Interaction between the ocean surface and underwater spherical blast waves

The object of this analysis is to obtain the solution for the flow field resulting from an explosion released at a small depth beneath the ocean surface. At time $\text{t\hspace{0.17em}=\hspace{0.17em}0}$ , the explosion produces a spherical blast wave. When the blast wave reaches the surface, a shock wave is transmitted into the atmosphere and an expansion wave is reflected back into the water region. The solution for the flow variables is given by Taylor series expansions in the three independent variables: $r$ , distance in the radial direction; $z$ , distance in the axial direction; and time, ${\mathrm{t\hspace{0.17em}-\hspace{0.17em}t}}_0$ . The series expansions are different in each of the following flow regions: (1) the air region disturbed by the transmitted shock wave, (2), the water region between the expansion wave and the ocean surface, and (3) the expansion region. Coefficients for the series expansions in region 3 are determined by introducing a family of surfaces ${\mathrm{z\hspace{0.17em}=\hspace{0.17em}(r,\hspace{0.17em}t,\hspace{0.17em}N}}_0)$ expanded in series for $r$ and ${\mathrm{(t\hspace{0.17em}-\hspace{0.17em}t}}_0)$ with parameter $N_0$ . The coefficients of the series expansions for the flow variables along the surfaces are determined by the integration across the expansion wave of a system of differential equations obtained from wave relations and the equations of motion. Numerical results are obtained for the case of a pressurized sphere of gas with an initial radius to depth ratio of $\frac{1}{3}$ and an initial pressure of 9000 atm.