一种水平变化波导中声传播问题的耦合模态法

针对介质参数及海底边界水平变化波导中的声传播问题,本文基于多模态导纳法提出一种能量守恒且便于数值稳定求解的耦合模态方法.将声压表示为一组正交完备的本地本征函数之和,对声压满足的Helmholtz方程在本地本征函数上作投影,推导出关于声压模态系数的二阶耦合模态方程组.耦合矩阵直观描述水平变化因素对模态耦合的贡献.为避免直接求解二阶耦合模态方程组可能遇到的数值发散问题,将其重构为两个耦合的一阶演化方程组,引入导纳矩阵并使用Magnus数值积分方法获得稳定的声场解.利用该耦合模态方法数值计算水平变化波导中的声场,并与COMSOL参考解比较,结果表明该耦合模态理论能够精确求解水平变化波导中的点源及分布源传播问题.

A coupledj-mode method for sound propagation in range-dependent waveguides

The sound propagation problems in range-dependent waveguides are a common topic in underwater acoustics.The range-dependent factors,involving volumetric and bathymetric variations,significantly influence the propagation of sound energy and information.In this paper,a coupled-mode method based on the multimodal admittance method is presented for analyzing the sound propagation and scattering problems in range-dependent waveguides.The sound field is expanded in terms of a local basis with range-dependent modal amplitudes.The local basis corresponds to the transverse modes in a waveguide with constant physical parameters and constant cross section equal to the local cross section in the range-dependent waveguide.This local basis takes the advantage that it is easier to compute than the usual local modes which are the transverse modes in a waveguide with local physical parameters and constant cross-section equal to the local cross-section,especially for waveguides with complex environments.Projection of the Helmholtz equation that governs the sound pressure onto the local basis gives the second-order coupled mode equations for the modal amplitudes of the sound pressure.The correct boundary conditions are used in the derivation,giving rising to boundary matrices,in order to guarantee the conservation of energy among modes.The second-order coupled mode equations include coupled matrices and boundary matrices,which directly describe the effect of mode coupling due to contribution from volumetric variation(range-dependent physical parameters)and bathymetric variation(range-dependent boundaries).By introducing the admittance matrix,the second-order coupled mode equations are reduced to two sets of first-order evolution equations.The Magnus integration method is used to solve the first-order evolution equations.These first-order evolution equations allow us to obtain the numerical stable solutions and avoid the numerical divergence due to the exponential growth of evanescent modes.The numerical examples are presented for the waveguides with range-dependent physical parameters or rangedependent boundaries.The agreement between the results computed with the coupled mode method and COMSOL verifies the accuracy of the coupled mode method.Although the analysis and numerical implementation in this paper are based on two-dimensional waveguides in Cartesian coordinate system,it can be generally extended to study more complex waveguides.