Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed

NI Yunlin; ZHANG Xijiang; YU Jiangmei; SHEN Liangduo

doi:10.21656/1000-0887.440255

Volume 45 Issue 5

May 2024

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NI Yunlin, ZHANG Xijiang, YU Jiangmei, SHEN Liangduo. Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed[J]. Applied Mathematics and Mechanics, 2024, 45(5): 606-621. doi: 10.21656/1000-0887.440255

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Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed

doi: 10.21656/1000-0887.440255

NI Yunlin^{1
,
,},
ZHANG Xijiang^1
,,
YU Jiangmei^2
,,
SHEN Liangduo^{1
,
,}

1.
College of Marine Engineering Equipment, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, P.R.China
2.
Zhoushan Ecological Environment Low Carbon Development Center, Zhoushan, Zhejiang 316021, P.R.China

Received Date: 2023-08-22
Rev Recd Date: 2024-01-28
Publish Date: 2024-05-01

Abstract

Abstract

A finite difference model was established to solve the modified mild slope equation, and the reflection of waves on the slope with superimposed undulating seabed was studied. Firstly, the reflection problems of incident waves on straight slope terrain, singly periodic sinusoidal sand ripples and doubly periodic sinusoidal sand ripples superimposed on the horizontal seabed, were verified in excellent agreement with the numerical and analytical solutions and experimental data of others, to prove the correctness of the model. Then, the influences of the upper convex height and lower convex depth of the slope with superimposed parabolic terrain, the number, height, and length of the slope with superimposed singly periodic sinusoidal sand ripples, and the height of the slope with superimposed doubly periodic sinusoidal sand ripples on wave reflection coefficients, were explored. The results show that, the reflection coefficient decreases with the upper convex height and increases with the lower convex depth on the parabolic slope. The law of wave reflection of slope with superimposed singly periodic sinusoidal sand ripples is the same as that of the horizontal seabed. Compared with the horizontal seabed, where the amplitude of the resonance phase downshift decreases with the number of sand ripples and then stays unchanged, the amplitude of the resonance phase downshift of the slope with superimposed sand ripples firstly decreases and then increases with the number of sand ripples. In the slope with superimposed doubly periodic sinusoidal sand ripples terrain, the peaks of the Bragg resonance increase with the heights of the 2 superimposed sand ripples, respectively, where the resonance bandwidth is almost unaffected, and the phase downshift amplitude of the peak Bragg primary resonance decreases, which is contrary to the situation that the amplitude of resonance phase downshift increases with the height of the singly periodic sand ripples. With the fixed number, length, and height of sand ripples, the reflection intensity of doubly periodic sinusoidal sand ripples on waves and the resonance bandwidth excited by them are larger than those of singly periodic sinusoidal sand ripples, and the amplitude of resonance phase downshift is smaller than that of singly periodic sand ripples, despite the horizontal seabed or the slope seabed. Moreover, the reflection intensity of the slope with superimposed sinusoidal sand ripples is greater than that of the horizontal seabed, the phenomenon of zero reflection no longer exists, and the phase downshift amplitude of the peak Bragg primary resonance is larger than that of the horizontal seabed.
- modified mild slope equation,
- finite difference method,
- slope seabed,
- Bragg resonance reflection,
- phase downshift,
- resonance bandwidth

(Recommended by LIU Huanwen, M. AMM Editorial Board)

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References(25)

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