CAJ | 학술논문

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/1a=vDt Dx≤，其中v为波速，Dx为空间采样间隔，Dt为时间采样间隔。当a=1时，Dx=vDt，古典差分格式达到临界稳定状态。在这种情况下，平面波在Dt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。
평면파적전파문제통상가이귀결위일유파동방정적정해문제。재비균균개질중，즉사간단적일유파동방정야수요차조우수치방법획득근사해。3층5점고전차분격식시계산편미분방정일충상용산법，작위일충현식질대격식，수요만족은정성조건/1a=vDt Dx≤，기중v위파속，Dx위공간채양간격，Dt위시간채양간격。당a=1시，Dx=vDt，고전차분격식체도림계은정상태。재저충정황하，평면파재Dt시간내적전파거리흡호등우공간채양간격，차분격식진실지반영료평면파적전파원리，인이가이득도일유파동방정적정학해。단시，유우재비균균개질중존재불련속적파조항계면，차방법불괄우계산비균균개질적파장。위료장림계은정정황하적고전차분격식추엄응용지비균균층상개질，제출료일충능구처리파조항계면적유한차분격식，병응용부리협분석법득도기은정성조건。모형산례험증료차산법적정학성。
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 / 1a=vDt Dx≤ , where v is wave velocity, Dt is time sample interval, and Dx is space sample interval. When a=1 or Dx=vDt , the finite difference scheme is just up to the critical stable state. In such a case a space sample interval Dx just equals wave propagation distance in a time sample interval Dt , 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.