1.Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China 2.Key Laboratory of Marine Information Acquisition and Security (Harbin Enhineering University), Ministry of Industry and Information, Harbin 150001, China 3.College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: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 range-dependent 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. Keywords:coupled mode method/ multimodal admittance method/ range-dependent waveguides
首先计算分布源激发声场. 环境参数选取为${\rho _1} = 0.2$, ${\rho _2} = 0.2$, ${c_1} = 0.2$, ${c_2} = 0.4$, ${h_1} = - 0.1$. $H = 200$ m, ${x_{\rm{r}}} = 1600$ m. 声源为从$x = 0$处向右入射的分布源${p_i} = \varepsilon {\phi _1}(y;0) = \sin (1.5{\text{π}}y/200)$, 其中$\varepsilon = 10$为入射波归一化系数. 频率$f = 20$ Hz. 在该频率下, 波导在$x = 0$ m处有五阶可传播模态, 在水平不变区域中有三阶可传播模态. 图2(a)为利用双向CMM((10)式)计算得到的声压幅值(Pa)分布, 截断数选取为$N=10$, 图2(b)为使用COMSOL计算得到的声压幅值(Pa)分布, 图2(c)为深度$y = 20$ m处声压幅值沿x轴的分布, 图2(d)为深度$y = 50$ m处声压幅值沿x轴的分布. 可以看出水平变化区域中存在明显的后向散射效应, 模态间耦合作用剧烈, 两种方法的计算结果一致. 图 2 水平变化波导中的声场(声源为分布源, 频率为20 Hz) (a) 利用CMM计算得到的声压幅值分布, 截断数$N = 10$; (b) 利用COMSOL计算得到的声压幅值分布; (c) 深度为20 m处, 声压幅值的水平分布; (d) 深度为50 m处, 声压幅值的水平分布 Figure2. Sound fields in a range-dependent waveguide (the source is a distributed source at 20 Hz): (a) Sound field computed by CMM where the truncation number is $N = 10$; (b) sound field computed with COMSOL; (c) sound field distribution along x direction at depth 20 m; (d) sound field distribution along x direction at depth 50 m.
其中$ {{{Y}}_{{x_s}}} = \sqrt {{{A}}({x_s})\left( {{{C}}({x_s}) - {{K}}{{(}}{x_s}{{)}}} \right)} .$ 图3所示为点源激发声场. 水平变化波导的环境参数为${\rho _1} = 0.2$, ${\rho _2} = 0.2$, ${c_1} = 0.2$, ${c_2} = 0.4$, ${h_1} = - 0.05$. $H = 200$ m, ${x_{\rm{r}}} = 1600$ m. 声源位于$(0, 10)$ m处. 频率为$f = 20$${\rm{Hz}}$. 图3(a)为利用双向CMM((10)式)计算得到的声压幅值(Pa), 截断数选取为$N = 50$, 图3(b)为使用COMSOL计算得到的声压幅值(Pa), 图3(c)为深度$y = 71$ m处声压幅值沿x轴的分布, 图3(d)为深度$y = 101$ m处声压幅值沿x轴的分布. 可以看出明显的后向散射作用. 两种方法的计算结果一致, 说明双向CMM能够准确计算水平变化波导中的声场. 两种方法的计算偏差来源于两方面: 1) CMM在数值实现中对级数求和(3)式作截断处理, 导致的部分和与真值之间的误差; 2)本文中采用Clenshaw-Curtis数值积分方法[26]计算系数矩阵及耦合矩阵, 离散点采用Chebyshev插值点, 而COMSOL计算采用三角形网格, CMM与COMSOL离散方式不同导致两种结果出现偏差. 图 3 水平变化波导中的声场(声源为位于$(0, 10)$ m处的点源, 频率为20 Hz) (a) 利用双向CMM计算得到的声压幅值分布, 截断数$N = 50$; (b) 利用COMSOL计算得到的声压幅值分布; (c) 深度为71 m处, 声压幅值的水平分布; (d) 深度为101 m处, 声压幅值的水平分布 Figure3. Sound fields in a range-dependent waveguide generated by a point source at $(0, 10)$ m (the frequency is 20 Hz): (a) Sound field computed by CMM where the truncation number is $N = 50$; (b) sound field computed with COMSOL; (c) sound field distribution along x direction at depth 71 m; (d) sound field distribution along x direction at depth 101 m.
23.2.水平缓变波导: 单向近似与绝热近似 -->
3.2.水平缓变波导: 单向近似与绝热近似
水平缓变是指环境参数在一个波长内的水平变化量远小于波长, 对于本节的计算模型, 水平缓变代表${\rho _1} \ll 1$, ${c_1} \ll 1$及${h_1} \ll 1$. 考虑水平缓变波导算例, 假设${\rho _1} = 0.01$, ${\rho _2} = 0.2$, ${c_1} = 0.01$, ${c_2} = $$ 0.4$, ${h_1} = - 0.001$, $H = 200$ m, ${x_{\rm{r}}} = 1600$ m. 声源为从$x = 0$处向右入射的分布源${p_i} = \varepsilon {\phi _1}(y;0) = $$ \sin (1.5{\text{π}}y/ $$ 200)$. 频率$f = 20$ Hz. 图4(a)—(c)分别为利用双向CMM理论((10)式)、单向近似CMM理论((17)式)和绝热近似CMM理论((18)式)计算得到的声压幅值(Pa)分布, 图4(d)和图4(e)分别为接收点在60 m和120 m深度处, 声压幅值的水平分布. 三种方法的数值实现采用相同的离散点和截断数N = 10. 可以看出, 单向近似和绝热近似耦合模态理论均可较为准确地计算水平缓变波导中的近距离声场, 但计算误差随水平距离的递推逐渐累积. 图 4 水平缓变波导中的声场(声源为分布源, 频率为20 Hz) (a) 利用双CMM计算得到的声压幅值分布; (b) 利用单向近似CMM计算得到的声压幅值分布; (c) 利用绝热近似CMM计算得到的声压幅值分布; (d) 深度为60 m处, 声压幅值的水平分布; (e) 深度为120 m处, 声压幅值的水平分布 Figure4. Sound fields in a waveguide with weak range dependence generated by a distributed source (the frequency is 20 Hz): (a) Sound field computed by two-way CMM; (b) sound field computed with one-way CMM; (c) sound field computed with adiabatic CMM; (d) sound field distribution along x direction at depth 60 m; (e) sound field distribution along x direction at depth 120 m.