Acoustic wave equation is a partial differential equation (PDE) which describes propagation of acoustic waves through a material. In general, the solution to this PDE is nonunique. Therefore, it is necessary to impose initial conditions in the form of Cauchy conditions for obtaining a unique solution. Theoretically, solving the wave equation is equivalent to representing the wavefield in terms of a radiation source which possesses finite energy over space and time.The radiation source is represented by a forcing term in the right-hand-side of the wave equation. In practice, the source may be represented in terms of normal derivative of pressure or normal velocity over a surface. The pressure wavefield is then calculated by solving an associated boundary-value problem via imposing conditions on the boundary of a chosen solution space. From ananalytic point of view, this manuscript aims to review typical approaches for obtaining unique solution to the acoustic wave equation in terms of either a volumetric radiation source, or a surface source in terms of normal derivative of pressure or normal velocity. A numerical approximation of the derived formulae will then be explained. The key step for numerically approximating the derived analytic formulae is inclusion of source, and will be studied carefully in this manuscript.

翻译：声波方程是描述声波在材料中传播的偏微分方程。一般而言，该偏微分方程的解不具有唯一性。因此，需以柯西条件形式施加初始条件以获得唯一解。理论上，求解波动方程等价于将波场表示为在时空中具有有限能量的辐射源项。该辐射源由波动方程右侧的强迫项表征。实际应用中，源项可表示为压力法向导数或表面法向速度。压力波场则通过施加选定解空间边界上的条件，求解相应的边值问题来计算。从理论分析角度，本文旨在综述通过体积辐射源或基于压力法向导数/法向速度的面源获得声波方程唯一解的典型方法，并进一步阐述推导公式的数值近似过程。数值近似推导公式的关键步骤在于源项的引入，本文将对此进行深入研究。