Bragg Resonance and Phase Upshift of Linear Water Waves Excited by a Finite Periodic Array of Parabolic Trenches
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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。
Abstract:The Bragg resonant reflection excited by a finite periodic array of parabolic trenches was analytically studied. First, the modified mild-slope equation (MMSE) with implicit coefficients was transformed into an ordinary differential equation with explicit coefficients through variable substitution. Second, an analytical solution to the MMSE was established in terms of the Frobenius series, and the convergence condition for the series solution was given. Finally, by means of the mass-conservation matching conditions, an analytical formula for the reflection coefficient was built. With the analytical formula, the effects of the number, the depth and the width of trenches on the peak value, the phase and the band width of the resonance, were investigated. The results show that, when the depth and width of trenches keep constant, and the number of trenches increases, the Bragg resonance peak value will increase up to 1, while the resonance bandwidth will narrow down and approach a fixed value. When the number and width of trenches keep constant, the Bragg resonance peak value will increase with the depth of trenches. When the number and depth of trenches keep constant, the Bragg resonance peak value will increase at first and then decrease with the width of trenches, which implies that there exists a certain width of trenches to make the Bragg resonance peak value reach the maximum, laying a theoretical base for the optimization of Bragg resonance vs. the trench width. Particularly, the phase upshift of the Bragg resonance wave reflection peak value recently observed over finite periodically arranged cycloidal trenches, was confirmed again over the parabolic trenches. That implies that, the phase upshift of the Bragg resonance reflection peak value is a common phenomenon excited by finite periodic trenches with arbitrary cross sections. Consequently, for sinusoidal ripples and periodic artificial bars, the phase of the Bragg resonance reflection will shift downward, while for an array of periodic trenches, regardless of the shape of the trench cross section, the phase of the Bragg resonance reflection will shift upward. In addition, starting from the initial definition of the Bragg resonance, the mathematical mechanism of the phenomenon of phase upshift is well explained.
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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) $ 和$ \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 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
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