| 非均匀层状介质一维波动方程精确解的有限差分算法 |
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| 引用本文： | 范留明.非均匀层状介质一维波动方程精确解的有限差分算法[J].岩土力学,2013,34(9):2715-2720. |
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| 作者姓名： | 范留明 |
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| 作者单位： | 西安理工大学 土木建筑工程学院，西安 710048 |
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| 基金项目： | 陕西高校省级重点实验室重点科研计划项目(No.2010JS085)。 |
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| 摘 要： | 平面波的传播问题通常可以归结为一维波动方程的定解问题。在非均匀介质中，即使简单的一维波动方程也需要借助于数值方法获得近似解。3层5点古典差分格式是计算偏微分方程一种常用算法，作为一种显式迭代格式，需要满足稳定性条件 ，其中 为波速， 为空间采样间隔， 为时间采样间隔。当 时， ，古典差分格式达到临界稳定状态。在这种情况下，平面波在 时间内的传播距离恰好等于空间采样间隔，差分格式真实地反映了平面波的传播原理，因而可以得到一维波动方程的精确解。但是，由于在非均匀介质中存在不连续的波阻抗界面，此方法不适于计算非均匀介质的波场。为了将临界稳定情况下的古典差分格式推广应用至非均匀层状介质，提出了一种能够处理波阻抗界面的有限差分格式，并应用傅里叶分析法得到其稳定性条件。模型算例验证了此算法的正确性。
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| 关 键 词： | 一维波动方程 有限差分法 古典差分格式 层状介质 精确解 |
| 收稿时间： | 2012-07-01 |
A new kind of finite difference scheme for exact solutions of one-dimensional wave equation in heterogeneous layer media |
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| FAN Liu-ming.A new kind of finite difference scheme for exact solutions of one-dimensional wave equation in heterogeneous layer media[J].Rock and Soil Mechanics,2013,34(9):2715-2720. |
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| Authors: | FAN Liu-ming |
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| Institution: | College of Civil Engineering and Architecture, Xi’an University of Technology, Xi’an 710048, China |
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| Abstract: | The plane-wave propagation can be generalized as a definite-solution problem of one-dimensional wave equation. In spite of the simple formality, solutions of one-dimensional wave equation in inhomogeneous media have to be solved with the aid of numerical methods. The classic three-level five-point finite difference scheme is a usual numerical method to calculate partial differential equations, which must meet the stable condition as an explicit iteration method. The stable condition is , where is wave velocity, is time sample interval, and is space sample interval. When or , the finite difference scheme is just up to the critical stable state. In such a case a space sample interval just equals wave propagation distance in a time sample interval , so the classic difference scheme exactly expresses plane-wave propagation theory and can be used to obtain exact solutions of one-dimensional wave equations. However, because of existence of wave impedance interfaces, the algorithm is unable to calculate wave fields in heterogeneous layer media. In order that the classic difference scheme in the critical stable state can be generalized to apply to heterogeneous layer media, an improved scheme is put forward, which can deal with impedance interfaces. Its stable condition is also given by Fourier transform analysis and the correctness is proved by some numerical model tests. |
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| Keywords: | one-dimensional wave equation finite difference method classic finite difference scheme layer media exact solutions |
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