A stochastic nonlinear differential propagation model for underwater acoustic propagation: Theory and solution

Abstract

• The perturb-coefficient nonlinear propagation equation is derived from the continuity equation, the Euler’s equation and the adiabatic equation. • The physical elements are divided into two types, and the location parameter to present the location-affected elements is designed. The stochastic parameter is designed to model the random-occur physical elements. The small parameters are used to demonstrate the weakly-nonlinear theory. • The initial and boundary conditions are analyzed. The solution existence of the SNDP model is proved. • The operator splitting procedure is proposed to solve the model numerically. The principle of underwater acoustic signal propagation is of vital importance to realize the “digital ocean”. However, underwater circumstances are becoming more complex and multi-factorial because of raising human activities, changing climate, to name a few. For this study, we formulate a mathematical model to describe the complex variation of underwater propagating acoustic signals, and the solving method are presented. Firstly, the perturb-coefficient nonlinear propagation equation is derived based on hydrodynamics and the adiabatic relation between pressure and density. Secondly, physical elements are divided into two types, intrinsic and extrinsic. The expression of the two types are combined with the perturb-coefficient nonlinear propagation equation by location and stochastic parameters to obtain the stochastic nonlinear differential propagation model. Thirdly, initial and boundary conditions are analyzed. The existence theorem for solutions is proved. Finally, the operator splitting procedure is proposed to obtain the solution of the model. Two simulations demonstrate that this model is effective and can be used in multiple circumstances.