The Effects of Bubble Scattering on Sound Propagation in Shallow Water

Abstract

As sea waves break, a bubble layer forms beneath the sea surface. The bubble scattering affects sound propagation, thus influencing the accuracy of sound field prediction. This paper aims to investigate the effects of bubble scattering on the statistical characteristics of the sound field, the distribution of transmission loss (TL), and the average scattering attenuation in shallow water. A bubble layer model based on the bubble spectrum and a parallel Parabolic Equation (PE) model are combined to calculate and analyse the sound field in the marine environment with bubbles. The effects of the bubble layer are then compared with those of the fluctuant sea surface. The results show that the bubble scattering causes additional energy loss and random fluctuations of the sound field. The TL distribution properties and the average scattering attenuation are related to the wind speed, range, frequency, and source position relative to the negative gradient sound speed layer in shallow water. The comparison demonstrates that the random variation caused by the fluctuation of the sea surface is more significant than that caused by bubbles, and the energy loss caused by bubble scattering is more significant than the fluctuant sea surface under strong wind conditions.

1. Introduction

With the movement of sea waves, a bubble layer forms beneath the sea surface, and its resulting scattering has a significant effect on sound propagation in shallow water. When the sea wave breaks up, a large number of bubbles are pulled into the water, and the mixture of water and bubbles forms an approximate cone under the sea surface extending downward, known as the bubble plume. The upper water layer, which contains many bubble plumes, is the bubble layer. The sound speed changes significantly in the bubble layer, where the sound rays are scattered, leading to additional energy loss and random variation of the sound field. Hence, the results of underwater acoustic experiments are affected, the accuracy of sound field prediction is decreased, and the application performance of underwater acoustic equipment is degraded. In shallow water, multiple surface–bottom interactions of the sound rays intensify the effects of the bubble layer underneath the sea surface. Our study of the effects of bubble scattering presented in this paper aims to contribute to the, thus far, limited research carried out in relation to these phenomena.

Establishing a reliable bubble model is crucial for investigating the effects of the bubble layer on sound propagation. Prior to the 1990s, the bubble layer was treated as a uniform medium due to the lack of understanding of the quantitative relationship between the bubble layer and the range, depth, and time. Based on the measured data, Hall proposed a comprehensive bubble spectrum related to wind speed and bubble radius [1]. In Hall’s model, the vertical change of the bubble layer was considered, but the horizontal inhomogeneity was ignored. In the 1990s, researchers became more focused on the inhomogeneity of the bubble layer and attempted to resolve the problem of insufficient knowledge of the bubble distribution under the broken sea. The available descriptions in the literature at the time were generally inconsistent with the empirical observations. Moreover, there were only a few direct, experimental measurements of the sound speed in the bubble layer, which is the most fundamental knowledge necessary for analyzing sound propagation in the bubble layer. In 1994 Lamarre made a breakthrough by measuring the sound velocity in the bubble plume, which was consistent with the results derived from the bubble density and void fraction [2]. Monahan proposed a comprehensive bubble model including different development stages of bubble plume, describing the entire process from wave breaking to bubble dissipation and the background environment [3,4,5]. However, Monahan’s model was still not fully parameterized and did not satisfy the needs of acoustic calculation. In 1998, Novarini [6] proposed a wind-caused microbubble set distribution model based on Monahan’s [3,4,5] and combined it with the bubble spectrum from the literature [7] and a few direct measurements. Novarini’s work realized the parametric description of the spatial distribution, different development stages, and the geometry of the bubble sets and corrected the penetration depth of the bubble plume. This model also provided a calculation method of sound velocity and sound absorption of the bubble layer. Therefore, Novarini’s model is adopted to simulate the bubble environment in this paper.

Following these developments, researchers have made further progress in studying bubble effects on sound propagation. Hall studied the influence of the bubble layer on the sound field of different frequencies [2]. Other researchers have attempted to explain some abnormal findings, such as that the backscattering intensity of sound rays with a small grazing angle was higher than the predicted value [8] in reverberation experiments. Novarini pointed out that the attenuation of the sound field in a bubble environment is mainly caused by β-plume (one of the bubble set development stages) [6]. Norton compared the influence of the fluctuant sea surface and bubble layer and pointed out that the sea surface led to more significant disturbance of the sound field while also leading to less energy loss. At the same time, Norton found that the main attenuation mechanism is scattering from the bubble plumes, rather than refraction or absorption in bubble plume, within the frequency range of 2 kHz to 4 kHz [9].

There have been several studies recently analysing the dynamics and influences between bubbles and rough sea surface since the bubble layer always appears with the rough sea surface. In 2011, Boyles noted that under the same wind speed, both fluctuant sea surface scattering and bubble scattering affected the sound propagation of 12.5 kHz. Under strong winds, the main influence factor was the bubble scattering [10,11]. In the literature [9,10,11], only the case of high-frequency and near-field was studied, and only one simulation result was compared without considering the randomness of the sea surface and bubble layer. In 2020, Yao [12] gave the increment of propagation loss generated by wind-caused bubble layer under different wind speeds but still adopted a simplified bubble layer model independent of range. Future studies should focus more on the randomness of the bubble layer, and the cases of low-frequency, far-field, and different hydrological environments should be discussed. More systematic research on the factors that decide the scattering effects of the bubbles is needed. In addition, further comparison between the influence of fluctuant sea surface and that of bubbles should be considered based on a range-dependent and randomly variable bubble layer model.

The purpose of this paper is to study the influence of bubble scattering on the statistical characteristics of the low-frequency sound field in shallow water under different sea states. In this paper, a random construction model of the wind-caused bubble layer is established. The parallel Parabolic Equation (PE) model is used to simulate the sound field. With the Monte Carlo method, the sound fields in randomly varying bubble environments are simulated, and then the statistical characteristics are analysed. The main conclusion is that the bubble layer attenuates and fluctuates the sound field, and the effects are dependent on the wind speed, propagation distance, frequency, and the position of the source relative to the negative gradient sound speed layer. Additionally, under strong winds, the scattering attenuation of the sound field caused by the bubble layer is greater than the rough sea surface, while the random variation caused by the rough sea surface is more significant.

This study contributed to the existing body of work with the investigation of the randomness of the bubble layer caused by the sea waves. To our knowledge, this is the first time the study of the probability distribution of transmission losses (TLs) in bubble environments have been carried out. Additionally, the mechanism of bubbles causing energy loss of the sound field is explained by the ray method, and the scenario with a negative gradient sound speed layer was also studied. We discovered that bubbles might have different effects on the sound field depending on whether the sound source is above or below the thermocline and explained the reason for this phenomenon. Finally, this paper suggested that the effects of rough sea surface and bubbles should be considered jointly.

The remainder of the article is organized as follows. Section 2 presents the bubble model and sound field simulation methods. Section 3 and Section 4 discuss the calculation results of the sound field in the bubble environment. Section 5 demonstrates that the three-dimensional scattering from a bubble plume is insignificant. The conclusions are summarized in Section 6.

2. Wind-Caused Bubble Layer Model and Sound Field Calculation Model in Bubble Environment

2.1. Bubble Distribution Model in Bubble Layer

As the sea waves move, the bubble plumes form consecutively with a time interval ranging from 3 to 15 s. The total lifetime of a bubble plume is more than 100 s. Therefore, the sound propagation is continuously affected by the bubbles.

From wave breaking to bubble dissipation, a bubble plume experiences several development stages, α, β, and γ [3,4,5]. When the sea wave slope exceeds the limit, the wave peak breaks and a whitecap appears (the first stage of whitecap), shown in Figure 1a. The α-plume is the downward extension of the whitecap under the sea surface. The α-plumes are very small in size and have a very short lifetime (less than 1 s). The α-plume rapidly decays into β-plume, while simultaneously, the whitecap evolves into a foam patch (the second stage of whitecap). The β-plumes have a larger scale and a longer lifetime (about 4 s). Then, β-plumes develop into γ-plumes, which exist for 100 to 1000 s. Although their scale is largest and their lifetime is longest, their void fraction is the lowest, three orders of magnitude less than β-plumes. Eventually, γ-plumes decay into a weak stratified background layer with a few bubbles. Different stages of the bubble plume and the background bubble layer have different effects on sound propagation. The above description implies that bubble effects on the sound field are mainly caused by the β-plumes, as was shown in the literature [6]. Therefore, this paper only focuses on the β-plumes.

The β-plume is modelled as a conical microbubble set embedded into seawater, and the width of its lengthwise section decreases exponentially with depth. For example, a three-dimensional schematic diagram of β-plumes (excluding the whitecaps on the top of the plumes) when the wind speed is 12 m/s is presented in Figure 1c. The bubble layer in this stage is described by the bubble spectrum and some spatial variables proposed in the literature [6]. The bubble spectrum is also known as the bubble density spectrum, representing the number of bubbles in the unit volume of the medium, which is related to bubble radius, wind speed and depth.

The bubble spectrum of β-plume is written according to Novarini’s bubble model [6],

$$n_{\beta }\left(a,z,v\right)=N_{0,\beta }{G}_{\beta }\left(a\right)Z_{\beta }\left(z\right)U\left(v\right)$$

(1)

where $n_{\beta }$ is the number of bubbles per unit volume, in m⁻³ μm⁻¹, $a$ is the bubble radius, $z$ is the depth, and $v$ is the wind speed 10 m above the sea surface. $N_{0,\beta }$ is a constant, and Novarini set it to be 2.0 $\times$ 10⁷ m⁻³ μm⁻¹ to correspond with the measured void fraction and bubble density of β-plumes. $G_{\beta }$ is the bubble spectrum shape function, and $U$ is the wind speed-dependent function, with the approximately same form as in literature [1] is adopted,

$$U\left(v\right)={\left(\frac{v}{13}\right)}^3.$$

(2)

In the Equation (1) $Z_{\beta }$ is the depth-dependent function. Monahan [3,4,5] assumed the bubbles were uniform, thus:

$$Z_{\beta }\left(z\right)=\{\begin{array}{ll}1,\hfill & z\le z_{{\beta }_{\max }}\hfill \\ 0,\hfill & z>z_{{\beta }_{\max }}\hfill \end{array},$$

(3)

where $z_{{\beta }_{\max }}$ is the maximum penetration depth. It equals

$$z_{{\beta }_{\max }}=\left(1.23\times {10}^{-2}\right)v^2.$$

(4)

Using this method, a bubble spectrum at a depth of 0.2 m with a wind speed of 12 m/s is plotted in Figure 1b.

Next, the spatial description of the β-plumes [6] is introduced. The cross-sectional area of the bubble plume $A_{\beta }$ decreases exponentially with depth described as,

$$A_{\beta }\left(z\right)=A_{0,\beta }\exp \left(-z/d_{\beta }\right),$$

(5)

where $A_{0,\beta }$ is the area of the bubble plume at the sea surface (i.e., the top of the bubble plume), whose unit is m², corresponding to the size of the whitecap in the second stage. According to Bondur’s measurement [13], it equals

$$A_{0,\beta }=17.0+0.0307{\left(v-5\right)}^2.$$

(6)

In the range-depth two-dimensional coordinate system, the equivalent horizontal length $L_{\beta }$ (equal to the square root of the area) is used to represent the size of the bubble plume as:

$$L_{\beta }\left(z\right)=L_{0,\beta }\exp \left(-z/2d_{\beta }\right),$$

(7)

where $L_{0,\beta }=\sqrt{A_{0,\beta }}$, and

$$d_{\beta }=z_{{\beta }_{\max }}/5.99.$$

(8)

The distance between two β-plumes is used to describe the distribution of β-plumes over the range axis. Because β-plumes are attached to whitecaps, the distance between two β-plumes $S$ is equal to the distance between the whitecaps [14]

$$S=237\times v^{-1.07}.$$

(9)

According to these parameters, a bubble layer consisting of β-plumes is reconstructed and shown in Figure 1d.

2.2. Sound Velocity in Bubble Layer

The equivalent sound velocity in the bubble plume is [6],

$$c_{eff}{}^{-2}=\left[\left(1-V\right){\rho }_w+V{\rho }_{gas}\right]\left[\left(1-V\right)K_W+\Delta K\right],$$

(10)

where $V$ is the fractional volume occupied by bubbles in the bubble plume, and ${\rho }_w$ and ${\rho }_{gas}$ are the density of water and the bubble, respectively. $K_w$ is the compressibility of water, and $\Delta K$ is the compressibility difference between the bubble plume and water:

$$\Delta K\left(f,z\right)=\frac{1}{{\rho }_w\pi f^2}{\displaystyle {\int }_{a_{\min }}^{a_{\max }}a{n}_{\beta }\left(a\right){\left[{\left(f_r/f\right)}^2-1+i\delta \right]}^{-1}\mathrm{d}a}.$$

(11)

where $f$ is the frequency of the sound wave, $\delta$ is attenuation coefficient, $f_r$ is resonance frequency, and $n_{\beta }\left(a\right)$ is given in Equation (1). The minimum bubble radius $a_{\min }$ equals 10 μm, and the maximum $a_{\max }$ equals 1000 μm. The $c_{eff}$ is a complex number, of which the real part is the phase velocity $c_{b,p}$,

$$c_{b,p}=\mathrm{Re}\left\{c_{eff}\right\}.$$

(12)

The sound absorption ${\alpha }_b$ (the unit of which is dB/m) caused by bubbles is deduced by the imaginary part of $c_{eff}$,

$${\alpha }_b=\frac{40\pi f}{\ln 10}\mathrm{Im}\left\{\frac{1}{c_{eff}}\right\}.$$

(13)

We take the phase velocity $c_{b,p}$ as the sound velocity in the bubble plume $c_b$.

The sound speed distribution in a β-plume is calculated according to Equations (7) and (12). For example, Figure 2a shows the sound speed distribution for 1000 Hz with a wind speed of 12 m/s. The presence of bubbles makes the sound speed in the mixed medium less than that in seawater (roughly equal to 1500 m/s). Since the penetration depth of the bubble plume is only a few meters, and the sound speed $c_b$ and the equivalent width $L_{\beta }$ change significantly with depth $z$, (Figure 2a shows that the gradient of $c_b$ along the depth direction is relatively large, and Equation (7) shows that $L_{\beta }$ decreases exponentially with the depth), the vertical step $\Delta z$ should be relatively small when the sound speed data $c_b\left(r,z\right)$ are generated. In this paper, the vertical step is set as 0.1 m, and the corresponding horizontal step is set as 0.25 m.

The following is the reconstructing method of the random two-dimensional sound speed field. There are a large number of bubble plumes in the bubble layer, distributing along the horizontal direction. These bubble plumes are different from each other, with these variables varying randomly within certain intervals: the bubbles density $N_{0,\beta }$, the maximum penetration depth $z_{{\beta }_{\max }}$, the equivalent length at the sea surface $L_{0,\beta }$, and the distance between adjacent plumes $S$. These variables are assumed to obey the Gaussian distribution. $N_{0,\beta }$ is a constant in Section 2.1, multiplied by a random variable $R_1$ to represent its variation. $R_1$ obeys the Gaussian distribution with a mean value of 100% and a root mean square (RMS) of ${\sigma }_1$. According to the measurements, $N_{0,\beta }$ is considered to vary in the range of $x_1$($x_1$ = 20%), and therefore the Gaussian distribution of $R_1$ is truncated in the range of $\left(1-x_1,1+x_1\right)$. In order to determine the value of the RMS ${\sigma }_1$ subjected to the truncation constraint condition, we plotted the probability diagram of 1000 samples of the random variable $R_1$, shown in Figure 2b. As both ${\sigma }_1=x_1/2$ and ${\sigma }_1=x_1/3$ meet the requirements, we let ${\sigma }_1=x_1/2$. Similarly, the random variables, $R_2$ and $R_3$, for $L_{0,\beta }$ and $S$, are obtained based on their variation range, 30% and 20%. $R_4$ is the random variable for $z_{{\beta }_{\max }}$. In addition, it should also be considered that at the same wind speed, a larger-scale bubble plume has more bubbles and a larger penetration depth accordingly, i.e., there is a positive correlation between $R_1$ and $R_4$. We assume a linear relationship between them, $R_4=\left(R_1-1\right)x_4/x_1+1$ where $x_4$ is the variation range equal to 30%. Then a two-dimensional sound speed field in the bubble environment is produced. For example, a sound speed field of 1000 Hz with a wind speed of 12 m/s is shown in Figure 2c, where the white colour represents 1537 m/s.

According to Equations (10) and (11), the sound speed in β -plume depends on the bubble density, which is related to the wind speed according to Equation (1). Thus, the sound speed in β- plume is affected by the wind speed, shown in Figure 3a. The sound speed difference between bubble plume and seawater increases (corresponding to the decrease of sound speed in Figure 3a) with wind speed. According to Equations (1) and (2), as there is a positive correlation between the number of bubbles in the plume and wind speed, the sound speed difference caused by bubbles increases with the increment of wind speed.

Additionally, there is a strong correlation between the sound velocity and the sound frequency, according to Equation (11). In the sound speed dispersion graph (Figure 3b), the sound speed changes dramatically with frequency, within the frequency band from 0.1 to 20 kHz, and the sound speed reaches its minimum at the frequency of 3.5 kHz. Within the low-frequency range, i.e., 0.1 to 3.5 kHz, the sound speed in the bubble plume decreases with the increase of frequency.

The above analysis shows that wind speed and frequency should be considered when the influence of the bubble layer on sound propagation is discussed.

2.3. Sound Field Calculation Model in Bubble Environment

In order to calculate the sound field in shallow water with a bubble layer, Monterey-Miami Parabolic Equation (MMPE) model [15] is used, which can deal with the range-dependent environment. This model is based on Parabolic Equation,

$$\frac{\partial \psi }{\partial r}=-i{k}_0\psi +i{k}_0{Q}_{op}\psi ,$$

(14)

where $\psi \left(r,z\right)$ is the PE field function and $k_0$ is the reference wave number. A wide-angle approximation of the operator $Q_{op}$ and the split-step Fourier (SSF) algorithm is employed to solve Equation (14). With the reconstructed sound velocity field in the bubble layer from Section 2.2 being the input parameters, the sound field in the bubble environment is calculated using the MMPE model.

In the analysis in Section 2.2, since the bubble layer under the same wind speed varies randomly, the propagation loss (TL) of sound energy from the source to the receiver is randomly fluctuant. To obtain the random distribution of TL, the Monte Carlo method is used to simulate the random sound field in the bubble environment. By employing random bubble density, bubble plume size, bubble plume penetration depth, and the spacing between plumes based on the random variables $R_1$, $R_2$, $R_3$, and $R_4$ (subject to Gaussian distribution) in the bubble model, diverse bubble layers are generated. Then the corresponding two-dimensional sound speed fields are numerically computed. Finally, numerous sound fields are obtained to simulate the random changing sound field in a bubble environment.

In order to obtain a smooth TL curve, we average the sound energy of $N_f$ discrete frequencies within 1/3 Octave bandwidth around the centre frequency,

$$T{L}_j=-10\log \left(\frac{1}{N_f}{\displaystyle \sum _{i=1}^{N_f}{\left|p_{i,j}\left(r,z\right)\right|}^2}\right),$$

(15)

where $p_{i,j}\left(r,z\right)$ is the sound pressure at the i-th frequency in the j-th reconstructed bubble environment and $N_f$ = 10.

The convergence of the Mento Carlo experiment was verified. $N_b$ is the number of generated bubble layers, i.e., the number of random simulation experiments. When $N_b$ ≥ 20, the average TL converges. When $N_b$ ≥ 50, the TL probability distribution converges. To reduce the amount of calculation, we take $N_b$ = 50 when it is necessary to draw a probability diagram and $N_b$ = 20 in other cases.

To investigate the effects of bubble scattering, the simulation under the same sea state conditions contains dozens of sound field calculations, with time intervals consuming up to several hours. A parallel MMPE model is used to reduce the computation time by 71% on a quad-core computer.

3. The Fluctuation and Energy Loss of Sound Field Caused by Bubble Scattering

We analysed the sound field fluctuation and energy loss in the presence of bubble layers in two typical shallow-water environments, a homogenous waveguide and a waveguide with a negative gradient sound speed layer. To compare the effects of the bubble scattering and the rough sea surface scattering, the same environments as those in literature [16] were used. According to Figure 3b, there are evident sound speed changes caused by bubbles in the frequency band of 0.1 to 20 kHz. Since previous studies have investigated high-frequency cases [9,10,11], this paper focuses on low-frequency ones. In order to compare with the effects of the fluctuant sea surface [16], we discussed the sound field at the frequencies of 200 Hz and 1000 Hz here.

3.1. Bubble Effects in the Isothermal Environment

A typical isothermal environment in the South China Sea is used, displayed in Figure 4. The environmental parameters were measured in an experiment in 2004 [17]. The water depth is 88 m, and the sound speed in the water is 1537 m/s. The bottom consists of an 8-m-thick sediment layer and a homogeneous half-space basement. The sound speed, density and attenuation of the sediment layer are 1606 m/s, 1.65 g/cm^3, and 0.517 ∗ (f/1000)^1.07 dB/λ, respectively. Those parameters of the basement are 1704 m/s, 1.9 g/cm³, and 0.517 ∗ (f/1000)^1.07 dB/λ, respectively. Source and receiver depths are 7 m and 10 m, respectively.

To investigate the distribution of transmission loss under different sea states, numerical simulation is carried out for three wind scales, four, five, and seven, corresponding to the wind speeds of 7.5 m/s, 10 m/s, and 15 m/s. Fifty bubble layers are randomly generated at each wind speed, using the bubble layer model given in Section 2. The sound field in each bubble environment is calculated with the MMPE model and averaged within the one-third Octave bandwidth according to Equation (15). The 50 samples of transmission loss from the source to the receiver are grouped and counted (divided into groups every 0.4 dB), and then the probabilities of the TLs are calculated using the following equation:

$$q\left(TL\right)=\frac{N\left(TL\right)}{N_b},$$

(16)

where $TL$ is the midpoint of the corresponding interval range of a sample group, $N\left(TL\right)$ is the number of samples in the group, and $N_b$ is the total number of samples.

The probability distribution of the transmission loss in Figure 5 was calculated using Equation (16). In Figure 5a, the TL probability distribution of receivers at different ranges at 1000 Hz when the wind speed is 15 m/s indicates that the TLs of each receiver is distributed randomly over a specific interval because the size and the spatial distribution of the bubble plumes and the bubble density in the bubble plumes are random in the bubble layer. Hence, the sound field in the bubble environment changes randomly as well. Besides, with the range rising, the magnitude of the TLs significantly increases.

The probability distribution of the transmission loss (at a depth of 10 m and the range of 20 km) for different wind speeds is shown in Figure 5b. The magnitude and dispersion of the TLs increase with the increase in wind speed. The increase of TL magnitude is explained by the fact that the bubble density and size of the bubble plumes increase with wind speed. The bubble density, size, and spatial properties of the plume vary by 20% to 30% at the same wind speed. These variables are more significant at higher wind speeds, and thus the absolute variation ranges are relatively more extensive, making the sound fields more fluctuant, i.e., the probability distributions of the sound field are more dispersed.

The results of different frequencies at the wind speed 15 m/s are compared in Figure 5c. It is shown that the probability distribution at 200 Hz is more concentrated than 1000 Hz. This result is very similar to the probability distribution of TL under rough sea surface conditions [16]. Since the wavelength of 200 Hz is larger than 1000 Hz, the perturbation of the bubble plumes on the same scale is minor.

The sound energy loss induced by the bubble layer is analysed as well. The transmission losses at 1000 Hz in the isothermal environment (shown in Figure 4) without bubbles and with a bubble layer when the wind speed is 15 m/s are shown in Figure 6. The transmission loss in the bubble environment (Figure 6b) is more significant than in the bubble-free environment (Figure 6a). The mechanism of this phenomenon is analysed by ray theory in Figure 7, which includes the trace of sound rays in the bubble-free (Figure 7a) and the bubble environment (Figure 7b). The scattering of the rays by the bubble plumes increases the grazing angles of the rays, even making some rays propagate backwards. The rays with the larger grazing angles suffer more sea bottom interaction and absorption. Therefore, the bubble plumes increase the energy loss of the sound field. Furthermore, the backscattering rays result in additional energy loss for forward propagation, and the significant backscattering caused by bubbles also confirms the abnormal reverberation phenomenon mentioned in the literature [8].

Then the energy loss caused by bubbles (i.e., scattering attenuation caused by bubbles) is quantitatively analysed. The average sound energy at a specific range is represented by the mean of the sound intensities of all receiver depths at the range. Suppose that the receivers are set with a vertical interval of 10 m, with $l$ representing the serial number of the receivers and $z_l$ representing the receiving depth. We average the sound intensity for all frequency points within one-third Octave bandwidth and then average the sound intensity for all samples. The average transmission loss at the range of $r$ under the wind speed $v$ is

$${\overline{\mathit{TL}}}_{z,b}\left(r,v\right)=-10\log \left(\frac{1}{N_z}{\displaystyle \sum _{l=1}^{N_z}\left(\frac{1}{N_b}{\displaystyle \sum _{j=1}^{N_b}\left(\frac{1}{N_f}{\displaystyle \sum _{i=1}^{N_f}{\left|p_{i,j}\left(r,z_l\right)\right|}^2}\right)}\right)}\right),$$

(17)

where $N_z$ is the number of receivers. The energy loss caused by bubbles is defined as the difference between the transmission loss in the bubble environment and that in the bubble-free environment

$$\Delta TL\left(r,v\right)=\overline{T}{\overline{L}}_{z,b}\left(r,v\right)-\overline{T}{\overline{L}}_{z,b}\left(r,0\right),$$

(18)

The following is the discussion of the relationship between wind speed, propagation range, and scattering attenuation. In Figure 8, the scattering attenuation of 200 Hz and 1000 Hz increases with the wind speed. Both the sound speed difference and the penetration depth of the bubble plumes increase with the wind speed rising, corresponding to the increase of scattering intensity and roughness of the scattering interface. These reasons result in the reinforcement of the effect of bubble scattering. Furthermore, the scattering attenuation from 5 km to 20 km increases with the propagation range because the sound rays transmitting further go through the bubble layer more times. Thus, the accumulated attenuation of these rays is more significant, i.e., there is more significant scattering attenuation for longer ranges.

The relation between scattering attenuation and frequency is discussed below. Figure 9 shows the average sound field in the bubble environment at the wind speed of 12 m/s, obtained by averaging the sound intensity for all frequency points and all bubble environments versus the sound field in a bubble-free environment. There is little difference between a bubble case and a bubble-free case for 200 Hz, while there is a significant sound energy decrease in the bubble case compared with the bubble-free case for 1000 Hz. This phenomenon is explained in Figure 10. The bubble layer is considered an undulating interface, consisting of the outline of the bubble plumes and the shape of the horizontal sea surface (Figure 1d), with the penetration depth of the plume corresponding to the undulating height of the rough scattering interface. Analogous to the effect of the rough sea surface, there is a greater relative fluctuation height of the scattering interface in the bubble layer, with higher frequency (i.e., less wavelength). Therefore, the scattering effect is greater. The relative undulating height of the interface is quantitatively described by the Rayleigh parameter:

$$R=4\pi {\sigma }_s\sin \theta /\lambda ,$$

(19)

where ${\sigma }_s$ is the mean square root of displacement of the scattering interface equal to the penetration depth $Z_{\max }$ of the plume, $\lambda$ is the wavelength, and $\theta$ is the grazing angle of the plane wave. In Figure 10, within the low-frequency range, i.e., below 2000 Hz, the Rayleigh parameter (with $\theta =10°$) increases linearly with frequency, and the sound speed difference (between the bubble plume and water) also increases with frequency. Thus, the scattering effect of the higher frequency is more significant.

3.2. Bubble Effects in the Environment with a Negative Gradient Sound Speed Layer

The shallow water environment with a negative gradient sound speed layer from the Yellow Sea Experiment in 1996 is shown in Figure 11 [18]. The sea is 75 m deep, and the negative gradient sound speed layer lies from 13 m to 28 m. The sound speed above and below the negative gradient sound speed layer is 1537 m/s and 1480 m/s, respectively. The sound speed gradient of the layer is −3.8 s⁻¹. The cases for the sound source located above the negative gradient sound speed layer (at a depth of 7 m) and below the negative gradient sound speed layer (at a depth of 50 m) are separately discussed.

When the sound source is located above the negative gradient sound speed layer at a depth of 7 m, the relationships between the scattering attenuation caused by the bubbles and frequency, wind speed, and propagation distance are similar to those in the isothermal environment. The scattering attenuation of 1000 Hz is shown in Figure 12, which increases with the wind speed and distance. The average sound fields in a bubble-free environment and bubble environment at the wind speed of 12 m/s are shown in Figure 13, where bubble scattering significantly lowers the energy of the sound field at 1000 Hz; however, with a relatively small effect on the sound field at 200 Hz.

The average sound fields for the source below the negative gradient sound speed layer at 50 m are shown in Figure 14. In this case, there is no significant difference between the sound fields at 200 Hz in a bubble-free environment and a bubble environment for the wind speed 12 m/s (Figure 14a,b). At 1000 Hz, there is a slight difference in sound energy above the negative gradient sound speed layer in a few kilometres (Figure 14c,d). The maximum penetration depth of the bubble plume at the wind speed of 12 m/s is 1.8 m, i.e., all are above the negative gradient sound speed layer. Most of the sound rays only reversed and reflected between the upper boundary of the negative gradient sound speed layer and the sea bottom, called refracted bottom-reflected (RBR) rays. Therefore, the bubble plumes have little effect on the sound field in this case. A small number of rays with larger grazing angles reach the area above the negative gradient sound speed layer, and their scattering attenuation caused by the bubble plumes increases the propagation loss above the negative gradient sound speed layer. They propagate in the whole depth, reflected by surface and bottom, called surface-reflected bottom-reflected (SRBR) rays. Since the reflection coefficient of these rays at the sea bottom is small, leading to the fact that the rays rapidly decay over a few kilometres. Hence, Figure 14c,d merely show a near-field difference. When the source is above the negative gradient sound speed layer, all rays are SRBR rays, which are affected by bubbles. Therefore, Figure 13c,d show a noticeable difference.

4. Comparison of the Influence of Bubble Scattering and Fluctuant Surface Scattering

The effects of bubble layer and fluctuant surface on sound propagation are compared in terms of the probability distribution of TLs and energy loss. The method to construct a rough surface is introduced in detail in the literature [16]. The one-dimension wind-wave spectrum is used to generate stochastically fluctuant surface,

$${W^{\prime }}_1(k)=L_{corr}{(1+L_{corr}^2{k}^2)}^{-(\beta -1)/2},$$

(20)

where $k$ is the spatial wavenumber, $\beta$ is the weighted index of the spectrum, and $L_{corr}$ is the correlation length. The Formula of root mean square (RMS) of the wave height of the rough surface is:

$$\sigma =v^2/20g,$$

(21)

where $v$ is the wind speed and $g$ is the acceleration of gravity. When the wind speeds are 7.5 m/s, 10 m/s, and 15 m/s, RMSs of the wave height are 0.28 m, 0.50 m, and 1.13 m, respectively. The direction of sea waves is the same as the acoustic track direction.

Figure 15 gives the probability distribution of TLs for 1000 Hz (at a depth of 10 m and the range of 20 km) in the isothermal environment. It shows that at the same wind speed, the TL corresponding to the maximum probability of bubble layer (solid line) is larger than that of the fluctuant sea surface (dashed line). The distribution interval of TLs in the bubble environment is smaller. For example, when the wind speed is 15 m/s, the maximum probability of a fluctuant surface is at 75.5 dB, and the variation interval is 8 dB, while the corresponding values of the bubble layer are 85.8 dB and 2 dB, respectively. That is, under the same wind speed, the amplitude of the transmission loss in the bubble environment is larger, and the distribution is more concentrated, compared with those in the environment with fluctuant sea surface, i.e., the random variation caused by the fluctuant sea surface and the energy loss induced by bubbles are more significant.

A further comparison of scattering attenuation is based on the results in Figure 16. In the isothermal waveguide and the thermocline waveguide with the sources above the negative gradient sound speed layer, the sound fields both accord with the fact that for 200 Hz, the scattering attenuation of bubbles is weaker than that of the fluctuant surface under slight-wind condition and greater than fluctuant surface under strong wind condition, and the critical point is at about 10 m/s. For 1000 Hz, the scattering attenuation caused by bubbles is greater than that caused by the fluctuant surface at any wind speed. This is because the Rayleigh parameters (i.e., the interface fluctuations relative to the sound wavelength) of the undulating sea surface and the scatter interface in the bubble layer are magnified by the same extent with increasing frequency. In addition, the sound speed difference in the bubble plumes also increased with frequency within 100 Hz to 2000 Hz. It grows by 7 m/s from 200 Hz to 1000 Hz (Figure 10). Therefore, with the rise of frequency, two kinds of scattering attenuation increase simultaneously, while the bubble scattering attenuation increases more significantly. When the frequency increases from 200 Hz to 1000 Hz, the solid and dashed lines in Figure 16a,c move upward together. Since the solid line moves up more remarkably, the two curves no longer cross, as shown in Figure 16b,d. That is, the bubble scattering attenuation caused by bubbles is greater than that caused by the fluctuant sea surface under any wind speed.

For 200 Hz, the energy loss caused by the fluctuant surface at low wind speeds is more significant than that caused by bubbles. Although the energy loss caused by the sea surface at 1000 Hz is smaller than bubbles, it can reach several dBs. Therefore, the combined effect of both fluctuant surface and bubbles should be considered when the energy loss is analysed.

5. Three-Dimensional Scattering from a Bubble Plume

Since the bubble plumes are approximate cones, it is necessary to study the possible three-dimensional (3D) effects caused by bubble plumes. The following is the discussion of the 3D scattering from one bubble plume. The environment is a homogeneous waveguide, the same as in Section 3.1. The source is set at x = 0, y = 0, and z = 7 m in the 3D coordinates, and the frequency is 1000 Hz. There is a β-plume located at x = 30 m and y = 0. The plume is generated at the wind speed of 15 m/s. Its diameter is 5.8 m, and its depth is 3.4 m.

The effects of 3D scattering are assessed by comparing 2D and 3D solutions for the transmission loss. Figure 17 shows the transmission losses at a depth of 10 m. The results are calculated by BELLHOP-N × 2D model and BELLHOP-3D model, respectively. The hollow circle represents the position of the bubble plume, and the dotted lines represent $\theta =0°$, $\theta =2.75°$, and$\theta =5.56°$, respectively, where $\theta$ is the horizontal azimuth. Figure 18 shows the comparison of the transmission loss at these lines. Figure 17 and Figure 18 show little difference between the results of N × 2D and 3D. Therefore, the three-dimensional scattering effect by bubble plumes is insignificant.

The bubble layer and the ice canopy are both irregular scatterers near the sea surface, but their 3D effects are different. Ballard has demonstrated the significant 3D scattering effects of the sea ice ridges. He found that the 3D solution is characterized by a 20% greater variance in the depth-averaged transmission loss than the 2D solution [19]. However, we find no significant 3D effects of the bubble plumes, even though the scenario is at the maximum wind speed discussed in this paper (15 m/s). This difference is caused by their differences in size and texture. Firstly, the scale of the ice ridges is much larger than bubble plumes. In literature [19], the lengths of the ice ridges are more than 1 km, and the widths are dozens of meters. Their penetration depths are approximately 10 m to 20 m, which is 7% to 13% of the water depth (150 m). In this paper, the maximum diameter of bubble plumes is merely several meters. The penetration depth is 2.5 m to 3.6 m, which is 3% to 4% of the water depth (88 m). The considerable difference in size makes the 3D scattering from bubble plumes much less than that from the ice ridges. Secondly, the ice is much harder than bubble plumes, and thus its scattering effects is larger than bubble plumes. The density of ice is 0.9 $\times$ 10³ kg/m³, the compressional speed is 2900 m/s to 3500 m/s, and the characteristic impedance is 2.61 $\times$ 10⁶ N$·$s/m³ to 3.15 $\times$ 10⁶ N$·$s/m³. The density of bubble plumes is about 1 $\times$ 10³ kg/m³. With the increase of depth, the sound speed difference between the bubble plumes and the seawater gets smaller. At the frequency of 1000 Hz, the sound speed is about 400 m/s at the top of the bubble plume, and it is about 700 m/s at the maximum depth. Thus, the characteristic impedance is about $4\times$ 10⁵ N$·$s/m³ to $7\times$ 10⁵ N$·$s/m³. The texture of the bubble plumes is softer, and the characteristic impedance is smaller. Thus, the 3D scattering effects are less significant than the ice ridges.

6. Conclusions

A wind-caused random bubble layer reconstruction model and parallel PE sound field calculation model are combined to simulate the sound field in the bubble environment. The statistical characteristics of sound propagation loss and scattering attenuation for low frequencies are calculated and discussed in two typical shallow water environments, an isothermal environment and an environment with a negative gradient sound speed layer. The mechanism of bubbles influence on sound propagation is explained by ray theory.

The results show that the effects of bubbles on the low-frequency sound field obey the following rules: the bubble scattering in the shallow water makes the sound field attenuate and fluctuate within a certain range. The statistical results of the propagation loss and scattering attenuation are affected by the wind speed, propagation distance, frequency, and the position of the sound source relative to the negative gradient sound speed layer in shallow water. With the increase of the wind speed, the value of TLs increases, and the probability distribution of TLs becomes more dispersed. The scattering attenuation of the bubble layer increases with the wind speed and propagation distance, and it is greater for higher frequency. In the environment with a negative gradient sound speed layer, the scattering attenuation of the sound field for the source above the negative gradient sound speed layer is significant, while it is only slight when the source is below the negative gradient sound speed layer. These effects are explained theoretically by the change of scattering angle, the relative fluctuation height of the scattering interface, the interaction intensity of the sound field and the bubble layer, and the sound speed difference in bubble plumes. In addition, the random variation caused by the fluctuant sea surface is more prominent, while the scattering attenuation caused by the bubble layer is larger under strong wind conditions. Therefore, the influence of both on low-frequency sound propagation in shallow water should be considered. Besides, the three-dimensional scattering from the bubble plumes is insignificant.

The study of the influence of bubble layer on sound propagation under different sea states in this paper is beneficial to fully understand the sound propagation laws in shallow water and improve the accuracy of sound field prediction. It also provides a reference for underwater acoustic research such as sonar performance prediction, source localization and acoustic inversion of marine environmental parameters under different sea states.