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:A method for analytically studying sound focusing in inhomogeneous waveguides is presented. From the viewpoint of acquiring the maximum acoustic pressure at an arbitrary position with normalized energy flux injection, optimal incident waves can be derived based on the multimodal admittance method. The method involves two steps. The first step is to expand the wave solution onto a complete orthogonal basis set so that the Helmholtz equation can be transformed into two sets of first-order coupled differential equations in the modal domain. The second step is to solve the coupled equations numerically by introducing admittance matrices and propagators, which can be used to derive reflection matrices and transmission matrices. Using the multimodal admittance method, one can circumvent the contamination caused by exponentially diverging evanescent modes and acquire stable wave solutions. Then the mapping between the acoustic pressure at an arbitrary position and that of the incident wave can be constructed, and this mapping changes the problem of wave focusing into solving the extrema of inner products in Hilbert space. The optimal incident waves that generate wave focusing at an arbitrary position can be readily computed together with the corresponding wave solutions. In this paper, we study the sound focusing in waveguides with varying cross-sections, scatterers and sound-speed profiles. The results show that the optimal incident waves will take full advantage of wave scattering caused by the boundaries and inhomogeneities during propagation to achieve the maximum pressure at foci, leading to good single-point and multi-point sound focusing performance. In addition, we find when injecting the spatially sampled optimal incident waves or the optimal incident waves with random perturbations, the resultant wave focusing phenomena will be still apparent. The focusing behaviors are highly robust to the perturbations of the moduli of the incident waves and slightly less robust to that of the arguments of the incident waves. Our method is also available for analyzing wave focusing in other kinds of inhomogeneous waveguides. We believe that our research can provide guidance on designing acoustic lenses or metamaterials to focus sound waves in complex media, and can offer inspiration in wave communications, imagings and non-destructive testing. Keywords:sound focusing/ inhomogeneous waveguides/ multimodal admittance method
综上, 可采用多模态导纳法将复杂波导中的声场求解问题转化为计算模态域的展开系数问题, 再引入导纳矩阵${{Y}}\left( x \right)$和传播算子${{M}}\left( x \right)$, 可以直接构建总声场与入射波的展开系数的映射关系, 这种直接明确的映射关系将空间任意位置处的声压表示为映射向量${{{Q}}^*}\left( {{x_0}, {y_0}} \right)$与入射波模态展开系数${{{p}}_i}\left( 0 \right)$的内积的形式, 求解声聚焦的问题最终转化为求解使该内积((15)式)达到极值的最佳入射波问题. 需要指出的是, 对于变截面波导, 本文选取的局部本征函数${\psi _n}\left( {x, y} \right)$并不满足实际的上边界条件, 也就是${\psi _n}\left( {x, y} \right)$在上边界满足的边界条件是${{\partial {\psi _n}} / {\partial y}} = 0$, 而不是真正的边界条件${{\partial {\psi _n}} / {\partial {n}}} = 0$, 这使(8)式的收敛速度不高(${1 / {{N^2}}}$). 文献[26]提出了改进多模态导纳法, 用以提高声场解的收敛性(提高为${1 / {{N^4}}}$). 方法为构造一阶新的本征函数${\psi _{ - 1}}\left( {x, y} \right)$, 使之既与原局部本征函数${\psi _n}\left( {x, y} \right) $$\left( {n = 0, 1, \cdots, N - 1} \right)$正交, 又在上边界满足非齐次Neumann边界条件. 但是, 由于构建的本征函数${\psi _{ - 1}}\left( {x, y} \right)$在传播方向上表现为衰逝模态[26], 且由(4)式, 发现衰逝模态对声聚焦的作用很小, 因此可忽略${\psi _{ - 1}}\left( {x, y} \right)$对声聚焦的贡献, 故本文选取的局部本征函数${\psi _n}\left( {x, y} \right)$足够满足声聚焦分析的要求. 总而言之, 对于非均匀波导, 只要求得散射区域内的导纳矩阵${{Y}}\left( x \right)$和传播算子${{M}}\left( x \right)$, 进而得到反射矩阵R及透射矩阵T, 即可给出声压与入射声压的模态展开系数的映射关系, 最终计算在任意点产生聚焦的最佳入射波. 图3给出了利用上述方法计算变截面波导中, 在不同位置处产生聚焦的声场. 入射波频率为$k = 29.1{\text{π}}$, 波导上边界表达式为$h(x) = 0.8 + $$0.2\cos \left( {{{2{\text{π}}x} / 3}} \right) $, ${x_{\max }} = 3$. 图3(a)和3(b)中的聚焦点分别位于透射区域和散射区域, 坐标为$\left( {{x_0}, {y_0}} \right) = $$\left( {3.2, 0.9} \right) $及$\left( {1.6, 0.2} \right)$. 图3(c)及图3(d)插图中的蓝色实线分别对应图3(a)及图3(b)中的入射波形, 均为由(16)式计算得到的最佳入射声波, 黑色点线为平面入射波; 主图中的蓝色实线为固定${x_0}$时, 声压幅值随高度的变化曲线, 即$\left| {p\left( {{x_0}, y} \right)} \right|$; 黑色点线为入射波是平面波时, 对应的$\left| {p\left( {{x_0}, y} \right)} \right|$. 如图3所示, 当入射声波是最佳入射波时, 不论聚焦点在散射区域还是透射区域, 声波利用边界的散射作用, 均在对应点处发生了聚焦, 并且聚焦点处的声压幅值明显大于入射波是平面波时的声压幅值, 聚焦效果良好. 图 3 (a)和(b)为变截面波导分别在$\left( {{x_0}, {y_0}} \right) = (3.2, 0.9)$(透射区域)及$(1.6, 0.2)$(散射区域)处产生聚焦的声场; (c)和(d)主图中的蓝色实线分别为(a)和(b)中${x_0}$处的声压幅值随高度方向的分布, 黑色点线为${p_i} = \varLambda {\psi _0}\left( y \right)$(平面波)时${x_0}$处的声压幅值分布; 插图中的蓝色曲线和黑色点线分别为最佳入射波及平面波的幅值曲线 Figure3. Acoustic focusing field in the waveguide as calculated by the present method. The foci are located at (a) $\left( {{x_0}, {y_0}} \right) = $ (3.2, 0.9) in transmission region and (b) $(1.6, 0.2)$ in scattering region, respectively. The blue solid lines in (c) and (d) are $\left| {p({x_0}, y)} \right|$ corresponding to (a) and (b), respectively, and the black dotted lines are $\left| {p({x_0}, y)} \right|$ generated by ${p_i} = \varLambda {\psi _0}\left( y \right)$(plane wave). The insets plot the modulus of the corresponding incident waves.
根据线性叠加原理, 可以实现非均匀波导中的多点声聚焦. 选取多个聚焦点位置, 利用(16)式分别获得对应的最佳入射波. 然后将这些入射波求和构建新的入射波并将其输入至波导中, 对应声场则产生多点声聚焦效应. 图4(a)给出多点声聚焦的声场, 其中频率的选取及波导结构与图3一致, 聚焦点位于$(3.2, 0.9)$及$(3.2, 0.1)$. 图4(b)画出了叠加后的总入射声压的幅值分布. 图4(c)中的黑色虚线、红色点划线和蓝色实线分别为只在$(3.2, 0.1)$处产生单点聚焦、只在$(3.2, 0.9)$处产生单点聚焦和同时在$(3.2, 0.1)$和$(3.2, 0.9)$处产生双点聚焦时$x = 3.2$处的声压幅值随y的分布. 可以看出双点聚焦时各个聚焦点处的声压幅值均低于单点聚焦的情况, 这是符合能量守恒定律的. 双点聚焦时各个聚焦点处的声压幅值均明显大于其他位置处的声压幅值, 说明输入计算得到的总入射波, 可以实现良好的多点声聚焦效果. 图 4 (a) 变截面波导中的双点聚焦声场, 聚焦点为$(3.2, 0.9)$及$(3.2, 0.1)$; (b) 最佳入射声压幅值分布; (c) 蓝色实线为(a)中声场在$x = 3.2$处的声压幅值分布; 红色点划线表示声波在$(3.2, 0.9)$处单点聚焦时的声压幅值分布, 与图3(c)中蓝色曲线一致; 黑色虚线为声波在$(3.2, 0.1)$处单点聚焦时的声压幅值分布. 频率和波导几何参数与图3一致 Figure4. (a) Sound two-point focusing field in the waveguide with varying cross-section, the foci are located at $(3.2, 0.9)$ and $(3.2, 0.1)$; (b) modulus of the optimal incident pressure; (c) the blue solid line represents $\left| {p(3.2, y)} \right|$ in (a); the red dot-dashed line shows $\left| {p(3.2, y)} \right|$ when the wave focus only at $(3.2, 0.9)$, which is same as the blue solid line in Fig. 3(c); and the black dashed line shows $\left| {p(3.2, y)} \right|$ when the wave focus only at $(3.2, 0.1)$. The frequency and geometries of the waveguide are same as Fig. 3.
23.2.含散射体波导中的声聚焦 -->
3.2.含散射体波导中的声聚焦
简单起见, 考虑波导中仅存在一个散射体时的声聚焦问题, 波导模型如图5所示, 散射体边界为$\left[ {y = a\left( x \right), y = b\left( x \right)} \right]$, 波导主介质的密度及声速为${\rho _1}, {c_1}$, 散射体内部介质的密度及声速为${\rho _2}, {c_2}$, 且${\rho _1}, {\rho _2}, {c_1}, {c_2}$均为常量. 波导介质区域与散射体占据区域分别用${\varOmega _1}$和${\varOmega _2}$表示. 声压满足的无量纲亥姆霍兹方程为 图 5 含散射体刚硬波导示意图 Figure5. Configuration of rigid waveguides involving a scatterer.