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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a v t/x≤1,其中v为波速,x为空间采样间隔,t为时间采样间隔。当a 1时,x v t,古典差分格式达到临界稳定状态。在这种情况下,平面波在t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。
The propagation of plane waves can usually be attributed to the problem of one-dimensional wave equations. In inhomogeneous media, even a simple one-dimensional wave equation needs to be approximated by means of numerical methods. The 3-level 5-point classical difference scheme is a common algorithm for calculating partial differential equations. As an explicit iterative scheme, the stability condition avt / x≤1 is required, where v is the wave velocity, x is the spatial sampling interval, t is the time sampling interval. When a 1, x v t, the classical difference scheme reaches a critical steady state. In this case, the propagation distance of the plane wave exactly equals to the spatial sampling interval at time t, and the difference scheme truly reflects the propagation principle of the plane wave. Therefore, an exact solution to the one-dimensional wave equation can be obtained. However, this method is not suitable for calculating the wave field of inhomogeneous media because of the discontinuous wave impedance interface in the inhomogeneous media. In order to generalize and apply the classical difference scheme in the case of critical stability to non-uniform layered media, a finite difference scheme is presented to deal with the wave impedance interface. The stability condition is obtained by Fourier analysis. The example of the model verifies the correctness of the algorithm.