Spatial domain division based on superposition pattern of seismic waves and spatial variation of motion：Case of SV wave incidence
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摘要: 针对地震动空间差异问题，以半无限空间内平面SV波入射为例，采用波动理论和叠加原理相结合的方法研究地震动的空间特征。首先，分析入射波和反射波在空间内形成质点运动的叠加模式差异，并依此对空间域进行划分；其次，针对不同泊松比和SV波入射角情形研究叠加区与分离区的分界线控制情况；最后，对比分析不同空间域内的质点运动在峰值、持时等方面的特征。研究结果显示：当SV波入射时，叠加区与分离区的分界线通常由z3 （反射P波与反射SV波的分离线）控制；同时存在两种特殊情况，当SV波垂直入射时，分界线由z1 （入射SV波与反射SV波的分离线）控制，当反射SV波幅值为零时，分界线由z2 （入射SV波与反射P波的分离线）控制。在入射波和反射波的影响下，质点运动时程的形状具有水平不变性。三波贡献时段只出现在深度小于z1的质点的运动时程中，且持时随着深度的增加线性减少；双波贡献时段出现在位于叠加区内的质点的运动时程中，持时沿深度先增加后减少；单波贡献时段随着深度的增加而逐渐加长，在分离区达到最大值。质点运动总持时随深度逐渐增加，在分离线z2和z3深度处存在两个拐点。在质点运动峰值方面，靠近自由面的叠加区质点运动峰值变化较大，深度较大的叠加区和分离区的质点运动峰值一般不变。Abstract: For the investigation of spatial variation of seismic motion, the wave theory and superposition principle are used to illustrate the spatial variation of motions caused by seismic waves. And plane SV wave’s incidence in semi-infinite space is used as a case in this paper. Firstly the difference in superposition pattern of waves in different position is investigated and the space region is divided into different domains according to superposition pattern. Then the border line between superposition domain and separation domain with different Poisson’s ratio and SV wave’s incident angle is discussed. Finally the difference in peak value and duration time of particle motion in different domains is investigated. The result shows that, the border line is usually controlled by z3 (separation line generated by reflected P wave and reflected SV wave). Particularly, there exist two special cases, one is the vertical incidence of SV wave in which the border line is controlled by z1 (separation line generated by incident SV wave and reflected SV wave), the other is the zero amplitude of reflected SV wave in which the border line is controlled by z2 (separation line generated by incident SV wave and reflected P wave). The time history of different particle motion in one depth has the same shape with the influence of incident and reflected waves. In time history of particle motion, the period contributed by three waves only exists in depth lower than z1 and the duration time decreases with depth. The period contributed by two waves exists in all superposition domain and the duration time increases firstly and then decreases with depth. The period contributed by one single wave increases with depth and reaches its maximum in separation domain. The increase of particle motion duration has two slope change in depths of z2 and z3. The peak value of particle motion changes dramatically in superposition domain near free surface and remains constant in deep superposition domain and separation domain.
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图 1 地震波叠加模式与质点运动空间差异示意图
（a） 问题说明；（b） A点和B点的水平向运动
Figure 1. Superposition pattern of seismic waves and spatial variation of particle motion at different point
（a） Description of problem；（b） Horizontal motion at points A and B
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图 3 不同泊松比时叠加区范围系数a1 （a），a2 （b）和a3 （c）随平面SV波入射角的变化
Figure 3. Variation of superposition range coefficient a1 （a），a2 （b） and a3 （c） with incident angle of SV wave
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图 4 泊松比为0.15 （a），0.25 （b）和0.35 （c）时叠加区范围系数对比
Figure 4. Comparison of superposition range coefficient with Poisson’s ratio of 0.15 （a），0.25 （b） and 0.35 （c）
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图 5 分界线由Si与Pr分离线控制的对应情况
Figure 5. Border controlled by separation line generated by Si and Pr
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图 7 不同深度h下特征位置质点的水平向运动时程
Figure 7. Time history of horizontal particle motions in different depth h
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图 8 不同地震波叠加类型下持时沿深度分布
Figure 8. Duration of different superposition pattern of particle motion in different depth
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图 10 质点运动水平向（a）、垂直向（b）峰值沿深度变化
Figure 10. Peak displacement of particle motion in horizontal （a） and vertical （b） displacement along depth
表 1 不同深度质点地震波贡献时段
Table 1 Contribution time interval of waves on particle motion at different depth
质点 深度/km 单波贡献时段/s 双波贡献时段/s 三波贡献时段/s Si Pr Sr （Si，Pr） （Pr，Sr） （Si，Pr，Sr） D1 0 0—2.00 D2 1.0 0—0.89 2.89—3.17 0.89—1.17 2.00—2.89 1.17—2.00 D3 1.8 0—1.61 2.00—2.11 3.61—4.11 1.61—2.00 2.11—3.61 D4 3.0 0—2.00 2.68—3.52 4.68—5.52 3.52—4.68 D5 9.0 0—2.00 8.05—10.05 10.57—12.57 
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