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周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移

潘俊杰 刘焕文 李长江

潘俊杰,刘焕文,李长江. 周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移 [J]. 应用数学和力学,2022,43(3):237-254 doi: 10.21656/1000-0887.420123
引用本文: 潘俊杰,刘焕文,李长江. 周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移 [J]. 应用数学和力学,2022,43(3):237-254 doi: 10.21656/1000-0887.420123
PAN Junjie, LIU Huanwen, LI Changjiang. Bragg Resonance and Phase Upshift of Linear Water Waves Excited by a Finite Periodic Array of Parabolic Trenches[J]. Applied Mathematics and Mechanics, 2022, 43(3): 237-254. doi: 10.21656/1000-0887.420123
Citation: PAN Junjie, LIU Huanwen, LI Changjiang. Bragg Resonance and Phase Upshift of Linear Water Waves Excited by a Finite Periodic Array of Parabolic Trenches[J]. Applied Mathematics and Mechanics, 2022, 43(3): 237-254. doi: 10.21656/1000-0887.420123

周期排列抛物形系列沟槽引起的线性水波Bragg共振及共振相位上移

doi: 10.21656/1000-0887.420123
基金项目: 国家自然科学基金(11572092; 51879237);浙江省高校“钱江学者”人才基金
详细信息
    作者简介:

    潘俊杰(1995—),男,硕士(E-mail:865171184@qq.com)

    刘焕文(1963—),男,教授,博士,博士生导师(通讯作者. E-mail:liuhuanwen@zjou.edu.cn)

  • 中图分类号: O357.41

Bragg Resonance and Phase Upshift of Linear Water Waves Excited by a Finite Periodic Array of Parabolic Trenches

  • 摘要:

    该文解析研究了有限个周期排列的抛物形沟槽激发的水波Bragg共振。首先, 利用变量替换, 先将系数为隐函数的修正缓坡方程(MMSE)转化为系数为显函数的显式方程。然后,构造了修正缓坡方程的Frobenius级数解, 并给出了级数解的收敛条件。最后,利用质量守恒的耦合条件, 建立了反射系数的解析公式。根据反射系数的解析公式, 分析了沟槽个数、沟槽深度与宽度对Bragg共振峰值、共振相位和共振带宽的影响。当沟槽深度和宽度固定而沟槽个数增加时, 共振峰值逐渐增大并趋向于1, 而共振带宽则逐渐变窄并趋于固定值。当沟槽个数和宽度固定时, Bragg共振峰值随沟槽深度增加而增加。当沟槽个数和深度固定时, Bragg共振反射峰值随沟槽宽度增加而先增后减, 预示了沟槽存在某个宽度使得共振峰值达到最大, 为Bragg共振反射针对沟槽宽度的优化奠定了理论基础。特别地, 前不久在有限个周期排列旋轮线形沟槽上刚刚观察到的Bragg共振反射峰值相位的上移现象, 再次在该文考虑的抛物形沟槽上得到确认, 表明针对有限周期排列的沟槽地形, Bragg共振反射峰值的相位上移是一个普遍现象。也因此说明, 凡是正弦沙纹和周期人工沙坝所激发的Bragg共振反射, 其主振相位将会下移, 而凡是周期系列沟槽所激发的Bragg共振反射, 无论沟槽形状如何, 其主振相位都将上移。另外,我们从Bragg共振的原始定义出发,定量地解释了相位上移发生的数学机理。

  • 图  1  有限个周期排列的全等抛物形沟槽示意图

    Figure  1.  A schematic diagram for a finite periodic array of identical parabolic trenches

    图  2  方程(13)除$ K\tanh\; K=K_1\tanh\; K_1 $的虚根外的所有奇异点

    注 为了解释图中的颜色,读者可以参考本文的电子网页版本,后同。

    Figure  2.  Singularities of eq.(13) except for the imaginary roots of $ K\tanh \; K = K_1\tanh \; K_1 $

    图  3  级数$ \xi_1(K) $$ \xi_2(K) $的收敛区域(即阴影区域)以及两个例子的收敛性分析

    Figure  3.  Convergent regions of $ \xi_1(K) $ and $ \xi_2(K) $ and the convergence analysis in 2 examples

    图  4  采用本文解析模型和基于Laplace方程的边界元模型[43]分别计算得到的两个解的比较

    Figure  4.  Comparison between the present solution and the BEM solution[43] to Laplace’s equation

    图  5  本文解析解与应用Miles的通用公式(55)的比较

    Figure  5.  Comparison between the present solution and the solution based on Miles’ formula (55)

    图  6  抛物形沟槽个数对Bragg共振反射及共振带宽的影响

    Figure  6.  Influences of the number of trenches on the Bragg resonance and the resonance bandwidth

    7  沟槽深度对Bragg共振的影响

    7.  Influences of the trench depth on the Bragg resonance

    图  8  沟槽宽度对Bragg共振的影响

    Figure  8.  Influences of the trench width on the Bragg resonance

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出版历程
  • 收稿日期:  2021-05-04
  • 录用日期:  2021-05-04
  • 修回日期:  2021-09-30
  • 网络出版日期:  2022-01-20
  • 刊出日期:  2022-03-08

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