## 3 Analytical and Numerical Solutions
### Case of Variable Density and Compressible Hull  
Here, we perform a simplified analysis of submersion of a body equipped with BE considering depth-wise variation in density and hull compression. Depth-wise density distribution can be evaluated through corresponding profiles of temperature, salinity and pressure. In Figures 15 and 16, the typical density profiles for equatorial waters and Northern seas are presented. Taking account of the hydrostatic law of increase in pressure depth-wise, the linear dependence of the hull compression on \(z\) can be  
\[\eta_{c}(z)=\frac{\Delta V_{c}(z)}{V_{0}}=\eta_{c_{\rm max}}\cdot\frac{z}{H}, \tag{24}\]  
where \(H\) is the design maximal depth, and \(\eta_{c_{\rm max}}\) is the maximal relative hull compression. Quantity \(\eta_{c_{\rm max}}\) can be determined using the well-known boiler formulae of structural mechanics. Figure 14: Speed of submergence of the cylinder versus time for \(l_{c}=1\) m, \(H=200\) m, \(\eta=0.005\), \(t_{0}=20c\). For a cylinder,  
\[\eta_{c_{\max}}=\frac{p_{H}\cdot r_{c}}{E\cdot\delta_{c}}\left(\frac{5}{2}-2\
u_ {P}\right). \tag{25}\]  
For a sphere,  
\[\eta_{c_{\max}}=\frac{3p_{H}\cdot r_{s}}{E\cdot\delta_{s}}\left(\frac{1-\
u_{P} }{2}\right). \tag{26}\]  
In Formulas (25) and (26): \(p_{H}\) is the design pressure at depth \(H\); \(r_{c}\) and \(r_{s}\) are the radii of the cylinder and the sphere, correspondingly; \(\delta_{c},\delta_{s}\) are the thicknesses of the hull shell for the cylinder and sphere, correspondingly; \(E\) is the module of elasticity; \(\
u_{P}\) is the Poisson coefficient. It is assumed in this paper that for a given design pressure, the relative thickness of the hull shell is calculated based on yield strength \(\sigma_{T}\) using the formula  
Figure 16: Typical depth-wise density distribution in Northern seas. Figure 15: Typical depth-wise density distribution in equatorial waters. \[\frac{p_{H}\cdot r}{\delta}\approx\sigma_{T}. \tag{27}\]  
Some results of calculations for a steel cylinder and steel sphere are presented in Figures 17-20. Note that in the calculations it was assumed that the sphere has initial volume identical to that of the cylinder (equivalent sphere). It follows from equation of motion show in Equation (2) that, in a general case, there can exist hanging modes for which both acceleration and speed of the vehicle are equal to zero. Assuming in Equation (2) that \(\dot{z}=0\), \(\ddot{z}=0\) n \(f(t)=1\), we obtain the following equation of state of equilibrium of the vehicle. \[1-\overline{\rho}(z_{0})[1-\eta_{c}(z_{0})\mp\eta]=0 \tag{28}\]  
where \(z_{0}\) is the depth of hanging, if such a mode is realized, i.e., if there exists a solution to Equation (26). Restricting our analysis by the case of submersion, one comes to the  
Figure 17: Submersion versus time: equatorial waters, steel cylinder \(l^{*}=l_{c}=2\), \(C_{D}=0.82\), \(\eta=0.007\). Figure 18: Speed of submersion versus time: equatorial waters, steel cylinder, \(l^{*}=l_{c}=2\), \(C_{D}=0.82\), \(\eta=0.007\). following relationship between the relative (in this case, negative) volume of replacement, density profile and hull compression parameters in hanging mode. Figure 14:0  
where \(\Delta\rho(z)=\rho(z)-\rho_{0}\) is the increment of density at depth \(z\) with respect to the water density at the free surface \(\rho_{0}\). Figure 14:1 Submersion versus time: equatorial waters, steel sphere \(C_{D}=0.4\), \(\eta=0.007\). Figure 14:2 Speed of submersion versus time: equatorial waters, steel sphere, \(l^{s}=l_{s}=1/3\),\(C_{D}=0.4\), \(\eta=0.007\). Formally, it is possible to extract from Equation (27) a quantity \(\eta=\eta_{\max}^{*}(z_{0}^{*})\) corresponding to the maximal depth of hanging \(z_{0}^{*}\) for a given density profile and hull compression parameters. The maximal depth of hanging \(z_{0}^{*}\) can be found from the following condition. Figure 14:3  
To illustrate the hanging mode, in Figures 21 and 22, the corresponding dependences of submersion and its speed versus time for a steel cylinder are shown.