## 4 Results
Next, we studied the pressure  
Figure 4: Impact of TSS on the flow past the body: dynamics of the density fields, M\({}_{\infty}=2\), without energy source—_blue_, with TSS, \(\alpha_{j}=0.3\); \(N=2.5\)—_red_; (**a**) \(t=9.0\), isochores; (**b**) \(t=20.0\), isochores; (**c**) interaction of TSS with the BSW (enlarged, isochores). The distance between isochores is 0.05; (**d**) without TSS, \(t=20.0\), colors and isochores; (**e**) with TSS, \(t=20.0\), colors and isochores. profiles for \(t=20.0\) at the level of \(r=4.0\) (near-field pressure signatures--NFPS) when these pressure profiles are close to stationary in time. The relative pressure profiles \(\Delta p/p_{\infty}\) at the level \(r=4.0\) (NFPS) for TSS with different values \(\alpha_{j}\) are presented in Figure 6a. It can be seen from the enlarged image (Figure 6b) that the smaller \(\alpha_{j}\) in the TSS (the higher the temperature in the layers), the greater the effect of this TSS on the BSW front. One can also conclude that this effect causes a decrease in pressure at the BSW front. Figure 5: Dynamics of profiles of relative pressure \(\Delta p/p_{\infty}\) for \(r=4.0\), M\({}_{\infty}=2\), in TSS \(\alpha_{j}=0.3\), \(N=2.5\). Figure 6: \(Cont.\)  
The density fields in isochores of the steady flows at \(t=20\) and \(\alpha_{j}=0.15\) under the action of the TSSs with the number of layers \(N=3.5\) (\(R_{s}=0.5R\)), \(N=5.5\) (\(R_{s}=0.8R\)), and \(N=7.5\) (\(R_{s}=1.1R\)) are presented in Figure 7a,b, Figure 8a,b and Figure 9a,b, accordingly (in red). The distance between isochores is 0.05. The enlarged images of the density fields during the interaction of different TSSs with the BSW are also shown there. For comparison, the corresponding dynamics of the density fields without the impact of the TSS are shown (in blue) as well. In Figure 7c, the density field for the steady flow mode (\(t=20.0\)) in the absence of TSS, in colors and isochores is presented. In Figures 7d, 8c and 9c, the corresponding density fields under the TSS impact, in colors and isochores are presented. In the behavior of the density fields for a different number of layers in the TSS, the same trend can be obtained as for \(N=2.5\) (Figure 4); namely, the influence of the TSS is localized in the area immediately adjacent to the body (and behind it). Figure 7: Cont. Figure 8: _Cont._  
Figure 7: Impact of TSS on the flow past the body: dynamics of the density fields, M\({}_{\infty}\) = 2, without energy source—_blue_, with TSS, \(\alpha_{j}\) = 0.15; \(N\) = 3.5—_red_; (**a**) \(t\) = 20.0, isochores; (**b**) interaction of TSS with the BSW (enlarged, isochores); (**c**) without TSS, \(t\) = 20.0, colors and isochores; (**d**) with TSS, \(t\) = 20.0, colors and isochores. The enlarged images in Figures 4c, 7b, 8b and 9b show that the front of the BSW in the TSS region is almost completely blurred (see the red contours). This is taken place due to the influence of three factors that distinguish the effect of TSS from the effect of homogeneous energy sources. Firstly, this is a manifestation of the RMI in multiple points at the initial stage of interaction, which smears the BSW [15,16]. Secondly, the influence of the heated layers interspersed with cold gaps between them changes the shape of the BSW and gives it a wavy appearance, which also contributes to the blurring of its front. The third one is, of course, that the BSW also weakens simply due to a change in the Mach number inside the heated layers. This smearing of the front under the action of the TSS is further influenced the pressure amplitudes in the near-field signature and, finally, may possibly affect the pressure amplitudes in the ground signature and the PLdB on the ground. Figure 8: Impact of TSS on the flow past the body: dynamics of the density fields, M\({}_{\infty}\) = 2, without energy source—_blue_, with TSS, \(\alpha_{j}\) = 0.15; \(N\) = 5.5—_red_; (**a**) \(t\) = 20.0; (**b**) interaction of TSS with the BSW (enlarged); (**c**) with TSS, \(t\) = 20.0, colors and isochores.