## 4 Applications of 3D PFFT-BEM
### Prediction of Sound Scattering by Traveling Internal Waves  
Tidal interactions with bottom features often generate internal solitary waves (ISWs) in a shallow water environment. Previous experiments [26; 27; 28] found that acoustic signals fluctuate when sound waves are transmitted through ISWs. The amplitudes and phases of these fluctuations are highly correlated with those of ISWs. Quantifying the interactions between the acoustic waves and ISWs is important for shallow acoustics. Here, we compare our numerical simulation of sound propagate through ISW with the experiment data from SWRAM'95 (shallow-water acoustics in a random medium) which is conducted in the New Jersey continental shelf in 1995. The experiment setup and data are presented in detail by previous studies [26; 27; 28; 29; 30]. The purpose of including this example is to demonstrate the capability of using large-scale direct simulations in the prediction of acoustic scattering by internal waves, which is critically needed in shallow water acoustics.  
We plot the positions of the source, receiver and ISWs fronts during experiment in Figure 18a. During the interval 18:00 to 20:00 GMT on 4 August 1995, the source is fixed at \(z_{s}=12\) m below the sea surface at the position approximately 15 km from the receiver, as shown in Figure 18b [28]. The receiver is placed at \(z_{R}=45\) m below the sea surface. The broadband source frequency \(f\) was 30-160 Hz with three peak frequencies at 30, 60, and 90 Hz.  
Using the PFFT-BEM, we simulate the propagation of the acoustic waves through the internal waves. As shown in Figure 18b, a soliton wave packet model based on Duda and Preisig [31] is used in the 3D PFFT-BEM direct numerical simulation. The properties of the water column and sea bottom are shown in Figure 18b. To consider the change in the sound speed, we use four layers in the gradient region where the sound speed changes from 1530 to 1480 m/s. The properties of the bottom are \(c_{b}=1750\) m/s, \(\rho_{b}=1.8\) g/cm\({}^{3}\) with an attenuation parameter of \(\alpha=0.4375\) dB/\(\Lambda\). The velocity of the ISWs (in the y direction) is 0.65 m/s. Here, we ignore the differences between the speeds of separate solitary waves. As a result, the entire internal wave train is assumed to move at the same speed. The sound source frequencies used in the 3D PFFT-BEM are 30, 60 and 90 Hz.  
To demonstrate the accuracy of PFFT-BEM in simulating the acoustical scattering effects by the internal waves, we compare the acoustic pressure calculated using the 3D PFFT-BEM at the receiver with the field measurement during a 30 min period [28]. As shown in Figure 19, the numerical prediction by the PFFT-BEM agrees fairly well with the field experimental data for the time variation trend and the magnitude of the fluctuation of the received sound pressure. The difference is mainly because the internal waves are irregular during the experiment, while the regular wave model is used in our PFFT-BEM numerical model. The root mean square between the numerical prediction and experimental data is 3 %. For underwater sound measurements, the uncertainty of the receiver system  
Figure 18: (**a**) Field experimental arrangement of the source/receiver and their relative positions with respect to the internal wave fronts; (**b**) parameters of the PFFT-BEM simulation.  
in field conditions is approximately 0.5 dB [32]. As a result, a relative error of 3 % (i.e., 0.13 dB) would be sufficient for the underwater acoustic numerical simulation.