S波

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平面剪切波

二维网格中球面S波的传播（经验模型）

S波（S-wave，secondary wave）是二種體波（體波的命名是因為此波穿越地球內部，相對於體波的是表面波）中之一。它是因地震而產生的，被地震儀記錄下來。命名為S波（二次波，secondary wave）是因為它的速度僅次於P波（最快的地震波）。S波也可以代表剪切波（shear wave），因為S波是一種橫波，地球內部粒子的震動方向與震波能量傳遞方向是垂直的。S波與P波不同的是，S波無法穿越外地核。所以S波的陰影區正對著地震的震源。

S波移动时是剪切波或横波，因此其运动方向与波的传播方向是垂直的，若要形象地描述S波，可以认为S波是挥动绳子时，绳子上传播的波，这与P波是不同的。P波是一种纵波，纵波就如振动的弹簧上传播的波，其形态就像蠕虫一样。S波通过弹性介质移动，而主要的恢复力来自於剪切效应。这些波是不发散的，遵守不可压缩介质的连续性方程：

\nabla \cdot \mathbf {u} =0

原理

P波阴影区。S波不会穿过外核，因此在远离震中超过104°的全部区域S波都处在阴影区中（来源：USGS）

S波预测来自於1800年代的理论，最初来自於各向同性固体的應力－应变关系：

\tau _{ij}=\lambda \delta _{ij}e_{kk}+2\mu e_{ij}\

其中 $\tau$ 是应力， $\lambda$ 和 $\mu$ 是拉梅参数（ $\mu$ 是剪切模量）， $\delta _{ij}$ 是克罗内克函数，而应变张量定义为

e_{ij}={\frac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)

其中u是应变位移。将後式代入前式得到

\tau _{ij}=\lambda \delta _{ij}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)

这种情况下的牛顿第二定律给出了地震波传播的运动齐次方程：

\rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\partial _{j}\tau _{ij}

其中 $\rho$ 是质量密度。代入上面的应力张量得到：

\rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=\partial _{i}\lambda \partial _{k}u_{k}+\partial _{j}\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)=\lambda \partial _{i}\partial _{k}u_{k}+\mu \partial _{i}\partial _{j}u_{j}+\mu \partial _{j}\partial _{j}u_{i}.

利用向量恒等式并取一定的近似可得到均匀介质中的地震波方程：

\rho {\ddot {\boldsymbol {u}}}=\left(\lambda +2\mu \right)\nabla (\nabla \cdot {\boldsymbol {u}})-\mu \nabla \times (\nabla \times {\boldsymbol {u}})

其中牛顿标记（英语：Newton's notation）用於表示时间导数。取方程的旋度并利用向量恒等式最终得到：

\nabla ^{2}(\nabla \times {\boldsymbol {u}})-{\frac {1}{\beta ^{2}}}{\frac {\partial ^{2}(\nabla \times {\boldsymbol {u}})}{\partial t^{2}}}=0

这一方程是一个只包含了u的旋度和速度 $\beta$ 的波动方程，其中 $\beta$ 满足

\beta ^{2}={\frac {\mu }{\rho }}\

这一公式描述了S波的传播。若用均匀介质中的地震波方程的散度代替旋度，则会得到描述P波传播的方程。

參見

P波
橫波

参考文献

Shearer, Peter. Introduction to Seismology 1st ed. Cambridge University Press. 1999. ISBN 0-521-66023-8.
Aki, Keiti; Richards, Paul G. Quantitative seismology 2nd ed. University Science Books. 2002. ISBN 0-935702-96-2.
Fowler, C. M. R. The solid earth. Cambridge, UK: Cambridge University Press. 1990. ISBN 0-521-38590-3.

地震要素

地震观测

现代度量

震级度量

曾用度量

理论研究

地震波（P波 · S波 · 面波 · 勒夫波 · 自由震盪）

轉換波分析

彈性回跳理論

震源机制解

古登堡—里克特定律

斷層（活断层 · 逆斷層 · 盲斷層 · 潛移斷層 · 板块边缘）

地震帶（环太平洋火山带 · 阿尔卑斯带 · 大西洋洋中脊）

连发式地震带

爆炸地震效应

地震類型

防灾對策

相關條目

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