Sound intensity fluctuations caused by internal solitary wave train in the South China Sea

Dynamic internal solitary wave (ISW) can cause significant underwater sound fluctuations. In this paper, the periodic characteristics of sound intensity fluctuations and related physical mechanisms are investigated in the scenario that an ISW train passing through the acoustic path from source to receivers. From simulation results and experimental data, it is shown that the sound intensity fluctuates quasi-periodically, and the predominant fluctuation frequency is associated with modal interference and modal intensity. The initial mode coefficients at the source and the mode coupling are responsible for the fluctuations of modal intensities, and one of the predominant fluctuation periods is the same as that of the ISWs.


Introduction
Internal waves are common dynamical processes in the ocean that lead to temporal and spatial variation of the structure of temperature and salinity fields, resulting in sound field fluctuations [1][2][3][4][5][6] .The characteristics of sound fluctuations in dynamic ISW environment is a hot topic, which has attracted the attention of both oceanographers and acousticians.
Some previous studies have investigated the periodic characteristics of sound intensity fluctuations when a single ISW propagates along the acoustic path [7][8][9] .In this case, the dynamic internal solitary wave can be regarded as a mode coupling matrix moving along the acoustic path, and the mainly physical mechanisms causing the sound field fluctuations is modal interference.In this paper, we focus on the periodic characteristics and related physical mechanisms of sound fluctuations in dynamic ISW train environment in the South China Sea.

Acoustic experiment descriptions
An underwater acoustic propagation experiment was conducted in the South China Sea (Figure 1(a)) from September 13 to 17, 2015, to measure the sound field fluctuations in the internal wave environment.A detailed introduction of the acoustic experiment is given in reference 10 [10] .This section provides a brief description of the acoustic experiment.
As shown in Figure 1(b), the acoustic source and the first vertical line array of temperature-depth sensors (VLA-TDs) were deployed at point S, and the acoustic source transmitted linear frequencymodulated signals with a centre frequency of 200 Hz and bandwidth of 50Hz.The vertical line array receivers (VLA-Rs) and the second VLA-TDs were deployed at point R. The third VLA-TDs was deployed at point H.Using the data measured by the three VLA-TDs, the Angle θ between the propagation direction of internal wave and the acoustic path SR was calculated to be 11.17°, and the average internal wave velocity is 0.76 m/s.The bottom of SR is flat, with depths ranging from 100m to 110m (Figure 1(c)).The acoustic source was deployed at a depth of 108m.

Environment model
Based on the above acoustic experimental descriptions and internal waves analysis, an ideal environment model for acoustic simulation was constructed, shown in Figure 3.The seafloor is set to be flat at a depth of 110m.Two-layers of seafloor are considered and geoacoustic parameters are set to be range-independent.The thickness of the sediment layer is 30m.The upper boundary of the ideal ISW   (, ) has a time-and range-dependent form where ∧= 30m ,  = 0.76 ×  = 0.75/ ,  0 and ∆= 200 m are the amplitude, propagation speed, centre range and width of the ISW.  0 is the upper boundary of the thermocline.In order to construct an ISW train, the horizontal distance between the two ISW centres is set to be 1km.The thickness of the thermocline layer is set at 40m, and the sound speed in the thermocline decreases uniformly from 1542m/s to 1522m/s.Thus, the vertical structure of the sound speed profile is described as

Simulation in an ideal ISW train environment
According to the environmental parameters set in section 2.2, the ideal background sound speed fields can be obtained, and Figure 4 shows the sound speed fields at two moments.At moment = 0, the acoustic path (0-14.8km) is time-and range-independent.Then, the ISWs begin to pass through the source (at range  0 = 0km) and propagate along the acoustic path with a constant shape and a propagation speed of 0.75m/s.
The time-domain sound pressure  of source frequency  at point (, ) was calculated using the parabolic equation theory model (Ram) with a time resolution of 30 seconds.During this period, the ISW train travelled a distance of 22.5m.The intensity of the sound field is defined as (, , , ) = |(, , , )| 2 .In order to study the periodic characteristics of the sound intensity fluctuations in dynamic ISW train environments, the time-domain acoustic field intensity can be converted into a frequency-domain spectrum   using the Fourier transform method where  ̅ is the time-averaged intensity where  is the angular frequency and the fluctuation frequency  is defined as  =  2 ⁄ in ℎ (cycle per hour).The frequency that plays a major role in the intensity fluctuations is called the dominant frequency  * , which corresponds to the peak in the power spectrum   .It is seen that there are some discontinuous highlight strips in Figure 5, which represent the predominant fluctuation frequencies  * .The highlight strips of the broadband source (50-250Hz) can be classified into types: the vertical highlight strips and the sloping highlight strips.In the vertical strips,  * does not vary with  and there are three predominant fluctuation frequencies  * = 1.64 ℎ, 2.7 ℎ and 5.47 ℎ (Figure 5(a), 5(c) and 5(e)).In the sloping highlight strips,  * decreases with increasing , and the sloping highlight strips is more obvious in the shallow layer (Figure 5 To present insight into the fluctuation characteristics at different time period, the spectrums were calculated using the depth-averaged intensity, by Fourier transform over a 1.85-hour time period (Figure 7).It is seen that there are also three highlight strips at  * = 1.64 ℎ, 2.7 ℎ and 5.47 ℎ.In addition, the highlight strip at  * = 2.7 ℎ is the most noticeable.As shown in Figure 3, the environment is time-independent at range   until the ISW train arrives at the VLA-Rs.Thus, the mode function ∅  (  , ) is independent of time.The modal amplitude   (  , ) can be obtained combining ∅  (  , ) and the time-domain sound pressure (  , , )

Simulation with measured environment and verification with experimental data
To construct the measured environment model, the measured bathymetry data of the acoustic path SR and the TD data from the VLA-TDs at point S are utilized.The time-domain TD data and a constant salinity profile are used to calculate the sound speed (Figure 9(a)).The shape and propagation speed of the ISWs are assumed to be constant.Therefore, the sound speed measured at different time correspond to different range.The depth-averaged sound intensity processed using the measured acoustic data, and its corresponding spectrums are presented in Figure 10.It can be seen that the depth-averaged sound intensity oscillates quasi-periodically (Figure 11(a)).In the time range of 10:19:00-12:19:00, a significant highlight strip (marked with an ellipse) also appears around  * = 2.7 ℎ , which corresponds well with the simulation results in Figure 9(a) and Figure 10, verifying the consistency of the predominant period between the sound intensity fluctuations and the ISWs.

Discussion
The complex sound pressure at the range   can be described as a coherent superposition of a series of normal modes (  , , ) = ∑   (  , )∅  (  , )   (6)   where   and ∅  are the local modal amplitude and mode function, respectively.According the coupled normal mode theory, the modal amplitude can be written as [11]   (  , ) = 1 ICFOST where   , ℎ  are the local eigenvalue and Hankel function, respectively. , and   are transmission matrix and initial mode coefficient, which are time-dependent in the ISW train environment.And, the mode coupling matrix are contained in  , .

Interpretation for predominant frequencies of intensity fluctuations
The fluctuations of the sound intensity are mainly due to the modal interference and modal intensity fluctuation (Equation ( 6)).The interference frequency of two acoustic modes at range   can be given be  , = ∆ , (  ) 2 ⁄ (8) where ∆ , (  ) =   (  ) −   (  ) is the eigenvalue difference of two acoustic modes  and . is the propagation speed of the ISW train.It is found that the frequency of modal interference is proportional to the propagation speed of the ISW train and independent of its the shape.The modal interference frequency curves and the fluctuation spectrums of the sound intensity at range   of two different depths are presented in Figure 12.
Comparing the modal interference curves and the highlight strips of the intensity fluctuation spectrums, it is seen that the modal interference frequency curves correspond well with the sloping highlight strips, which suggests that the predominant frequencies represented by the sloping highlight strips are mainly caused by the modal interference.The incoherent sound field are reconstructed as: (1) the modal amplitudes are calculated (according to Equation ( 5)) and their absolute values are taken; (2) the incoherent sound fields are calculated using the first 7 modes based on Equation (8).The intensity fluctuation spectrums of the incoherent sound field at depth 40m is presented in Figure 13.It is seen that there are three vertical highlight strips appear at  * = 1.64 ℎ, 2.7 ℎ and 5.47 ℎ, which is consistent with the vertical highlight strips in Figure 12(b).From Equation ( 8), it can be seen that the temporal variation of the modal amplitude is the only factor contributing to fluctuations of the incoherent sound field.Thus, it can be concluded that the modal intensity fluctuation is the mainly factor responsible for the predominant frequencies represented by the vertical highlight strips.

Relationship between modal intensity fluctuations and internal waves
As analyzed in section 4.1, the ISW train alters the acoustic modal interference structures and causes the modal intensity fluctuations, which are the two main reasons for the intensity fluctuation of the sound fields.The frequency of the acoustic modal interference is independent of the shapes of the ISWs and is proportional to their propagation speed.So, what is the relationship between the modal intensity fluctuations and the ISW train?
In the ideal ISW train environment, the distance between two adjacent ISW crests is 1km, and the propagation speed of the ISW is 0.75 m/s.Therefore, the length of time required for a single ISW to fully enter the acoustic path is approximately 22.22 minutes, i.e., the period of the ISW is 22.22 minutes (or a frequency of 2.7 cph).During the time period 0-5.48 h, 15 ISWs passed through the source (at range 0 km) and propagated along the acoustic path, and the crest position of each ISW is marked with symbol " X " in Figure 14(a).In the same time period, the sum of the modal intensities of the 200Hz sound field  1− (at range   ) oscillate quasi-periodically, and there are also 15 peaks corresponding to the crests of the ISWs (Figure 14(b)).There are some highlight strips in the normalized spectrums   1− (Figure 14(c)), and it is seen that  * = 2.7 ℎ is the most noticeable predominant fluctuation frequency.
During the time period 11:19:00-13:49:00, eight relatively regular and strong ISWs (measured data from the VLA-TDs) passed through the source (Figure 15(a)), and the averaged period of them is about 22 minutes. 1− were obtained from the simulated sound field in the measured ISW train environment, and it is seen that there are also 8 peaks also appear on the  1− curve (Figure 15

Figure 1 .
Figure 1.Schematic diagram of the acoustic experiment.(a) acoustic experiment area and bathymetry contours, (b) geometry of the acoustic source and receiver deployment locations (top view), and (c) geometry of the underwater acoustic source and receiver locations (side view).During the acoustic experiment, strong ISW trains propagated from the source to the VLA-Rs.Figure 2 shows the temperature data measured at point S over a period of time.It is seen that there are three strong ISW trains passed through the point S to propagated toward the point R along the acoustic path SR (Figure 2(a)), and the time interval between the ISW trains is approximately 12 hours.The ISW trains resulted in significant perturbations to the temperature structure.The depth variations of 29℃ isotherms can be up to 30m (Figure 2(b)).

Figure 2 .
Figure 2. Time-domain temperature data measured by the VLA-TDs at the point S.

Figure 3 .
Figure 3. Schematic diagram of the ideal environment model for acoustic propagation simulation.

Figure 4 .
Figure 4.The background sound speed fields at = 0 (a) and = 4.44 hours (b) in the ideal ISW train environment.3.1.1.Broadband simulation.According to the propagation speed of the ISW train v = 0.75m/s and the horizontal distance of the acoustic path r e = 14.8km, it takes 5.48 hours for the ISW train to propagate from the source to the VLA-Rs.The fluctuation spectrums were obtained using the timedomain sound intensity sequences over the entire 5.48-hour time period.Then, the fluctuation spectrums of different source frequencies were normalized to [0, 1], and the simulation results are presented in Figure 5.
(a)).For narrowband sound fields (175-225Hz, Figure5(b), 5(d) and 5(f)), the vertical strips predominate, which means that  * remains almost constant as  increases.3.1.2.200Hz simulation and modal analysis.The fluctuation spectrums of 200Hz sound shown in Figure 6, were calculated over the entire 5.48-hour time period.It is seen that there are three obvious discontinuous highlight strips at F * = 1.64 cph, 2.7 cph and 5.47 cph (Figure 6(a)), and there are also three inflexion points at the same fluctuation frequencies on the depth-averaged curve (Figure 6(b)).The discontinuous characteristics in the highlight strips are mainly due to acoustic modal interference.And the highlight strips disappear at shallow depths, which is because that the acoustic intensity is small compared to the middle and deep depths.

Figure 7 .
Figure 7. Depth-averaged sound intensity spectrums versus time and fluctuation frequency.
(  , ) = ∫ ∅  (  , )   0 (  , , ) (5) where  is the acoustic mode number. and  are depth and density, respectively.The modal intensity   (  , ) = |  (  , )| 2 , and the sum of modal intensity  1− (  , ) = ∑   (  , )  .As more and more ISWs pass through the source and propagate along the acoustic path,  1− oscillates quasi-periodically with time (Figure 8(a)).Meanwhile, the overall variation trend of  1− is decreasing with time, which is because more ISWs result in drastic mode coupling and energy transfer.The energy of the sound field at range   are mainly from the first 6 modes.It is seen that the spectrums of the modal intensity also have three inflexion points at  * = 1.64 ℎ, 2.7 ℎ and 5.47 ℎ (Figure 8(b)), which is consistent with the sound intensity fluctuations.

Figure 8 .
Figure 8.The sum of modal intensity versus time (a) and intensity spectrums of different mode (b).

Figure 9 (
b) and 9(c) show two snapshots of the sound speed fields constructed from the measured data.

Figure 10 .
Figure 10.Depth-averaged sound intensity spectrums versus time and fluctuation frequency.The depth-averaged sound intensity processed using the measured acoustic data, and its corresponding spectrums are presented in Figure10.It can be seen that the depth-averaged sound intensity oscillates quasi-periodically (Figure11(a)).In the time range of 10:19:00-12:19:00, a significant highlight strip (marked with an ellipse) also appears around  * = 2.7 ℎ , which corresponds well with the simulation results in Figure9(a) and Figure10, verifying the consistency of the predominant period between the sound intensity fluctuations and the ISWs.

Figure 11 .
Figure 11.Depth-averaged intensity in the time-domain of the 200 Hz measured sound field (a) and its corresponding spectrums (b).(Measured acoustic data).

Figure 12 .
Figure 12.Modal interference frequency curves and sound intensity fluctuation spectrums at depths  = 10 (a) and  = 40 (b).The subplots (a) and (b) correspond to Figure 5(a) and Figure 5(c),respectively.The modal phase leads to the modal interference.In order to exclude the effect of modal interference on sound field fluctuations, the incoherent sound field is reconstructed using the absolute value of the modal amplitudes

Figure 13 .
Figure 13.Intensity fluctuation spectrums of the noncoherent sound field at range   and depth 40m.
(b)), which correspond well to the ISWs' crests.The highlight strips in Figure 15(c) are comparatively complex, but it still can be seen that the noticeable highlight strip at  * = 2.7 ℎ.