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.
在负声速梯度的作用下, 声波在传播时会向海底弯曲, 并且由于海底山状散射体的反向散射作用, 部分声能会被限制在$x \in [0, 20 h]$区域, 而选择的聚焦点$(35 h, 0.1 h)$既位于透射区域, 又处在海面附近, 直觉上声波似乎难以传播至聚焦点所在区域, 但输入计算得到的对应目标点的最佳入射波时, 实现了在该点良好的声聚焦现象. 由于散射体不再近似刚硬, 在图7(a)中可以发现散射体内部存在声压分布, 且密度和声速的不同改变了声波的波长. 最佳入射声波在实现聚焦的传播过程中充分地利用了边界的反射、散射体的散射以及声速梯度变化引起的介质折射率变化等因素, 最终在目标点处获得了最大声压. 4.讨 论基于多模态导纳法, 本文提出在非均匀波导中的任意位置处实现声聚焦的理论分析方法. 多模态导纳法可以简单直观地将亥姆霍兹方程变换为模态域的耦合偏微分方程组, 并能够有效地规避数值积分时衰逝模态可能带来的指数发散问题, 是一种准确高效求解非均匀波导声场的手段. 由于实际中难以把计算得到的精确最佳入射波作为输入, 故讨论当稀疏输入最佳入射波时的声聚焦效果. 以图3(a)中变截面波导的声聚焦为例, 分别以半波长和单倍波长为间隔对理论得到的最佳入射波(图3(c))进行空间采样, 获得稀疏入射波(图8(c)). 通过数值积分及适当补零得到对应的模态展开系数, 进而计算声场, 结果分别对应图8(a)及图8(b). 两个声场在聚焦点${x_0} = 3.2$处的幅值分布在图8(d)中给出, 并与理论值进行对比. 由图8可知, 半波长离散得到的稀疏输入产生的聚焦效果与理论情况基本一致, 原因是半波长离散后的输入可包含绝大部分原输入声波的信息(奈奎斯特采样定律), 从而能够产生良好的聚焦效果. 而以单倍波长离散得到的稀疏输入作为声源时, 该声源只能包含理论最佳入射波的部分信息, 故产生的聚焦效果对比理论情况有所下降, 但依然可观察到明显的聚焦现象. 因此, 在y方向稀疏输入最佳入射波时, 通过合理的离散, 仍可实现良好的声聚焦效果. 图 8y方向稀疏输入对聚焦结果的影响 (a) 对最佳入射波进行半波长采样后的聚焦声场;(b) 对最佳入射波进行单倍波长采样后的聚焦声场;(c)采样后的入射波幅值分布; (d) 聚焦点${x_0}$处的声压幅值分布. 蓝色实线为理论值, 与图3(c)中的蓝色曲线一致 Figure8. Sound focusing fields when the optimal incident wave is discretized: (a) Half-wavelength spacing; (b) single-wavelength spacing; (c) the moduli of the two spaced incident waves; (d) the red dashed line and the black dot-dashed line are the corresponding $\left| {p(3.2, y)} \right|$ generated by the incident waves in (c). The blue solid line is the theoretical result which is same as that in Fig. 3(c).
此外, 考虑当最佳入射波的幅值或相位存在误差时对声聚焦效果产生的影响. 依然以图3(a)为例, 首先固定最佳入射波的相位, 构造一个关于y的取值在$[0.5, 1.5]$均匀分布的随机函数, 将其与最佳入射波的幅度相乘(相当于对最佳入射波的幅度叠加了一个上限为$ \pm 50\% $的随机偏差)作为输入(图9(c)红色虚线), 进而求解声场(图9(a)). 接着, 固定最佳入射波的幅值, 类似地, 构造一个在$[ - {{\text{π}} / 2}, {{\text{π}} / 2}]$区间内均匀分布的随机函数叠加到最佳入射波的相位上作为输入(图9(c)黑色点划线)并求解声场(图9(b)). 两个声场在聚焦点${x_0} = 3.2$处的幅值分布在图9(d)中给出, 并与无扰动时的理论值对比. 从图9(a)和图9(d)可以看出, 当最佳入射波的幅值存在较大范围随机扰动时, 其产生的聚焦效果与理论情况基本一致. 而根据图9(b)和图9(d), 当最佳入射波的相位存在随机扰动时, 聚焦效果有所下降. 声源存在随机扰动时依然可产生聚焦的原因是波导中的衰逝模态能够抑制声波高阶振荡(与波长对比)成分的传播, 所以即便入射声波的幅值或相位存在随机误差, 经过衰逝模态的修正, 扰动后的入射波依然会在目标位置处实现声聚焦效应. 图9表明声聚焦对幅度的扰动具有强鲁棒性, 对相位的扰动兼具一定的稳健性. 图 9 (a) 最佳入射波的幅值存在随机扰动时的聚焦声场; (b) 最佳入射波的相位存在随机扰动时的聚焦声场; (c) 红色虚线与黑色点划线分别为幅值扰动与相位扰动后的入射波幅值分布; (d) 红色虚线与黑色点划线分别为(c)中的入射波在聚焦点${x_0}$处产生的声压幅值分布, 蓝色实线为理论值, 与图3(c)中的蓝色曲线一致 Figure9. Sound focusing fields when (a) the moduli and (b) the arguments of the optimal incident wave are perturbed; (c) the red dashed line is the incident wave with perturbed moduli, and the black dot-dashed line is that with perturbed arguments; (d) the red dashed line and the black dot-dashed line are the corresponding $\left| {p(3.2, y)} \right|$ generated by the incident waves in (c). The blue solid line is the result without perturbation which is same as that in Fig. 3(c).