Abstract:To address the high numerical complexity and substantial computational cost involved in modeling underwater three-wave mixing acoustic fields based on the KZK equation, this paper develops a semi-analytical and semi-numerical method using first-order Lie–Trotter operator splitting. Leveraging the separable structure of the three-wave coupled equations, the method decomposes the evolution process into three sub-operators—nonlinear, dissipative, and diffractive components. The nonlinear term is solved using the finite difference method to capture energy exchange among the three waves, while the dissipative and diffractive terms are advanced analytically through closed-form expressions and the Hankel transform, thereby significantly reducing the need for fine spatial grids and large-scale matrix operations.For the pump-wave sound field, the Rayleigh integral solution is used as the reference to assess the accuracy of the proposed method, and the results show excellent agreement. For the generated difference-frequency field, an axisymmetric analytical solution of the KZK equation is adopted as the reference standard, and the finite difference method is additionally included in the comparison. The results demonstrate that the proposed method maintains high consistency with both the reference solution and the finite difference method while achieving a substantial improvement in computational efficiency. This confirms that the method provides an efficient and reliable numerical tool for underwater nonlinear acoustic field modeling and parametric array design.
