# Array Gain of a Linear Array in an Ocean Waveguide
## 4 Numerical Simulation Results and Discussion
### Array Gain in the Upslope Waveguide  
Finally, \(AG\) is investigated in the upslope waveguide where the waveguide is range-dependent and the acoustic channels have large fluctuations. The waveguide is shown in Figure 6a, including an abyssal plain (distance of 2 km with water depth of 5000 m), a continental slope (in the 2 to 90 km range, the oblique angle of the bottom is 3.5\({}^{\circ}\)) and shallow water (distance of 20 km with water depth of 229 m). The SSP of shallow water is a negative gradient, and the SSP of the abyssal sea is the standard Munk curve with a deep sound channel axial at 1300 m. The SSPs of the continental slope region which are range-dependent are shown in Figure 6b. Figure 6: The upslope waveguide: (**a**) Simulation environment and parameters; (**b**) The SSPs of the continental slope region. The source is at a fixed depth of 550 m, radiating a narrow-band signal with a center frequency of 190 Hz. The \(ATL\)s are calculated by RAM program developed on parabolic equations method which is the well-known technique for solving range-dependent propagation problems [25] (the calculation is the same as in simulations B and C). \(ATL\)s within 100 km in the upslope waveguide are shown in Figure 7.  
The HLA is suspended at a depth of 120 m, 48 km away from the source (at the direction of end-fire). The \(ATL\)s at distance 48 km are calculated as the function of element number which are displayed in Figure 8a. As can be seen from the Figure 8a, \(ATL\)s have rapid fluctuation compared to simulations B (Figure 3a for shallow water) and C (Figure 5a for deep sea). The variance of \(ATL\)s' fluctuation is 30.55 dB, which is larger than those in simulations B and C.  
The correlation coefficients between two arbitrary acoustic channels corresponding to the elements are shown in Figure 8b. It is observed that when the number of elements is less than 112, the acoustic channels are completely coherent, and the correlation coefficient between any pair of acoustic channels is larger than 0.8. As the number of elements continues to increase (the space between the elements increases), the acoustic channel coherence decreases rapidly. We investigate \(AG_{\text{W}}\) and \(AG_{\text{U}}\) of the HLA in the upslope waveguide. Firstly, for Case 2, ignoring the fluctuation of \(ATL\)s, \(AG_{\text{U}}\) is calculated as the function of element number utilized in Equation (19) (equivalent to \(AG_{\text{U}}\)), as shown by the blue solid line in Figure 8c. Then, for Case 3, \(AG_{\text{W}}\) is calculated by substituting the acoustic transfer functions into Equation (20), and the result is displayed by the red dash line in Figure 8c. It is observed that \(AG_{\text{U}}\) are almost equal to the ideal value when the element number is less than 112. Besides, \(AG_{\text{W}}\) is also close to the ideal value, however, \(AG_{\text{W}}\) is smaller than \(AG_{\text{U}}\), due to the fluctuation of \(ATL\)s on the elements (the variance of \(ATL\)s' fluctuation on 112 elements is 11.6 dB). When the number of elements is more than 112, as the acoustic channel coherence degradation, both \(AG_{\text{W}}\) and \(AG_{\text{U}}\) are lower than the ideal value. In addition, it is observed that \(AG_{\text{W}}\) is less than \(AG_{\text{U}}\), and the deviation of \(AG_{\text{W}}\) from the ideal value is more severe. This is because the \(ATL\)s fluctuate more severely when the number of elements keep increasing (more than 112), as shown in Figure 8a. Figure 7: Average transmission loss in the upslope waveguide when the source depth is 550 m.  
From the above simulations, it can be inferred that \(AG\) is determined by both transmission loss and acoustic channel coherence. For the completely coherent acoustic channels, \(AG\) is almost equal to the ideal value. The result of traditional method for \(AG\) in ocean waveguide is too large. The reason is that the effect of the transmission loss fluctuation with elements on \(AG\) is ignored in the traditional method. The numerical results verify the theory in Section 3.