Quantitative Expression of the Modified Bragg's Law for Bragg Resonances of Water Waves Excited by Five Types of Artificial Bars

For Class I Bragg resonance excited by five types of finite periodic array of widely spaced bars (rectangular, parabolic, rectified cosinoidal, trapezoidal, and triangular bars), the average phase velocity between any two adjacent bars in each type of bar field has been specifically calculated by employing a mathematical transformation, and therefore, the excitation condition of Class I Bragg resonance excited by each bar field is clarified and the modified Bragg's law established by Xie and Liu becomes a quantitative form. On one hand, if the incident surface wavelength, L, is fixed in advance, then according to the excitation condition, we can directly obtain the critical bar spacing, d, at which Class I Bragg resonance occurs. On the other hand, if the bar spacing, d, is fixed, then according to the excitation condition, we can use an iteration scheme to find out the critical surface wavelength, L, at which Class I Bragg resonance will be excited. In comparison with existing experimental, numerical, and analytical results of Bragg resonance excited by the five types of bar fields, it is shown that the modified Bragg's law is much more accurate than the Bragg's law, and the phenomenon of phase downshift can be well explained. In the linear long-wave limit, owing to the simple dispersion relation, the modified Bragg's law becomes an explicit expression and coincides with Liu's approximate law very well, although the former is a straight line segment while the latter is a curved line segment. Finally, it is shown that the phase downshift of Bragg resonance becomes more significant as the cross-sectional area of bars increases.