## 4 Simulation of Low-Frequency Communication in Plasma Sheath
### Propagation Characteristics of Low-Frequency EM Waves in Plasma Sheath  
To improve the collision frequency in a short time, firstly, we find that the maximum energy absorption of plasma in the state of thermodynamic non-equilibrium and chemical non-equilibrium is reached at a certain fixed frequency (this frequency is set to the maximum absorption frequency of 24 GHz on the second section of the head stagnation line). Secondly, the power density of HPM should be improved. Using HPM technology that already exists, it is simpler to deploy equipment with increased power density [19]. Therefore, we set the frequency of the HPM to 24 GHz to match the maximum absorption frequency on the stagnation line. During the fluid simulation, the flow direction of the fluid is set to the \(y\)-direction. In this case, when the polarization direction of the HPM pulse is set to the direction perpendicular to the \(y\)-direction, the electron number density in the channel can be minimized. Therefore, the HPM pulse was set as \(x\)-direction polarized harmonic plane waves  
\[\textbf{E}=\hat{x}E_{0}(2\pi ft), \tag{20}\]  
where \(E_{0}\) and \(f\) are the amplitude and the frequency of the incident EM waves, respectively. For the selection of HPM energy density, we set three HPM with higher energy densities, namely, HPM-1 = \(1.0\times 10^{8}\) W/m\({}^{2}\), HPM-2 = \(2.0\times 10^{8}\) W/m\({}^{2}\), and HPM-3 = \(3.0\times 10^{8}\) W/m\({}^{2}\), to analyze how different energy densities affect the dielectric loss tangent of the plasma. In order to study the change of dielectric loss tangent of the plasma, it is better not to choose the front and rear ends of the plasma sheath when selecting the point of the plasma sheath. We choose the middle part of the stagnation zone. As shown in Figure 4, the point \(x=-0.17\) m is located in the middle part of the stagnation zone, and its internal EM parameters change relatively smoothly. As shown in Figure 8, it can be seen that with the passage of time, the larger the energy density, the more advantageous it is to reduce the dielectric loss tangent of the plasma. We take the HPM-2 in Figure 8 as an example to analyze the relationship among collision frequency, dielectric loss tangent of the plasma, and the increment value of electron temperature in the plasma. According to the values of the plasma frequency and the collision frequency of \(x=-0.17\) m, and in (19), it can be known that when the collision frequency is greater than 601.4 GHz, the plasma's dielectric loss tangent steadily diminishes. However, when the dielectric loss tangent of the plasma reaches the initial value with the increase in collision frequency, according to (4), the increment value of the plasma electron temperature converted by the HPM is 3037 K. This means that the dielectric loss tangent of the plasma will begin to decrease, and the effect of absorbing low-frequency EM waves will begin to weaken only when the electron temperature in the plasma is greater than 3037 K. As shown in Figure 8, where \(t=22\) ns is the time, the frequency of collisions hits 601.4 GHz. After the time \(t\), the dielectric loss angular tangent value begins to rapidly decrease compared to the initial value at this stage, and the loss of plasma to EM waves decreases over time. We select four points of time \(t_{1}=15\) ns, \(t=22\) ns, \(t_{2}=40\) ns, and \(t_{3}=60\) ns in Figure 8, respectively. \(\omega_{p}\) and the \(v_{c}\) of the plasma sheath at these four time periods are stored individually to investigate changes in the plasma sheath's transmittance and reflectance to EM waves of various frequencies following HPM irradiation. At this time, we select a point (not included in the plasma sheath) at the front and back of the stagnation area of the plasma sheath to study the reflection and transmission of plasma sheath. As a corollary, we add EM waves with frequencies ranging from 0.1 to 100 GHz to examine transmission and reflection at four times in time for EM waves with varied frequencies, as shown in Figure 9a,b.