# Characteristics of Magnetic Fields Induced by the Wake of an Underwater Vehicle
## 4 Results and Discussions
### Characteristics of the Wake Magnetic Field
#### 4.2.2 Effect of Ship Speed on the Wake Magnetic Field  
We studied the relationship between the speed and wake magnetic field in three groups of different speed conditions: 2, 2.5, and 3 m. Among the above five measuring points, we selected monitoring point C to analyze the magnetic field change in the submarine before and after it passed through point C.  
Figure 12 shows the change in the magnetic field in the submarine passing through monitoring point C at different speeds. The time-domain curve of the left magnetic field shows that the submarine starts at t = 0, and the magnetic field increases gradually. At this time, the magnetic field was mainly produced by the flow field at the head of the submarine. When the submarine passed directly below the monitoring point, the wake magnetic field weakened. When the submarine passed through the monitoring point, the wake magnetic field gradually stabilized. The right magnetic field power spectrum shows that the magnetic field frequency of the submarine before and after passing through the monitoring point is less than the low-frequency signal of 1 Hz, which is likely to be confused with the wave or other noise signals; thus, it is difficult to detect.  
Monitoring point A was selected to analyze the time-domain and frequency-domain characteristics of the submarine wake magnetic field. Figure 13 shows the change in the magnetic field at monitoring point A when the submarine passed through it at different speeds. The time-domain curve of the left magnetic field shows that the larger the speed of the submarine, the larger the amplitude of the wake magnetic field is; moreover, the latter is positively correlated with the navigation speed. The power spectrum of the right magnetic field indicates that the larger the submarine's sailing speed, the smaller the characteristic frequency is. To better analyze the frequency domain characteristics of the wake magnetic field, we introduced the barycenter frequency and half-energy bandwidth. From Table 3, we observed that as the navigation speed increases, the barycenter frequency of the wake magnetic field decreases, and the half-energy bandwidth increases.  
According to relevant data, the traveling speed of conventional submarines in the cruising state is about 5-6 knots (corresponding to the speed of 3 m/s), and the speed of submarines at high speed is generally 20-30 knots (corresponding to the speed of 10-15 m/s). Therefore, when the submarine moves at the speed of 10-15 m/s, we can obtain the best results for changes in the amplitude of the wake magnetic field.  
According to the wave theory, the wave number \(k\) of the wake of an underwater vehicle is determined by ship speed and gravity and can be expressed as \(k\propto g/v^{2}\). The frequency correlation term of the wake fluctuation is \(e^{-j(\omega_{0}t+kx)}\). According to the dispersion relationship \(\omega_{0}=kv\), when an underwater vehicle navigates at speed \(v\), the frequency correlation term can be expressed as \(e^{-j(\frac{2k}{v}t)}\), while the magnetic field and wake fluctuation  
Figure 12: Magnetic field time domain and power spectrum of the submarine passing through monitoring point C at different speeds. (**a**) Time domain and (**b**) power spectrum.  
have the same frequency correlation term; therefore, the larger the speed, the smaller the frequency of the wake magnetic field is. Notably, the simulation results are in accordance with the theoretical calculation.