## 3 Results and Discussion  
Upon the activation of the loudspeaker (120 Hz), the flame then started a periodic oscillation centered on the original position (Figure 3a). The flame envelope basically maintained the original shape at a sound pressure of 8 Pa. With an increase in the speaker power, taking \(p_{\mathrm{e}}=13\) Pa, for instance, the vibration amplitude (\(\xi_{t}\)) was increased, and the conical flame structure was destroyed (Figure 3b). [PERSON] et al. [17] attributed this phenomenon to the effect of the acoustic mean flow (also described as acoustic streaming). However, the acoustic mean flow could not explain why the deviation direction was  
Figure 1: Scheme of experimental setup. Figure 2: Procedures of the image processing to track the instantaneous flame: (**a**) single-frame image; (**b**) grayscale image; (**c**) binary map; (**d**) binary map. sometimes opposite to the direction of the sound wave. There must be something unnoticed awaiting disclosure. To quantitatively evaluate the vibration, we measured the position of the flame's right edge (\(x_{\text{max}}\)) under the same sound power, as defined in Figure 3a, at \(t\) = 3.9 ms, and the results are listed in Figure 4a. The vibration frequency was equal to that of the sound wave (Figure 4), which is consistent with the previous study [19]. The vibrational amplitude (\(\xi_{t}\)) of the flame decreased with the increase in sound frequency but increased linearly with sound pressure (\(p_{e}\)), as shown in Figure 5.  
Figure 4: Frequency\(f_{t}\) of flame oscillation induced by acoustic disturbance of different frequencies\(f_{\text{s}}\). Figure 3: Periodic oscillation of the flame aroused by the speaker with local sound (120 Hz) pressure: (a) \(p_{e}\) = 8 Pa and (b) \(p_{e}\) = 13 Pa. The coherence of the frequencies of the sound and the flame indicated that it is reasonable for the sound waves to produce flame oscillation. However, although qualitatively possible, the flame amplitude \(\xi_{\rm f}\) was quantitatively overlarge compared to that of the sound particle (\(\xi_{\rm a}\)), i.e., air molecule, which could be estimated by the following formula:  
\[\xi_{\rm a}=\frac{p}{2\pi fz} \tag{1}\]  
where \(\xi_{\rm a}\) is the amplitude of the sound particle, \(p=\sqrt{2}p_{\rm e}\) is the averaged sound pressure, and \(z=233\) Pa \(\cdot\) s is the specific acoustic impedance of hot gas at 1000 K [20] (estimated temperature of butane flame). As shown in Figure 6, the corresponding theoretical amplitude of the air molecule was very small, about one thirtieth of the amplitude of flame fluctuation, which was difficult to observe in experiments; this proved that the sound pressure was not the reason for flame fluctuation. As shown in Figure 7, the amplitude of the speaker membrane \(\xi_{\rm mm}\) was almost equal to \(\xi_{\rm f}\). This indicates that the oscillation of the flame here was driven by the vibrating membrane of the speaker via the effect of macroscopic flow which, usually named \"induced wind\", was of the same frequency as the sound wave. Figure 5: Flame displacement amplitude \(\xi_{\rm f}\) in sound fields of different frequencies and intensities. Figure 6: Comparison of \(\xi_{\rm f}\) and \(\xi_{\rm a}\) under the same sound pressure, and the reduction trend with increasing distance \(d\). As shown previously in Figure 3b, when the induced wind was strong enough, the flame would be quenched. Although this kind of extinction was caused by the induced wind, it was still called acoustic extinction here as the induced wind was closely related to the acoustic membrane, which was similar to the descriptions in other previous studies [5; 7; 11; 15; 21]. It does not mean that this description is reasonable and accurate. It is noted that the acoustic extinction mechanism of diffusion flames has not been unrevealed.