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 analytic 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用具有时空有限能量的辐射源表示波场。辐射源以波动方程右端的强迫项形式表示。在实际应用中，源可通过压力法向导数或表面法向速度表示。继而，通过在选定解空间的边界施加条件求解相应的边值问题，从而计算压力波场。从解析角度，本文旨在综述通过体积辐射源或基于压力/法向速度法向导数的表面源获得声波方程唯一解的典型方法，并阐述由此推导的公式的数值逼近。数值逼近所推导解析公式的关键步骤在于源项的纳入，本文将对此进行深入研究。