## 5 The Striations Interference Phenomenon and Its Formation Mechanism
### The Relationship between Striations Slope and Its Generation Mechanism  
In either the bistatic or monostatic configuration, the distribution of the interference striations was very regular, and there seemed to be a constrained relationship between the slope of the striations and the range and frequency of the source. Assuming the target was far away from the source, Figure 10a shows the relationship between the intensity of the sound field and the range when the frequencies were different. Figure 10b shows the relationship between the intensity of the sound field and the frequency of the source signal when the range changed.  
Figure 9: Comparison of interference striations and scattering function amplitudes: (**a**) the result of the monostatic configuration for 4–8 kHz; (**b**) the result of the bistatic configuration for 4–8 kHz; (**c**) the result of the bistatic configuration for 40–80 kHz.  
With the increase of the distance, the frequency increased. The value of \(e\) in Equation (5) was zero for the simulation results of the monostatic mode. The mathematical relationship for the waveguide invariant under the monostatic configuration is the same as that obtained for the passive sonar. For the monostatic simulation results in Figure 3, we obtained the values of the waveguide invariant under different distances and different frequencies as shown in Table 1.  
For the bistatic simulation results in Figure 4, the calculation results of the waveguide invariants are shown in Table 2. In the bistatic configuration, the slope value \(\Delta\omega/\Delta\tau\) is replaced by the slope of the tangent line for curved striations.  
Similar to the simulation, the value of the lake test striations waveguide invariant was calculated. Taking Figure 7b as an example, for the striations A (\(f\) was 40 kHz, r was 51 m, and \(\Delta\omega/\Delta\tau\) was 817 Hz/m), the calculated value of the waveguide invariant was 1.042. In the same way, the waveguide invariant value of the striations B was calculated to be 1.057. According to the above idea, in Figure 7, the value of the waveguide invariant for  
\begin{table}
\begin{tabular}{c c c c} \hline
**Distance (m)** & **Frequency \(\omega\) (Hz)** & **Slope \(\Delta\omega/\Delta\tau\)** & \(\beta\) \\ \hline
20 & 800 & 36.363 & 0.909 \\
28 & 800 & 26.667 & 0.933 \\
48 & 660 & 12.381 & 0.990 \\
26 & 8000 & 285.714 & 0.929 \\
43 & 8000 & 181.818 & 0.977 \\
48 & 5300 & 113.043 & 1.022 \\
17 & 80,000 & 4444.444 & 0.944 \\
29 & 80,000 & 2666.666 & 0.967 \\
48 & 61,000 & 1312.500 & 1.033 \\ \hline \end{tabular}
\end{table}
Table 1: The slopes and waveguide invariants for the monostatic configuration.  
Figure 10: The sound field intensity variation: (**a**) the variations at different distances; (**b**) the variations at different frequencies.  
the lake test was about 1.046 at 20\(-\)40 kHz, and in Figure 8, the value at 40\(-\)80 kHz was about 1.104.  
Comparing the data in the above table, it can be seen that the value of the waveguide invariant at long distances tended to be 1. When the frequency was the same, the slope \(\Delta\omega/\Delta\tau\) was small when the distance was large. When the distance was the same, the slope \(\Delta\omega/\Delta\tau\) was large when the frequency was large. That mean if any element between the distance and the frequency changed, the other element compensated for it, so that there was a restrictive relationship between the changes.