Three-dimensional acoustic vector field model using Gaussian beam tracing

Research on modeling sound pressure fields has reached a relatively mature stage (Jensen , 2011). In recent years, the advent of vector hydrophones and advances in vector hydrophone technology have enabled researchers to collect scalar and vector information of sound fields simultaneously at a single point. However, research on modeling vector sound fields is currently limited. Researchers have used Euler's formula to perform numerical differentiation on the sound pressure field obtained from parabolic equations. Zhang (2010) have found that the interval of differentiation must be smaller than the wavelength/30 for the results to converge reliably. Smith (2008) provided an expression for the radial particle velocity using Euler's formula and a parabolic approximation operator and solved the vertical particle velocity using the split-step Fourier method exchanging the spatial gradient function for a wavenumber-domain multiplication. Elastic parabolic equations can be used to calculate the expansion and vertical displacement (Shang and Lee, 1989; Jerzak , 2005), but research on three-dimensional elastic parabolic equations remains limited. Scientists have also considered the ray method. Sun and Zhou (2016) have used the ray method to analyse the complex sound intensities and arrival angles of rays in the deep sea. They have assumed that the expressions for horizontal and vertical particle velocity and sound pressure are essentially the same, but Deal and Smith (2016) pointed out that only the results for horizontal particle velocity and sound pressure are similar. However, these methods require obtaining the sound pressure first before calculating particle velocity. The vector normal mode theory suggests that the sound pressure and horizontal particle velocity of the sound field can be represented using eigenvalues and eigenfunctions (Deal and Smith, 2016). However, determining the vertical particle velocity requires additional calculations of the derivative of the eigenfunctions in the vertical direction.

Gaussian beam tracing (GBT) has been widely used for seismic, atmospheric, and underwater sound propagation. Although the accuracy is also limited by frequency, this method is simple to describe and the solution is unique everywhere, making the results clearer and effectively overcoming the problem of traditional ray acoustics not being applicable in the caustic region and the problem of rapid transition in the sound shadow zone (Gabillet , 1993; Mo , 2017). This approach provides an expression for the beam equation that depends only on the beam width and the eikonal. It can output both sound pressure and particle velocity like normal mode, even in complex environments without numerical differentiation. Moreover, this method provides a concise description and clear physical meaning, and it is suitable for high-frequency, three-dimensional, range-dependent problems where normal mode or parabolic models are not practical alternatives (Porter and Bucker, 1987). Therefore, in this letter, a three-dimensional acoustic vector field model is proposed based on GBT method.

The paper is organized as follows. In Sec. 2, the methodology is described. In Sec. 3, numerical implementations with deep-water long-range sound propagation experiment environments are presented. Finally, the results are summarized in Sec. 4.

When the beam interacts with the interface where the sound speed or its derivative is discontinuous they change. The most significant changes are the amplitude, width, and slowness of the beam, while the eikonal remains continuous at the interface. The formulae of these changes derived by Popov and Pšenčík (1978) were used to update P and Q when the ray reflected or transmitted at the interface.

A long-range ocean sound propagation experiment was conducted in the South China Sea. The vector hydrophone is deployed at a depth of 3146 m, and the explosion depth of the explosive source is 200 m. The seafloor is nearly horizontal and at a depth of approximately 4299 m, and the experiment distance is 450 km. Figure 1 shows a sound speed profile measured by expendable conductivity-temperature-depth profiler during the experiment, with measured data above 1000 m and sound speed from a relevant database below 1000 m.

Figure 2 presents the sound pressure transmission loss curves obtained from theoretical simulation and experimental measurements, which are in good agreement with each other. This result is also consistent with the simulation in normal mode. This proves that three-dimensional GBT the sound pressure field in deep-water long-distance propagation.

While receiving the sound pressure signal, the vector hydrophone also synchronously and co-located receives the sound vector signal. Figures 3(a) and 3(b) show the experimental and theoretical calculation results of the transmission loss for the X and Y vector channels. The results show that the convergence zone locations of experimental measurement are in good agreement with the theoretical prediction results. However, the measured results of the X-direction particle velocity in the first two convergence zones and the Y-direction in the 5th convergence zone are wider than the theoretical prediction. Regarding these issues, it is speculated that the horizontal rotation of the vector hydrophone caused the deviation. Figure 3(c) shows the transmission loss of the radial particle velocity. It is found that the width of the convergence zones in the measured and simulated results is basically the same, indicating that the change in the width of the convergence zone is caused by the change in the hydrophone's attitude.

Moreover, as shown in Fig. 3(c), the measured acoustic energy in the first shadow zone (47–68 km) is significantly higher than the theoretical value. Since the GBT method can only calculate a single layer of seafloor, this discrepancy is likely attributed to the layered structure of the seafloor. Two layers of seafloor are considered in the normal mode. One is the sedimentary layer with a thickness of 10 m, and the sound speed, density, and attenuation are 1555 m/s, 1.6 g/cc, and 0.2 dB/wavelength, respectively. The other is a semi-infinite seafloor with a sound speed of 1650 m/s, density of 1.8 g/cc, and absorption coefficient of 0.3 dB/wavelength. When the layered seafloor was taken into account, the agreement between the measured and theoretical results in the shadow zone improved significantly, indicating that the particle velocity in the shadow zone is strongly influenced by the layered seafloor.

In addition, Fig. 4 shows the transmission loss for the vertical particle velocity. When the transmission distance is greater than 100 km, compared to the horizontal particle velocity, the Z-direction particle velocity signal is almost completely lost in noise. This is because long-range rays are mainly emitted at small angles, and the tangential vector of the entire trajectory in the Z direction is much smaller than in the horizontal direction. According to Eq. (8), the particle velocity in the Z direction is also smaller. This figure also shows the vertical particle velocity calculated by the normal mode method. NM1 describes the results stemming from the numerical differentiation of eigenfunctions with a differential spacing of 2.5 m, which align harmoniously with both the GBT results as well as experimental measurements. However, when the differential spacing is increased to 10 m, the results described by NM2, up to 70 km, deviate significantly from the experimental measurements and the GBT results. This indicates that for the calculation of the vertical particle velocity using the normal mode, the spacing must be even smaller than wavelength/3 to ensure the accuracy of the results prior to the first convergence zone. This reason is that, at close range, the ray emitted at large angles makes a significant contribution to the vertical particle velocity, aligning with higher-order normal modes. As the eigenvalue order increases, the eigenfunction displays more rapid variations with depth. Therefore, the requirement for suitably minute differential spacing is to maintain accuracy.

In this study, a three-dimensional acoustic vector field calculation model using GBT was developed that does not require numerical differentiation of the sound pressure and can simultaneously output scalar and vector information. The accuracy and effectiveness of the model were validated in a high-frequency deep-water environment.

The processed and analyzed data of the acoustic scalar field and vector field measured in deep water were used to obtain the experimental results of sound pressure and particle velocity transmission loss curves, which were compared with the theoretical results of the model established in this paper. The experimental results of sound pressure and particle velocity transmission loss are consistent with the theoretical predictions, with similar convergence zone positions and widths, demonstrating the effectiveness of the model in predicting deep-water long-range acoustic scalar and vector fields. However, due to the lower signal-to-noise ratio of the vector channel compared to the acoustic pressure channel, signals in the far-field shadow zone are completely submerged by noise, especially for vertical particle velocity, whose energy attenuation is much greater than horizontal particle velocity. In addition, the layered seafloor has a significant effect on the particle velocity in the shadow zone. Since Gaussian beam theory only considers a semi-infinite seafloor, the theoretical transmission loss will be greater than the measured transmission loss.

This research was supported by the National Natural Science Foundation of China (Grant No. 12204128).