# Prediction of Acoustic Energy Radiated by Bubble Produced by Raindrops

Authors: Qi Li; Shu Liu; Dajing Shang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Underwater noise produced by rainfall is an important part of underwater ambient noise. The bubbles produced by raindrops are the main noise source of underwater noise. Generally, the sound pressure signal of individual bubbles is easily contaminated by tank reverberation, hydrodynamic flow, and laboratory electrical noise. In order to solve this problem, this study proposes a method for calculating the acoustic energy of the bubble produced by a raindrop when the latter falls onto a plane water surface. For this purpose, a series of experiments was conducted in a 15 m × 9 m × 6 m reverberation tank filled with tap water. The bubble produced by a raindrop behaves as a simple exponentially damped sinusoidal oscillator. Based on the dipole radiation pattern, a formula was derived to predict the sound energy of these bubbles. The damping coefficient of the bubble formed by raindrops is found to differ appreciably from the empirical value of the bubble formed by other mechanisms. The resonance frequency of the bubbles is found to decrease with time. It is due to the rapid increase in the distance between the bubble and the interface. Then, the formula is optimized by using these two improved variables. The experimental results agree well with the theoretical derivation.


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## Body

## 1. Introduction

Wind-generated and rain-generated ambient sound from the ocean surface represents the background baseline of ocean noise. Some studies have shown that when present, the sound produced by rainfall dominates the underwater sound field [1–3]. Two distinctive acoustic fields associated with rainfall have been described. The first, associated with light rain or drizzle, is characterized by a broad spectral peak near 15 kHz [2]. The second, associated with heavy rainfall, is nearly white (from 4–20 kHz) and is highly correlated with the rainfall rate [3]. The study of rainfall noise is important not only for ocean physicists studying (i) rainfall and wind over the ocean [4] and (ii) the usage of sonars [5–7] but also for biologists studying how anthropogenically generated sound impacts marine mammals [8]. Another important application of rainfall (bubble) noise is in the field of biogeochemistry, specifically in estimating the rate of exchange of gas between air and water [9] or sediments and water [10].The bubbles produced by raindrops are the main noise source of underwater noise due to rain. The dynamics of underwater bubbles have long been of considerable interest because of their importance in various physical and engineering problems, including cavitation on ship propellers [11–13], underwater explosions [14–16], and ultrasonic cleaning [17–19]. Although there has been some work on underwater bubble swarms [20, 21], there has been much less on the acoustic energy generated by bubbles produced by rainfall.The earliest reference to bubbles as sound sources was made by Leonardo Da Vinci, who described the sound of running water already in the 1500 s. Bragg [22] has attributed the murmuring of a brook and the “plunk” of droplets falling into water to entrained air bubbles. Minnaert [23] has since shown that the sound generated by gas bubbles in liquids is associated with simple volume pulsations of the bubble with no change of its shape. Minnaert then derived the resonance frequency of the bubble, which behaves as a simple exponentially damped sinusoidal oscillator with one degree of freedom. The number of cycles required for the amplitude of motion to reduce to e−π of its original value is known as the Q factor of the bubble system. The total damping coefficient δ is now defined as the reciprocal Q of the bubble system. Devin [24] has since shown that the damping may be explained by losses originating from three processes, namely, (i) thermal damping due to the thermal conduction between the gas in the bubble and the surrounding liquid, (ii) sound-radiation damping, and (iii) viscous damping due to viscous forces at the gas-liquid interface. The total damping is the sum of these three processes.Several attempts have been made to show what happens when a raindrop strikes a water surface. Worthington [25] made the first photographic study, and Franz [26], Laville [27], and Buckingham [28] identified two main sources of underwater sound, namely, (i) the initial impact sound as a sharp pulse and (ii) the bubble sound as a decaying sinusoid. Franz also observed that the impact sound and bubble sound seem to behave as simple dipole sources near the free surface. Furthermore, laboratory experiments have been conducted to describe the sound generated by a single raindrop [29–32]. These experiments identified four mechanisms for raindrop-generated sound, namely, (i) the initial impact, (ii) a bubble trapped by a small raindrop at the bottom of the impact crater (type I), (iii) a bubble trapped underwater by a turbulent jet created by the splash canopy of a large raindrop (type II), and (iv) a bubble created during the secondary splashes of drop aerosols thrown up during the initial raindrop impact (type III). In the case of small raindrops associated with light rain and drizzle, a bubble-entrapment mechanism has been described, namely, the pinch-off of the tip of the splash crater by surface tension and gravity [29, 33, 34]. The bubbles created by this mechanism are designated as being of type I [31]. The process whereby a type-I bubble forms is shown in Figure 1(a). The raindrop hits the water surface at time t = 0, and the bubble is produced at t = 21.8 ms. In heavier rainfall, larger raindrops are present. The process whereby a type-II bubble forms is shown in Figure 1(b). Frame 1 shows the drop immediately before impact. In frames 2–5, a canopy is formed when a spray of droplets is ejected by the upward-moving water mass. At the apex of the canopy, water convergence generates upward- and downward-moving turbulent jets of water. In frame 10, a bubble is trapped underwater as the downward-moving jet penetrates the crater bottom. Bubbles formed by this mechanism are defined as being of type II [31–33].Figure 1
Physical sequences of raindrop splashes: (a) small raindrop splash (redrawn from Figure 9 in [34]); (b) large raindrop splash (redrawn from Figure 3 in [32]).
(a)(b)Medwin and Beaky [35] found that the resonance frequencies of all those bubbles decrease during the short oscillation times of the latter, and they tried to explain this change. One possibility is that the bubble moves away from the surface rapidly, with subsequent displacement due to an increase in the dipole axis. Another possibility is that the bubble is initially ellipsoidal and that its frequency decreases as it becomes spherical. However, to date, no studies have addressed how the bubble could move away from the surface so quickly in less than 2 ms (the bubbles produced by raindrops usually last no more than 2 ms).There has been a lot of study on the physical properties of bubbles, but little research has been done on the acoustic energy radiated by a single bubble produced by a raindrop. The noise radiated by bubbles is a transient signal, which is easily contaminated by tank reverberation, hydrodynamic flow, and laboratory electrical noise in the water tank.In order to eliminate the contamination, this study proposes a method for predicting the acoustic energy of the bubble produced by a raindrop. Using this method, the sound energy can be accurately calculated by measuring the direct sound pressure amplitude and initial resonance frequency of the bubble.


## 2. Methods

### 2.1. Theory

The bubble is assumed to be spherical (or sufficiently close to spherical) so that, away from the free surface of the water, the resonance frequency of the bubble is given by the following equation [36]:(1)f0=12πa3γP0ρ,where a is the bubble radius, P0 is the ambient pressure surrounding the bubble, ρ=1000kg/m3 is the density of the surrounding water, and γ is the ratio of the specific heat of air (γ = 1.4).A near-surface oscillating bubble forms a phase-reversed image in the smooth, reflecting water-air interface, and thereby behaves as a dipole. The far-field theoretical radiation pressure of the bubble acting as a dipole [35] is proportional to cosθ, where θ is the angle perpendicular to the water surface. This sound-radiation model of bubbles is shown in Figure 2, where R is the distance from the point of impact to the hydrophone, R2 is the distance from the image to the hydrophone, and R1 is the distance from the bubble to the hydrophone. In general, the pool used in the experiment can be considered a reverberation pool. When the direct sound received by the measuring point is equal to the reflected sound, the distance between the measuring point and the sound source is defined as the reverberation radius rn. When the distance between the measuring point and the sound source is less than rn, the received signal is dominated by the direct sound. When the distance is greater than rn, the received signal is dominated by the reflected sound.Figure 2
Sound-radiation model of a bubble near the surface.IfR is much less than the reverberation radius rn, the reflected sound can be ignored. Then, the energy radiated by a surface dipole [32] is given by the following equation:(2)E=2πR23ρc∫pax2tdt,where R is the distance from the point of impact to the hydrophone, c = 1500 m/s is the speed of sound in water, and paxt is the axial acoustic pressure at range R. The acoustic energy can be calculated using equation (2) once paxt has been measured. The acoustic pressure signal is a transient signal, which is easily contaminated by tank reverberation, hydrodynamic flow, and laboratory electrical noise in the water tank. To eliminate the contamination, the acoustic energy of a bubble can be predicted as follows.The acoustic pressure signal is described by a damped sinusoid, namely,(3)paxt=p0sin2πf0te−2πf0δt,where p0 is the peak pressure at R0 = 1 m (the distance from the point of impact to the hydrophone) on axis (based on the dipole radiation pattern) and δ is the damping coefficient for an air bubble in water [36] and is approximated by(4)δ≅k⋅f010001/3.This is an empirical formula for the damping coefficient of bubbles produced by the bubble generator. Also,k=0.025 when the resonance frequency of the bubble is between 1 and 100 kHz.Substituting equation (3) into equation (2) gives(5)E=2πR02p023ρc∫1−cos4πf0t2e−4πf0δtdt=2πR02p023ρc∫1−ℜe−4jπf0t2e−4πf0δtdt=R02p0212ρcf0δδ2+1,which is the equation used to predict the energy radiated by a bubble produced by a raindrop. Here, because the damping coefficient is much less than unity according to equation (4), we have that δ2+1≈1, and then, the formula can be simplified as follows:(6)E=R02p0212ρcf0δ.


### 2.2. Experiment

To verify the accuracy of equation (6) for predicting the acoustic energy radiated by a bubble produced by a raindrop, we conducted a series of experiments in a 15 m × 9 m × 6 m reverberation pool filled with tap water up to its maximum height (6 m). The temperature of water is 20 degree Celsius, and the temperature of air is 18.5 degree Celsius. Figure 3 shows the system used to measure the sound signal generated by raindrops. The four acoustically distinct ranges of the drop diameter D are defined as (i) minuscule drops (D ≤ 0.8 mm), (ii) small drops (0.8 mm < D ≤ 1.1 mm), (iii) midsize drops (1.1 mm < D ≤ 2.2 mm), and (iv) large drops (D > 2.2 mm). We used a standard intravenous drip bottle with needles of different caliber to generate drops of tap water of five different diameters, namely, 0.7, 0.9, 1.5, 2.5, and 3.9 mm. The height of the drop bottle is five meters. At this height, each drop struck the water surface at the terminal velocity of the raindrops in natural rainfall. Various testing conditions are listed in Table 1.Figure 3
A system used to measure sound signal generated by raindrops.Table 1
Testing conditions for bubbles produced by raindrops.
No.D (mm)Drop typeNumber of testsBubble type and probability10.7Minuscule50No bubble20.9Small62Type I (100%)31.5Midsize50No bubble42.5Large92Type II (28.3%), type III (34.8%)53.9Large94Type II (57.4%), type III (21.2%)No. is the serial number.D is the raindrop diameter.We suspended a hydrophone (type 8103; Brüel and Kjær, Denmark) 20 cm below the impact point. This distance is much less than the pool’s reverberation radiusrn=1.7m measured by Dajing et al. [37]. At this distance, the effect of reflected sound can be avoided. The hydrophone signal was sent to a computer using a pulse collector (type 3560E; Brüel and Kjær). Meanwhile, high-speed video photography was used to observe the underwater movement of the bubble produced by the drop. The camera (HERO4; GoPro, USA) operated at 240 frames per second (fps) and was placed underwater near the impact point along with a ruler to measure the distance of the bubble from the surface. For each bubble, the result of this procedure was a single film comprising a sequence of photographs of the bubble in the water.


## 2.1. Theory

The bubble is assumed to be spherical (or sufficiently close to spherical) so that, away from the free surface of the water, the resonance frequency of the bubble is given by the following equation [36]:(1)f0=12πa3γP0ρ,where a is the bubble radius, P0 is the ambient pressure surrounding the bubble, ρ=1000kg/m3 is the density of the surrounding water, and γ is the ratio of the specific heat of air (γ = 1.4).A near-surface oscillating bubble forms a phase-reversed image in the smooth, reflecting water-air interface, and thereby behaves as a dipole. The far-field theoretical radiation pressure of the bubble acting as a dipole [35] is proportional to cosθ, where θ is the angle perpendicular to the water surface. This sound-radiation model of bubbles is shown in Figure 2, where R is the distance from the point of impact to the hydrophone, R2 is the distance from the image to the hydrophone, and R1 is the distance from the bubble to the hydrophone. In general, the pool used in the experiment can be considered a reverberation pool. When the direct sound received by the measuring point is equal to the reflected sound, the distance between the measuring point and the sound source is defined as the reverberation radius rn. When the distance between the measuring point and the sound source is less than rn, the received signal is dominated by the direct sound. When the distance is greater than rn, the received signal is dominated by the reflected sound.Figure 2
Sound-radiation model of a bubble near the surface.IfR is much less than the reverberation radius rn, the reflected sound can be ignored. Then, the energy radiated by a surface dipole [32] is given by the following equation:(2)E=2πR23ρc∫pax2tdt,where R is the distance from the point of impact to the hydrophone, c = 1500 m/s is the speed of sound in water, and paxt is the axial acoustic pressure at range R. The acoustic energy can be calculated using equation (2) once paxt has been measured. The acoustic pressure signal is a transient signal, which is easily contaminated by tank reverberation, hydrodynamic flow, and laboratory electrical noise in the water tank. To eliminate the contamination, the acoustic energy of a bubble can be predicted as follows.The acoustic pressure signal is described by a damped sinusoid, namely,(3)paxt=p0sin2πf0te−2πf0δt,where p0 is the peak pressure at R0 = 1 m (the distance from the point of impact to the hydrophone) on axis (based on the dipole radiation pattern) and δ is the damping coefficient for an air bubble in water [36] and is approximated by(4)δ≅k⋅f010001/3.This is an empirical formula for the damping coefficient of bubbles produced by the bubble generator. Also,k=0.025 when the resonance frequency of the bubble is between 1 and 100 kHz.Substituting equation (3) into equation (2) gives(5)E=2πR02p023ρc∫1−cos4πf0t2e−4πf0δtdt=2πR02p023ρc∫1−ℜe−4jπf0t2e−4πf0δtdt=R02p0212ρcf0δδ2+1,which is the equation used to predict the energy radiated by a bubble produced by a raindrop. Here, because the damping coefficient is much less than unity according to equation (4), we have that δ2+1≈1, and then, the formula can be simplified as follows:(6)E=R02p0212ρcf0δ.


## 2.2. Experiment

To verify the accuracy of equation (6) for predicting the acoustic energy radiated by a bubble produced by a raindrop, we conducted a series of experiments in a 15 m × 9 m × 6 m reverberation pool filled with tap water up to its maximum height (6 m). The temperature of water is 20 degree Celsius, and the temperature of air is 18.5 degree Celsius. Figure 3 shows the system used to measure the sound signal generated by raindrops. The four acoustically distinct ranges of the drop diameter D are defined as (i) minuscule drops (D ≤ 0.8 mm), (ii) small drops (0.8 mm < D ≤ 1.1 mm), (iii) midsize drops (1.1 mm < D ≤ 2.2 mm), and (iv) large drops (D > 2.2 mm). We used a standard intravenous drip bottle with needles of different caliber to generate drops of tap water of five different diameters, namely, 0.7, 0.9, 1.5, 2.5, and 3.9 mm. The height of the drop bottle is five meters. At this height, each drop struck the water surface at the terminal velocity of the raindrops in natural rainfall. Various testing conditions are listed in Table 1.Figure 3
A system used to measure sound signal generated by raindrops.Table 1
Testing conditions for bubbles produced by raindrops.
No.D (mm)Drop typeNumber of testsBubble type and probability10.7Minuscule50No bubble20.9Small62Type I (100%)31.5Midsize50No bubble42.5Large92Type II (28.3%), type III (34.8%)53.9Large94Type II (57.4%), type III (21.2%)No. is the serial number.D is the raindrop diameter.We suspended a hydrophone (type 8103; Brüel and Kjær, Denmark) 20 cm below the impact point. This distance is much less than the pool’s reverberation radiusrn=1.7m measured by Dajing et al. [37]. At this distance, the effect of reflected sound can be avoided. The hydrophone signal was sent to a computer using a pulse collector (type 3560E; Brüel and Kjær). Meanwhile, high-speed video photography was used to observe the underwater movement of the bubble produced by the drop. The camera (HERO4; GoPro, USA) operated at 240 frames per second (fps) and was placed underwater near the impact point along with a ruler to measure the distance of the bubble from the surface. For each bubble, the result of this procedure was a single film comprising a sequence of photographs of the bubble in the water.


## 3. Results and Discussion

### 3.1. Acoustic Energy of the Bubbles

The 0.7 mm drop produced only weak impact-pulse noise. The 0.9 mm drop produced weak impact-pulse noise and a type-I damped bubble. The 1.5 mm drop produced only impact-pulse noise. The 2.5 mm and 3.9 mm drops both produced impact-pulse noise and type-II and type-III damped bubbles.Using the physical mechanisms for the three types of bubble produced by raindrops, we clarify here why the 1.5 mm drop produces only impact noise and no bubble, whereas the 0.9 mm and 2.5 mm drops both produce bubbles. Because the resultant force of gravity and surface tension is not strong enough to pinch-off the tip of the splash crater, there is no type-I bubble when a midsize raindrop strikes the water surface. Because a midsize raindrop does not have enough kinetic energy, no canopy is formed when the upward-moving water mass ejects a spray of droplets, and no bubble is trapped underwater. There is no type-II bubble when a midsize raindrop strikes the water surface, nor is there a type-III bubble because there are no secondary splashes of drop aerosols thrown up during the initial raindrop impact.The sound pressure signal collected 20 cm below the impact point is converted to sound pressure at 1 m on axis according to the dipole radiation pattern. Figure4(a) shows the acoustic signal produced by the raindrop of 3.9 mm in diameter. As can be seen in Figure 4(a), the signal generated by the initial pulse is measured at time t = 7.3 ms, and the signal generated by the bubble is measured at time t = 52.4 ms. The bubble is detected around 45.1 ms after impact. The sinusoidally damped bubble signal in Figure 4(a) is shown in Figure 4(b) with an appropriate amplitude amplification. Here, the echo is clearly distinguished from direct sound.Figure 4
Acoustic signal produced by a raindrop of 3.9 mm in diameter: (a) entire signal; (b) sinusoidally damped bubble signal.
(a)(b)The amplitudes of the impact-pulse noise and bubble noise generated by raindrops are given in Table2, where D is the raindrop diameter, pP is the amplitude of the impact-pulse noise, and pI, pII, and pIII are the amplitudes of type I, II, and III bubble noise, respectively. The amplitude of the impact-pulse noise generated by raindrops of the same diameter is clearly uniform and increases with the raindrop diameter. The average amplitude of the impact-pulse noise generated by raindrops of 2.5 mm and 3.9 mm in diameter is, respectively, 0.65 Pa and 2.02 Pa. The amplitude distribution of the type-I bubble noise is concentrated and uniform. The average amplitude of the type-I bubble noise is 0.36 Pa, whereas the amplitude distributions of the type-II and type-III bubble noise are dispersive and random. As the raindrop diameter increases, more bubbles with high sound pressure are produced.Table 2
Ranges of amplitude of sound pressure.
D (mm)pP (Pa)pI (Pa)pII (Pa)pIII (Pa)0.9—0.25–0.54——2.50.51–0.74—0.08–1.40.06–1.223.91.82–2.44—0.06–2.00.05–3.8The actual acoustic energy of the bubble was calculated using equation (2), while the theoretical acoustic energy of the bubble was calculated using equation (6). p0 is the peak pressure at the distance R0 = 1 m from the point of impact to the hydrophone.Figure5 compares the actual acoustic energy and the theoretical values for the three types of bubble (for the convenience of comparison, the actual sound energy data are sorted from smallest to largest). The theoretical acoustic energy of type-I bubble is less than the experimental value by about 60% overall. The root mean square (RMS) of the error is 1.50×10−11J. The theoretical acoustic energy of type-II and type-III bubbles is less than the experimental value by about 50% overall. The root mean square of the errors is 2.17×10−10J and 3.55×10−10J, respectively. According to equation (6), the deviation of acoustic energy may be caused by the deviation of damping coefficient δ or resonance frequency f0.Figure 5
Actual acoustic energy of bubble calculated using equation (2) (red) and theoretical acoustic energy of bubble calculated using equation (6) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


### 3.2. Influence of Damping Coefficient

The empirical damping coefficient in equation (4) is used to calculate the theoretical acoustic energy of bubble produced by the bubble generator. The actual damping coefficient δ of each bubble produced by each size of drop in the present experiments is fitted using equation (3). The actual damping coefficient and empirical damping coefficient are shown in Figure 6. The actual damping coefficients are different from the empirical values. The reason is that the bubbles produced by rainfall are different in mechanism from those produced by bubble generators. Bubbles generated by bubble generators are just beginning to be static in water. Then, the bubbles rise slowly under the influence of buoyancy. However, as soon as the bubbles are produced by raindrops, they will move down rapidly under the influence of inertia. The heat generated by the relative moving between the bubbles and water reduces the outward thermal radiation of the bubbles. The thermal damping of the bubbles produced by raindrops is less than that of other origin. Therefore, the actual damping coefficients are found to be less than the empirical values. Gillot et al. [38] have pointed out that the values of the damping time obtained by theoretical predictions are far from the values measured in water. This deviation of damping coefficient δ results in the deviation of acoustic energy.Figure 6
Comparison of the actual damping coefficient of bubbles produced by raindrops with the empirical value.Therefore, a new empirical formula for the damping coefficient of an air bubble produced by a raindrop is given by(7)δ=k1⋅f010001/3,where k1=0.012. This is the empirical formula for the damping coefficient of bubbles produced by raindrops. The empirical damping coefficient in equation (4) is twice the actual damping coefficient. The new curve fitted is shown in Figure 6.The theoretical acoustic energy of the bubble was recalculated using the modified damping coefficient. Figure7 compares the actual acoustic energy and the corrected theoretical values for the three types of bubble. In each case, the theoretical acoustic energy is very close to the experimental value, but the theoretical value is still less than the experimental value by about 10% overall. The root mean square of the errors is 4.87×10−12J, 1.19×10−10J, and 2.05×10−10J, respectively.Figure 7
Actual acoustic energy of bubble calculated using equation (2) (red) and correctional theoretical acoustic energy of bubble calculated using equation (6) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


### 3.3. Influence of Resonance Frequency

The corrected theoretical values for the three types of bubble are still less than the experimental value by about 10% overall inSection 3.2. According to equation (6), in addition to damping coefficient, the bubble pulsation frequency may also affect the acoustic energy radiated by the bubble. In order to study the influence of bubble resonance frequency on bubble acoustic energy, the actual bubble acoustic pressure curve is analyzed. The sound pressure signal collected by the hydrophone was Fourier transformed to obtain the initial resonance frequency f0 of the bubble, and then, the actual damping coefficient δ of the bubble was calculated according to equation (3). According to the initial resonance frequency and damping coefficient, the sound pressure curve of the bubble is fitted.Figure8 compares the actual sound pressure curve (red) and fitting sound pressure curve (blue). The fitting curve is an exponentially damped sinusoidal curve. As can be seen from the figure, the actual curve completely coincides with the fitting curve in the first few cycles. After a few cycles, the two curves begin to separate. The period of the actual curve gets longer, which means the resonance frequency of the bubble is decreasing. As a result, the theoretical acoustic energy calculated using equation (6) is lower than the actual value.Figure 8
Actual sound pressure curve (red) and fitting sound pressure curve (blue): (a)f0=7.5kHz; (b) f0=15kHz. The fitting curve is an exponentially damped sinusoidal curve.
(a)(b)Medwin and Beaky [35] found that the resonance frequencies of all those bubbles decrease during the short oscillation times of the latter, and they tried to explain this change. One possibility is that the bubble is initially ellipsoidal and that its frequency decreases as it becomes spherical. Strasberg [39] proved that the frequency is only slightly dependent on e (e is the ratio of major to minor axis of the spheroid). For example, the frequency of an oblate spheroid with e=2 differs by only 2 percent from that of a sphere with the same volume. Because actually e is less than 2, the deviation is less than 2 percent. Another possibility is that the bubble moves away from the surface rapidly, with subsequent displacement due to an increase in the dipole axis. However, to date, no studies have addressed how the bubble could move away from the surface so quickly in less than 2 ms (the bubbles produced by raindrops usually last no more than 2 ms). In view of this, this paper puts forward an explanation as follows.


### 3.4. Explanation for Decrease in Resonance Frequency

Figure9 shows the process of bubble generation and movement underwater as recorded by the underwater camera running at 240 fps, showing a drop of 3.9 mm in diameter impacting at terminal velocity. The frames are in order, and the final three are sequential. The total time is around 71 ms. The initial impact sound occurs in frame 1 (Figure 9(a)), a turbulent jet is created by the splash in frame 16 (Figure 9(b)), and a bubble is trapped underwater by the turbulent jet in frames 17 (Figure 9(c)) and 18 (Figure 9(d)). The bubble sound lasting less than 4.2 ms is detected between frames 17 and 18 around 67 ms after impact; it begins at the moment when the bubble detaches.Figure 9
Frames from the underwater camera running at 240 fps showing a drop of 3.9 mm in diameter impacting at terminal velocity. The time interval between each frame is approximately 4.2 ms.
(a)(b)(c)(d)The bubble is connected to the atmosphere when it is generated at the bottom of the turbulent jet in frame 17 (also in frame 9 of Figure1(a) and in frame 9 of Figure 1(b)). Although the bottom of the jet is 9 mm from the water surface, the depth of the bubble is considered to be zero when the bubble is connected to the atmosphere. The depth is equal to the diameter of the bubble at the moment when the bubble separates from the jet. After this separation, the bubble keeps going down, while water fills the crater. The depth of the bubble increases sharply in a time that is less than the 4.2 ms time interval between the two adjacent frames 17 and 18. Although the duration of the bubble sound is less than 4.2 ms, the depth of the bubble increases by around 9 mm during the bubble pulsation period.For a spherical bubble at close separationz from the free surface, Strasberg [39] indicated that the resonance frequency will be higher, namely,(8)f=f∞F,where f∞ is the resonance frequency of the bubble in free field and F is a depth-dependent factor as follows:(9)F=1−a2z−a2z4−1/2,where z is the depth of the bubble. Gillot et al. [38] have calculated the corrected frequency using the correction coefficients due to the deformation of the bubble and to the proximity to the interface, respectively.Figure10 shows the relationship between the factor F and the bubble depth for the initial pulsation frequencies of 5 kHz, 15 kHz, and 25 kHz. In all three cases, F decreases by around 10% when the bubble depth increases to ten times the radius of the bubble from the initial depth, which means that the bubble’s pulsation frequency also decreases by 10%. The minimum depth of the bubble is its radius. Meanwhile, the bubble frequency decreases more rapidly as the bubble’s initial pulsation frequency is increased. In Figure 7, the theoretical acoustic energy is less than the experimental value by 10% overall, which is consistent with the trend associated with the decrease in the bubble’s pulsation frequency. This consistency indicates that the decrease in the bubble’s pulsation frequency results in the deviation of the acoustic energy.Figure 10
Variation in factorF in equation (9) with bubble depth for different initial pulsation frequencies. The initial depth is the diameter of the bubble.


### 3.5. The Final Prediction Formula

Considering the correction of damping coefficient and decrease in the bubble’s resonance frequency, the prediction formula of sound energy is as follows:(10)E=R02p0212ρcfδ,where p0 is the peak pressure at R0 = 1 m on axis and δ is the damping coefficient in equation (7). Also, f=f0+f1/2, f0 is the initial resonance frequency of the bubble, and f1 is the resonance frequency of the last cycle of the sinusoidally damped bubble signal. The resonant frequency is modified by adding a factor of 0.9, which makes the theoretical sound energy consistent with the actual sound energy. In practice, f approximates to 0.9f0. Figure 11 compares the actual acoustic energy and the theoretical values calculated using final prediction formula equation (10) for the three types of bubble. In each case, the theoretical acoustic energy is very close to the experimental value. The RMS of the errors of bubble sound energy of types I, II, and III reduced from 1.50×10−11J, 2.17×10−10J, and 3.55×10−10J to 3.25×10−12J, 9.11×10−11J, and 1.78×10−10J, respectively.Figure 11
Actual acoustic energy of bubble calculated using equation (2) (red) and theoretical acoustic energy of bubble calculated using equation (10) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


## 3.1. Acoustic Energy of the Bubbles

The 0.7 mm drop produced only weak impact-pulse noise. The 0.9 mm drop produced weak impact-pulse noise and a type-I damped bubble. The 1.5 mm drop produced only impact-pulse noise. The 2.5 mm and 3.9 mm drops both produced impact-pulse noise and type-II and type-III damped bubbles.Using the physical mechanisms for the three types of bubble produced by raindrops, we clarify here why the 1.5 mm drop produces only impact noise and no bubble, whereas the 0.9 mm and 2.5 mm drops both produce bubbles. Because the resultant force of gravity and surface tension is not strong enough to pinch-off the tip of the splash crater, there is no type-I bubble when a midsize raindrop strikes the water surface. Because a midsize raindrop does not have enough kinetic energy, no canopy is formed when the upward-moving water mass ejects a spray of droplets, and no bubble is trapped underwater. There is no type-II bubble when a midsize raindrop strikes the water surface, nor is there a type-III bubble because there are no secondary splashes of drop aerosols thrown up during the initial raindrop impact.The sound pressure signal collected 20 cm below the impact point is converted to sound pressure at 1 m on axis according to the dipole radiation pattern. Figure4(a) shows the acoustic signal produced by the raindrop of 3.9 mm in diameter. As can be seen in Figure 4(a), the signal generated by the initial pulse is measured at time t = 7.3 ms, and the signal generated by the bubble is measured at time t = 52.4 ms. The bubble is detected around 45.1 ms after impact. The sinusoidally damped bubble signal in Figure 4(a) is shown in Figure 4(b) with an appropriate amplitude amplification. Here, the echo is clearly distinguished from direct sound.Figure 4
Acoustic signal produced by a raindrop of 3.9 mm in diameter: (a) entire signal; (b) sinusoidally damped bubble signal.
(a)(b)The amplitudes of the impact-pulse noise and bubble noise generated by raindrops are given in Table2, where D is the raindrop diameter, pP is the amplitude of the impact-pulse noise, and pI, pII, and pIII are the amplitudes of type I, II, and III bubble noise, respectively. The amplitude of the impact-pulse noise generated by raindrops of the same diameter is clearly uniform and increases with the raindrop diameter. The average amplitude of the impact-pulse noise generated by raindrops of 2.5 mm and 3.9 mm in diameter is, respectively, 0.65 Pa and 2.02 Pa. The amplitude distribution of the type-I bubble noise is concentrated and uniform. The average amplitude of the type-I bubble noise is 0.36 Pa, whereas the amplitude distributions of the type-II and type-III bubble noise are dispersive and random. As the raindrop diameter increases, more bubbles with high sound pressure are produced.Table 2
Ranges of amplitude of sound pressure.
D (mm)pP (Pa)pI (Pa)pII (Pa)pIII (Pa)0.9—0.25–0.54——2.50.51–0.74—0.08–1.40.06–1.223.91.82–2.44—0.06–2.00.05–3.8The actual acoustic energy of the bubble was calculated using equation (2), while the theoretical acoustic energy of the bubble was calculated using equation (6). p0 is the peak pressure at the distance R0 = 1 m from the point of impact to the hydrophone.Figure5 compares the actual acoustic energy and the theoretical values for the three types of bubble (for the convenience of comparison, the actual sound energy data are sorted from smallest to largest). The theoretical acoustic energy of type-I bubble is less than the experimental value by about 60% overall. The root mean square (RMS) of the error is 1.50×10−11J. The theoretical acoustic energy of type-II and type-III bubbles is less than the experimental value by about 50% overall. The root mean square of the errors is 2.17×10−10J and 3.55×10−10J, respectively. According to equation (6), the deviation of acoustic energy may be caused by the deviation of damping coefficient δ or resonance frequency f0.Figure 5
Actual acoustic energy of bubble calculated using equation (2) (red) and theoretical acoustic energy of bubble calculated using equation (6) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


## 3.2. Influence of Damping Coefficient

The empirical damping coefficient in equation (4) is used to calculate the theoretical acoustic energy of bubble produced by the bubble generator. The actual damping coefficient δ of each bubble produced by each size of drop in the present experiments is fitted using equation (3). The actual damping coefficient and empirical damping coefficient are shown in Figure 6. The actual damping coefficients are different from the empirical values. The reason is that the bubbles produced by rainfall are different in mechanism from those produced by bubble generators. Bubbles generated by bubble generators are just beginning to be static in water. Then, the bubbles rise slowly under the influence of buoyancy. However, as soon as the bubbles are produced by raindrops, they will move down rapidly under the influence of inertia. The heat generated by the relative moving between the bubbles and water reduces the outward thermal radiation of the bubbles. The thermal damping of the bubbles produced by raindrops is less than that of other origin. Therefore, the actual damping coefficients are found to be less than the empirical values. Gillot et al. [38] have pointed out that the values of the damping time obtained by theoretical predictions are far from the values measured in water. This deviation of damping coefficient δ results in the deviation of acoustic energy.Figure 6
Comparison of the actual damping coefficient of bubbles produced by raindrops with the empirical value.Therefore, a new empirical formula for the damping coefficient of an air bubble produced by a raindrop is given by(7)δ=k1⋅f010001/3,where k1=0.012. This is the empirical formula for the damping coefficient of bubbles produced by raindrops. The empirical damping coefficient in equation (4) is twice the actual damping coefficient. The new curve fitted is shown in Figure 6.The theoretical acoustic energy of the bubble was recalculated using the modified damping coefficient. Figure7 compares the actual acoustic energy and the corrected theoretical values for the three types of bubble. In each case, the theoretical acoustic energy is very close to the experimental value, but the theoretical value is still less than the experimental value by about 10% overall. The root mean square of the errors is 4.87×10−12J, 1.19×10−10J, and 2.05×10−10J, respectively.Figure 7
Actual acoustic energy of bubble calculated using equation (2) (red) and correctional theoretical acoustic energy of bubble calculated using equation (6) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


## 3.3. Influence of Resonance Frequency

The corrected theoretical values for the three types of bubble are still less than the experimental value by about 10% overall inSection 3.2. According to equation (6), in addition to damping coefficient, the bubble pulsation frequency may also affect the acoustic energy radiated by the bubble. In order to study the influence of bubble resonance frequency on bubble acoustic energy, the actual bubble acoustic pressure curve is analyzed. The sound pressure signal collected by the hydrophone was Fourier transformed to obtain the initial resonance frequency f0 of the bubble, and then, the actual damping coefficient δ of the bubble was calculated according to equation (3). According to the initial resonance frequency and damping coefficient, the sound pressure curve of the bubble is fitted.Figure8 compares the actual sound pressure curve (red) and fitting sound pressure curve (blue). The fitting curve is an exponentially damped sinusoidal curve. As can be seen from the figure, the actual curve completely coincides with the fitting curve in the first few cycles. After a few cycles, the two curves begin to separate. The period of the actual curve gets longer, which means the resonance frequency of the bubble is decreasing. As a result, the theoretical acoustic energy calculated using equation (6) is lower than the actual value.Figure 8
Actual sound pressure curve (red) and fitting sound pressure curve (blue): (a)f0=7.5kHz; (b) f0=15kHz. The fitting curve is an exponentially damped sinusoidal curve.
(a)(b)Medwin and Beaky [35] found that the resonance frequencies of all those bubbles decrease during the short oscillation times of the latter, and they tried to explain this change. One possibility is that the bubble is initially ellipsoidal and that its frequency decreases as it becomes spherical. Strasberg [39] proved that the frequency is only slightly dependent on e (e is the ratio of major to minor axis of the spheroid). For example, the frequency of an oblate spheroid with e=2 differs by only 2 percent from that of a sphere with the same volume. Because actually e is less than 2, the deviation is less than 2 percent. Another possibility is that the bubble moves away from the surface rapidly, with subsequent displacement due to an increase in the dipole axis. However, to date, no studies have addressed how the bubble could move away from the surface so quickly in less than 2 ms (the bubbles produced by raindrops usually last no more than 2 ms). In view of this, this paper puts forward an explanation as follows.


## 3.4. Explanation for Decrease in Resonance Frequency

Figure9 shows the process of bubble generation and movement underwater as recorded by the underwater camera running at 240 fps, showing a drop of 3.9 mm in diameter impacting at terminal velocity. The frames are in order, and the final three are sequential. The total time is around 71 ms. The initial impact sound occurs in frame 1 (Figure 9(a)), a turbulent jet is created by the splash in frame 16 (Figure 9(b)), and a bubble is trapped underwater by the turbulent jet in frames 17 (Figure 9(c)) and 18 (Figure 9(d)). The bubble sound lasting less than 4.2 ms is detected between frames 17 and 18 around 67 ms after impact; it begins at the moment when the bubble detaches.Figure 9
Frames from the underwater camera running at 240 fps showing a drop of 3.9 mm in diameter impacting at terminal velocity. The time interval between each frame is approximately 4.2 ms.
(a)(b)(c)(d)The bubble is connected to the atmosphere when it is generated at the bottom of the turbulent jet in frame 17 (also in frame 9 of Figure1(a) and in frame 9 of Figure 1(b)). Although the bottom of the jet is 9 mm from the water surface, the depth of the bubble is considered to be zero when the bubble is connected to the atmosphere. The depth is equal to the diameter of the bubble at the moment when the bubble separates from the jet. After this separation, the bubble keeps going down, while water fills the crater. The depth of the bubble increases sharply in a time that is less than the 4.2 ms time interval between the two adjacent frames 17 and 18. Although the duration of the bubble sound is less than 4.2 ms, the depth of the bubble increases by around 9 mm during the bubble pulsation period.For a spherical bubble at close separationz from the free surface, Strasberg [39] indicated that the resonance frequency will be higher, namely,(8)f=f∞F,where f∞ is the resonance frequency of the bubble in free field and F is a depth-dependent factor as follows:(9)F=1−a2z−a2z4−1/2,where z is the depth of the bubble. Gillot et al. [38] have calculated the corrected frequency using the correction coefficients due to the deformation of the bubble and to the proximity to the interface, respectively.Figure10 shows the relationship between the factor F and the bubble depth for the initial pulsation frequencies of 5 kHz, 15 kHz, and 25 kHz. In all three cases, F decreases by around 10% when the bubble depth increases to ten times the radius of the bubble from the initial depth, which means that the bubble’s pulsation frequency also decreases by 10%. The minimum depth of the bubble is its radius. Meanwhile, the bubble frequency decreases more rapidly as the bubble’s initial pulsation frequency is increased. In Figure 7, the theoretical acoustic energy is less than the experimental value by 10% overall, which is consistent with the trend associated with the decrease in the bubble’s pulsation frequency. This consistency indicates that the decrease in the bubble’s pulsation frequency results in the deviation of the acoustic energy.Figure 10
Variation in factorF in equation (9) with bubble depth for different initial pulsation frequencies. The initial depth is the diameter of the bubble.


## 3.5. The Final Prediction Formula

Considering the correction of damping coefficient and decrease in the bubble’s resonance frequency, the prediction formula of sound energy is as follows:(10)E=R02p0212ρcfδ,where p0 is the peak pressure at R0 = 1 m on axis and δ is the damping coefficient in equation (7). Also, f=f0+f1/2, f0 is the initial resonance frequency of the bubble, and f1 is the resonance frequency of the last cycle of the sinusoidally damped bubble signal. The resonant frequency is modified by adding a factor of 0.9, which makes the theoretical sound energy consistent with the actual sound energy. In practice, f approximates to 0.9f0. Figure 11 compares the actual acoustic energy and the theoretical values calculated using final prediction formula equation (10) for the three types of bubble. In each case, the theoretical acoustic energy is very close to the experimental value. The RMS of the errors of bubble sound energy of types I, II, and III reduced from 1.50×10−11J, 2.17×10−10J, and 3.55×10−10J to 3.25×10−12J, 9.11×10−11J, and 1.78×10−10J, respectively.Figure 11
Actual acoustic energy of bubble calculated using equation (2) (red) and theoretical acoustic energy of bubble calculated using equation (10) (blue): (a) type-I bubble; (b) type-II bubble; (c) type-III bubble.
(a)(b)(c)


## 4. Conclusion

In order to eliminate the contamination from the tank reverberation, hydrodynamic flow, and laboratory electrical noise in the water tank, a formula has been proposed for predicting the acoustic energy of the bubble produced by a raindrop when the latter falls onto a plane water surface. In order to verify and improve this formula, a series of experiments was conducted. By filming the impact with a high-speed underwater camera and recording the sound with a hydrophone 20 cm below the impact point, the acoustic energy radiated by the bubble produced by raindrops of different sizes has been studied. The main findings of this study are as follows:(1)
The damping coefficient of the bubble produced by a raindrop is found to differ appreciably from the empirical value of the bubble produced by bubble generators. The empirical damping coefficient of bubbles is twice the actual value. The reason is that the bubbles produced by rainfall are different in mechanism from those produced by bubble generators. This deviation of damping coefficientδ results in a 50% deviation between the theoretical and actual values of acoustic energy in the formula.(2)
After the correction of damping coefficient of the bubble, there is still 10% deviation between the theoretical and actual values of acoustic energy in the formula. This deviation is caused by the decrease in resonance frequencyf0 of the bubble. The bubble moves away from the water surface, thereby decreasing its resonance frequency rapidly.(3)
After the correction of damping coefficient and resonance frequency of the bubble, the results predicted by the formula proposed in this paper for predicting the acoustic energy of the bubble produced by a raindrop agree well with the actual acoustic energy. The RMS of the errors of bubble sound energy of types I, II, and III reduced from1.50×10−11J, 2.17×10−10J, and 3.55×10−10J to 3.25×10−12J, 9.11×10−11J, and 1.78×10−10J, respectively. This formula can be used to accurately predict the sound energy of bubbles produced by rainfall, even if the sound pressure signal of individual bubbles is easily contaminated by tank reverberation, hydrodynamic flow, and laboratory electrical noise.



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*Source: 4581937-2020-12-14.xml*