# Array Gain of a Linear Array in an Ocean Waveguide
## 4 Numerical Simulation Results and Discussion
### Array Gain in the Deep Sea  
Simulation C considers the gain of the HLA in deep sea with water depth of 5000 m. The SSP of the deep sea is a Munk curve with an acoustic channel axis at 1300 m, as shown in Figure 4a. The source is at a fixed depth of 110 m, radiating a narrow-band signal with a center frequency of 190 Hz (the same as in the simulation B). The \(ATLs\) can be calculated by Kraken program (the calculation is the same as in simulation B), and the result within 100 km as a function of the sea depth in deep sea is shown in Figure 4b.  
The HLA is at the direction of end-fire, with a receiving depth of 120 m and 20 km away from the source. The sound channel transfer functions corresponding to the HLA are calculated by Kraken program. We calculate \(ATLs\) at a distance of 20 km as a function of element number, and the result is shown in Figure 5a. It can be observed from Figure 5a that \(ATLs\) fluctuate with element number greatly, and the fluctuation is greater than that in the simulation B (shallow water). The variance of \(ATLs\)' fluctuation on the HLA is calculated and equal to 1.92 dB, which is larger than that in simulation B. Submitting the acoustic channel transfer functions into Equation (12), the spatial correlation coefficients between all pairs of the acoustic channels are calculated as a function of element number, and the result is shown in Figure 5b. Comparing Figure 5b to Figure 3b, it is observed that the correlation coefficients of acoustic channels decrease rapidly as the element spacing  
Figure 3: Fluctuant acoustic channels and AGs for the receiving depth of 120 m and distance of 49 km in the shallow water: (**a**) \(ATLs\) on the elements; (**b**) Correlation coefficients; (**c**) \(AG_{\text{U}}\) (blue solid line), \(AG_{\text{W}}\) (red dash line) and the ideal value of \(AG\) (black solid line) varying with the element number.  
increases (the number of elements increases) and is smaller than those in the simulation B. Since \(AG\) is affected by \(ATL\)s and acoustic channel coherence, it can be inferred that the gain will deviate from the ideal value.  
Figure 4: Simulation parameters and sound propagation in deep sea: (**a**) The sound speed profile; (**b**) The average transmission loss.  
Figure 5: Fluctuant acoustic channels and AGs for the receiving depth of 120 m and distance of 20 km in the deep sea: (**a**) \(ATL\) s on the elements; (**b**) Correlation coefficients; (**c**) \(AG_{\text{U}}\) (blue solid line), \(AG_{\text{W}}\) (red dash line) and the ideal value of \(AG\) (black solid line) varying with the element number.  
Both \(AG_{\text{U}}\) and \(AG_{\text{W}}\) corresponding to the HLA are calculated as the function of element number, utilizing Equation (19) and Equation (20), respectively. The results are shown in Figure 5c. It can be observed that both \(AG_{\text{U}}\) and \(AG_{\text{W}}\) are less than the ideal value since there is a rapid degradation of acoustic channel coherence. Besides, \(AG_{\text{W}}\) is smaller than \(AG_{\text{U}}\) as a result of the fluctuation of \(ATL\)s, which is different from those in simulation B where \(AG_{\text{W}}\) is equal to \(AG_{\text{U}}\).