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

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.

Key words: one-dimensional wave equation, finite difference method, classic finite difference scheme, layer media, exact solutions