Probability Density Function of Underwater Bomb Trajectory Deviation Due to Stochastic Ocean Surface Slope

Ocean wave propagation causes random change in an ocean surface slope and in turn affects the underwater bomb trajectory deviation $(r)$ through a water column. This trajectory deviation is crucial for the clearance of obstacles such as sea mines or a maritime improvised explosive device in coastal oceans using bombs. A nonlinear six degrees of freedom (6DOF) model has been recently developed and verified at the Naval Postgraduate School with various surface impact speeds and surface slopes as model inputs. The surface slope $(s)$ randomly changes between 0 and $π/2$ with a probability density function (PDF) $p(s)$⁠, called the $s$-PDF. After $s$ is discretized into $I$ intervals by $s1,s2,…,si,…,sI+1$⁠, the 6DOF model is integrated with a given surface impact speed $(v0)$ and each slope $si$ to get bomb trajectory deviation $r̂i$ at depth $(h)$ as a model output. The calculated series of ${r̂i}$ is re-arranged into monotonically increasing order $({rj})$⁠. The bomb trajectory deviation $r$ within $(rj,rj+1)$ may correspond to one interval or several intervals of $s$⁠. The probability of $r$ falling into $(rj,rj+1)$ can be obtained from the probability of $s$ and in turn the PDF of $r$⁠, called the $r$-PDF. Change in the $r$-PDF versus features of the $s$-PDF, water depth, and surface impact speed is also investigated.