figure 1 predicted using Helmholtz's model of open resonator Eq. (13). Gray bars: histogram of the resonant frequency of acoustic emission of the floating bubbles predicted using Minnaert's model of open resonator (18).