This paper investigates the approximate solutions to the time-dependent acoustic scattering problem with a point-like scatterer under some basic assumptions and provides a simple method to reconstruct the location of the scatterer. The approximations of the solution to the forward scattering problem are analysed utilizing Green's function and the retarded single-layer potential. Then, based on the approximate solutions, a sampling method is proposed to solve the inverse scattering problem for the location of the scatterer. The proposed method is easy to implement since no equations or matrices have to be computed for the reconstruction. Numerical experiments are provided to show the effectiveness of the method.

本文研究了在一些基本假设下具有点状散射体的瞬态声学散射问题的近似解，并提供了一种简单的方法来重建散射体的位置。利用格林函数和延迟单层电位分析了前向散射问题的近似解。然后，在近似解的基础上，提出了一种求解散射体位置逆散射问题的采样方法。所提出的方法易于实施，因为不必为重建计算方程或矩阵。数值实验证明了该方法的有效性。