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

PAN Junjie; LIU Huanwen; LI Changjiang

doi:10.21656/1000-0887.420123

Volume 43 Issue 3

Mar. 2022

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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

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

doi: 10.21656/1000-0887.420123

1.
School of Naval Architecture and Maritime, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, P.R.China
2.
Zhejiang Weihao Engineering Technology Co. , Ltd. , Zhoushan, Zhejiang 316000, P.R.China

Received Date: 2021-05-04
Accepted Date: 2021-05-04
Rev Recd Date: 2021-09-30

Available Online: 2022-01-20

Publish Date: 2022-03-08

Abstract

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.
- modified mild-slope equation,
- parabolic trench,
- series solution,
- Bragg resonant reflection,
- phase downshift,
- phase upshift

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