On the Downshift of Wave Frequency for Bragg Resonance

PENG Ji, TAO Ai-feng, FAN Jun, ZHENG Jin-hai, LIU Yu-ming

Abstract

For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples, strong wave reflection occurs under the Bragg resonance condition. The critical wave frequency, at which the peak reflection coefficient is obtained, has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction. It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects, but the intrinsic mechanism remains unclear. By a regular perturbation analysis, we derive the theoretical solution of frequency downshift due to thirdorder nonlinear effects of both bottom and free-surface boundaries. It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actual causes frequency upshift. The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness. Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom. In addition, direct numerical simulations based on the high-order spectral method are conducted to validate the present theory. The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.

摘要

当表面重力波传播到正弦形的水底地形时，在布拉格共振条件下会发生强烈的波反射。从物理实验和直接的数值模拟中都可以观察到，在获得峰值反射系数时的临界波频率比著名理论预测要低。长期以来，人们推测这种下降可能是由于高阶波底效应和自由表面边界效应，但其内在机制尚不清楚。通过正则摄动分析，导出了底部边界和自由表面边界三阶非线性作用下频率降移的理论解。结果表明，底层非线性对频率降移起主导作用，而自由表面非线性对频率上升起主导作用。频率下降/上升在底部/自由表面陡度上具有二次关系。多频相对于单频具有较大的频率降移。此外，基于高阶谱方法进行了直接数值模拟，验证了理论的正确性。该降频理论解与数值模拟和现有实验数据结果较为一致。