## 3 Analytical and Numerical Solutions  
### Case of Constant Density and Incompressible Hull  
Consider first the case of depth-wise constant density \(\rho(z)\equiv\rho_{0}\) and incompressible pressure hull \(\Delta V_{c}(z)\equiv 0\). Therewith, Equation (3) takes the form  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f(t)-\frac{C_{D}}{2l^{*}}\dot{z}(t )|\dot{z}(t)|];z(0)=0,\dot{z}(0)=0, \tag{4}\]  
At the first stage, separately consider the processes of acceleration and deceleration of the body. Assume that it begins submersion from the free surface from the state of equilibrium. Then, the problem for accelerated submersion takes the form  
\[\ddot{z}(t)=\frac{1}{1+\lambda}[g\eta\cdot f_{a}(t)-\frac{C_{D}}{2l^{*}}\dot{ z}^{2}(t)];z(0)=0,\dot{z}(0)=0, \tag{5}\]  
where the function \(f(t)\) at acceleration is adopted in the form \(f_{a}(t)=t/t_{0}\) for \(t\leq t_{0}\) and \(f_{a}(t)=1\) for \(t\geq t_{0}\), and \(t_{0}\) is the time of the buoyancy change. The acceleration of the body will continue until the moment it becomes equal to zero. Assuming that after this moment, \(t\geq t_{0},f(t)=1\), we obtain the speed \(\mathrm{v_{s}}\) at the beginning of the steady motion phase  
\[\mathrm{v_{s}}=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}. \tag{6}\]  
It follows from Equation (6) that for a given body, the speed of submersion during the stationary stage of motion is proportional to the square root from the relative volume of replacement. On the other hand, if the latter quantity is fixed, the speed of stationary submersion is proportional to the square root of the ratio \(l^{*}/C_{D}\). Assuming the buoyancy change is instantaneous (\(t_{0}=0\)), Equation (5) can be integrated in closed form. The dependences of the speed of submersion and vertical distance from the free surface were obtained in the form  
\[\dot{z}(t)=\sqrt{\frac{2 gl^{*}\eta}{C_{D}}}\tanh\left(\sqrt{\frac{g\eta C_{D}} {2l^{*}}}\frac{t}{(1+\lambda)}\right) \tag{7}\]  
\[z(t)=\frac{2l^{*}(1+\lambda)}{C_{D}}\ln\cosh\left(\sqrt{\frac{g\eta C_{D}}{2l^ {*}}}\frac{t}{(1+\lambda)}\right). \tag{8}\]  
To approximately determine the moment of time when the body motion becomes steady, solve Equation (7) with respect to time  
\[t=(1+\lambda)\sqrt{\frac{2l^{*}}{g\eta C_{D}}}\mathrm{tanh}^{-1}\left(\frac{ \dot{z}}{\mathrm{v_{s}}}\right). \tag{9}\]where is defined by the formula (6). Denote as \(t_{a}\) the time of completion of accelerated submersion when the magnitude of speed constitutes 0.99 of the maximum speed. Then,  
\[t_{a}=(1+\lambda)\sqrt{\frac{2l^{*}}{g\eta C_{D}}}\text{tanh}^{-1}(0.99)\cong 2.65(1+\lambda)\sqrt{\frac{2l^{*}}{g\eta C_{D}}}. \tag{10}\]  
The distance covered by that moment of time, determined with the same accuracy, can be found by means of the formula  
\[z_{ma}=\frac{2l^{*}(1+\lambda)}{C_{D}}\ln\cosh[\text{tanh}^{-1}\frac{\dot{z}(t _{ma})}{\text{v}_{\text{s}}}]=\frac{2l^{*}(1+\lambda)}{C_{D}}\ln\cosh(\text{ tanh}^{-1}(0.99))\cong 3.92\frac{l^{*}(1+\lambda)}{C_{D}}. \tag{11}\]  
As follows from Equation (11), the quantity \(z_{ma}\) does not depend on the volume of replacement. In Figures 1-4, the results obtained by means of Equations (7) and (8) are presented for the case of an elongated cylinder (\(l^{*}=l_{c},l_{c}/d_{c}=\kappa_{c}=10J_{c}=1\,\text{m}\), \(C_{D}=0.82\), \(\lambda_{c}<<1\)) and a sphere of identical volume. Figure 1: Submersion of the cylinder versus time in accelerated motion for an instantaneous change in buoyancy (\(t_{0}=0\)) and various magnitudes of \(\eta\), \(l_{c}=1\,\text{m}\). Figure 2: Speed of submersion of the cylinder versus time in accelerated motion for an instantaneous change in buoyancy (\(t_{0}=0\)) and various magnitudes of \(\eta\), \(l_{c}=1\,\text{m}\). It was assumed above that the buoyancy change takes place instantaneously. In reality, the buoyancy change occurs during a finite interval of time \(t_{0}\). Some results of the numerical solution of the acceleration problem shown in Equation (5) for the cylinder assuming the linear law of the buoyancy change are presented in Figures 5 and 6.