Finite Difference Solution of the Modified Mild Slope Equation for Wave Reflection by a Slope With Superimposed Undulating Seabed
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摘要: 基于修正缓坡方程，建立了方程求解的有限差分模型，研究了斜坡叠加起伏海床对波浪的反射.首先，对波浪入射平面斜坡以及水平海床上单周期、双周期正弦沙纹的反射问题进行了验证，结果与他人的数值解、解析解及实验数据吻合较好，证明了该模型的正确性.接着，讨论了抛物形斜坡的上凸高度及下凸深度，斜坡叠加单周期正弦沙纹的数量、高度及长度，斜坡叠加双周期正弦沙纹的高度对波浪反射系数的影响.结果表明，反射系数随抛物形斜坡上凸高度的增大而减小，随下凸深度的增大而增大.斜坡叠加单周期正弦沙纹对波浪反射的规律与水平海床基本一致，相比水平海床Bragg共振相位下移的幅度随沙纹数量增加，先减小后保持不变的情况，斜坡叠加沙纹共振相位下移的幅度随沙纹数量增加先减小后增大.波浪入射斜坡叠加双周期正弦沙纹地形，随叠加的两个沙纹高度分别增加，Bragg共振峰值均相应地增加，共振带宽几乎不受影响，主振峰值的相位下移幅度减小，这与单周期沙纹中随沙纹高度增加，共振相位下移幅度更大的情况相反.对比发现：当固定沙纹的数量、长度、高度，不论是水平海床还是斜坡海床，双周期正弦沙纹对波浪的反射强度以及激发的共振带宽均大于单周期正弦沙纹，共振相位下移的幅度则小于单周期沙纹；斜坡叠加正弦沙纹对波浪的反射强度要大于水平海床，零反射的现象不再存在，主振峰值的相位下移幅度较水平海床更大.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.
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Key words:
- modified mild slope equation /
- finite difference method /
- slope seabed /
- Bragg resonance reflection /
- phase downshift /
- resonance bandwidth
edited-byedited-by1) (我刊编委刘焕文推荐) -
图 1 二维斜坡叠加起伏海床示意图
Figure 1. Schematic diagram of the 2D slope superimposed undulating seabed
图 4 水平海床单周期正弦沙纹地形示意图
Figure 4. Schematic diagram of singly periodic sinusoidal sand ripples on the horizontal seabed
图 6 本文MMSE数值解与Chamberlain和Porter[6]的MMSE数值解、Liu等[19]的MMSE解析解、Porter等[20]的MMSE数值解的对比
Figure 6. Comparison among the present numerical solution of the MMSE, the numerical solution of the MMSE by Chamberlain and Porter[6], the analytical solution of the MMSE by Liu et al. [19], and the numerical solution of the MMSE by Porter et al. [20]
图 8 水平海床双周期正弦沙纹地形示意图
Figure 8. Schematic diagram of doubly periodic sinusoidal sand ripples on the horizontal seabed
图 10 斜坡叠加抛物形地形示意图
Figure 10. Schematic diagram of a slope with a superimposed parabolic terrain
图 11 斜坡叠加抛物形地形的上凸高度(或下凸深度)B对波浪反射系数KR的影响
Figure 11. Influences of upper convex height (or lower convex depth) B of the slope with a superimposed parabolic terrain on wave reflection coefficient KR
图 12 斜坡叠加单周期正弦沙纹地形示意图
Figure 12. Schematic diagram of a slope with superimposed singly periodic sinusoidal sand ripples
图 13 沙纹数量N对Bragg共振反射的影响，其中h1=6 m, b=1.5 m, d=12 m
Figure 13. Influence of the number N of sand ripples on Bragg resonance reflection, where h1=6 m, b=1.5 m, d=12 m
图 14 沙纹高度b对Bragg共振反射的影响，其中h1=6 m, N=8, d=12 m
Figure 14. Influences of height b of sand ripples on the Bragg resonance reflection, for h1=6 m, N=8, d=12 m
图 15 沙纹长度d对Bragg共振反射的影响，其中h1=6 m, N=4, b=1.5 m
Figure 15. Influences of length d of sand ripples on the Bragg resonance reflection, for h1=6 m, N=4, b=1.5 m
图 16 斜坡叠加双周期正弦沙纹地形示意图
Figure 16. Schematic diagram of a slope with superimposed doubly periodic sinusoidal sand ripples
图 17 沙纹高度b1和b2对Bragg共振反射的影响，其中h1=6 m, N=4, d=8 m, m=1/2
Figure 17. Influences of heights b1 and b2 of sand ripples on the Bragg resonance reflection, for h1=6 m, N=4, d=8 m, m=1/2
表 1 水平海床上单周期正弦沙纹对波浪Bragg共振反射的实验参数[18]
Table 1. Experimental parameters of Bragg resonance reflection of waves by singly periodic sinusoidal sand ripples on the horizontal seabed[18]
h1/m N b/h1 d/m wave number k/m-1 relative water depth kh1 0.156 4 0.32 1 0.5π≤k≤2.5π 0.245 0≤kh1≤1.225 2 
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表 2 水平海床上双周期正弦沙纹地形对波浪Bragg共振反射的数值参数[22]
Table 2. Numerical parameters of Bragg resonance reflection of waves by doubly periodic sinusoidal sand ripples on the horizontal seabed[22]
h1/m N b1/h1 b2/h1 d/m m wave number k/m-1 relative water depth kh1 0.156 4 0.32 0.32 1 1/2 0.5π≤k≤2.5π 0.245 0≤kh1≤1.225 2
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表 3 单周期正弦沙纹数量N对Bragg共振反射影响的计算工况
Table 3. The calculation cases of the influences of number N by singly periodic sinusoidal sand ripples on the Bragg resonance reflection
case N b/m d/m h3/m seabed slope 1 1 1.5 12 6 0 2 2 1.5 12 6 0 3 3 1.5 12 6 0 4 4 1.5 12 6 0 5 5 1.5 12 6 0 6 6 1.5 12 6 0 7 7 1.5 12 6 0 8 8 1.5 12 6 0 9 9 1.5 12 6 0 10 10 1.5 12 6 0 11 1 1.5 12 5.76 1/50 12 2 1.5 12 5.52 1/50 13 3 1.5 12 5.28 1/50 14 4 1.5 12 5.04 1/50 15 5 1.5 12 4.80 1/50 16 6 1.5 12 4.56 1/50 17 7 1.5 12 4.32 1/50 18 8 1.5 12 4.08 1/50 19 9 1.5 12 3.84 1/50 20 10 1.5 12 3.60 1/50
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表 4 单周期正弦沙纹高度b对Bragg共振反射影响的计算工况
Table 4. The calculation cases of the influences of height b by singly periodic sinusoidal sand ripples on the Bragg resonance reflection
case N b/m d/m h3/m seabed slope 1 8 0.5 12 6 0 2 8 0.8 12 6 0 3 8 1.0 12 6 0 4 8 1.2 12 6 0 5 8 1.5 12 6 0 6 8 0.5 12 4.08 1/50 7 8 0.8 12 4.08 1/50 8 8 1.0 12 4.08 1/50 9 8 1.2 12 4.08 1/50 10 8 1.5 12 4.08 1/50
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表 5 单周期正弦沙纹长度d对Bragg共振反射影响的计算工况
Table 5. The calculation cases of the influences of length d by singly periodic sinusoidal sand ripples on the Bragg resonance reflection
case N b/m d/m h3/m seabed slope 1 4 1.5 8 6 0 2 4 1.5 9 6 0 3 4 1.5 10 6 0 4 4 1.5 11 6 0 5 4 1.5 12 6 0 6 4 1.5 8 5.36 1/50 7 4 1.5 9 5.28 1/50 8 4 1.5 10 5.20 1/50 9 4 1.5 11 5.12 1/50 10 4 1.5 12 5.04 1/50
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表 6 叠加的两正弦沙纹高度b1和b2对Bragg共振反射影响的计算工况
Table 6. The calculation cases of the influences by superimposed 2 sinusoidal sand ripple heights b1 and b2 on the Bragg resonance reflection
case N b1/h1 b2/h1 d/m h3/m seabed slope 1 4 0.32 0 8 6 0 2 4 0 0.32 8 6 0 3 4 0.32 0.32 8 6 0 4 4 0.32 0.48 8 6 0 5 4 0.48 0.32 8 6 0 6 4 0.32 0 8 5.36 1/50 7 4 0 0.32 8 5.36 1/50 8 4 0.32 0.32 8 5.36 1/50 9 4 0.32 0.48 8 5.36 1/50 10 4 0.48 0.32 8 5.36 1/50
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表 7 叠加的两正弦沙纹高度b1和b2对Bragg共振相位和峰值的影响
Table 7. Influences of superimposed 2 sinusoidal sand ripple heights b1 and b2 on the phases and peaks of the Bragg resonance
b1/h1 b2/h1 the Bragg primary resonance the Bragg subharmonic resonance seabed slope is 0 seabed slope is 1/50 seabed slope is 0 seabed slope is 1/50 phase 2d/L peak KR phase 2d/L peak KR phase 2d/L peak KR phase 2d/L peak KR 0.32 0 0.907 50 0.294 32 0.902 50 0.341 36 1.932 50 0.008 16 1.930 00 0.013 99 0 0.32 1.922 50 0.048 29 1.917 50 0.075 74 - - - - 0.32 0.32 0.940 00 0.617 77 0.932 50 0.653 00 1.967 50 0.173 14 1.967 50 0.237 05 0.32 0.48 0.992 50 0.794 10 0.977 50 0.823 04 1.997 50 0.514 66 1.997 50 0.627 69 0.48 0.32 0.947 50 0.808 67 0.933 50 0.846 34 1.982 50 0.393 53 1.982 50 0.498 45
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