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, initial conditions in the form of Cauchy conditions are imposed 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. In practice, the source may be represented in terms of pressure, 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 an 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 $s$, or a singlet surface source in terms of normal derivative of pressure $(\partial/\partial \boldsymbol{n})p$ or its equivalent $\rho_0 u^{\boldsymbol{n}}$ with $\rho_0$ the ambient density, where $u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$ is the normal velocity with $\boldsymbol{n}$ a unit vector outwardly normal to the surface. For some cases including a time-reversal propagation, the surface source is defined as a doublet source in terms of pressure $p$. 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. It will be shown that compared to an analytical or ray-based solutions using Green's function, a numerical approximation of acoustic wave equation for a doublet source has a limitation regarding how to account for solid angles efficiently.

翻译：声波方程是描述声波在介质中传播的偏微分方程(PDE)。一般而言，该偏微分方程的解具有非唯一性。因此，需施加柯西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具有时空有限能量的辐射源。实际应用中，源项可通过表面上的压力、压力法向导数或法向速度来表征。随后，通过在所选解空间的边界上施加条件，求解相应的边值问题来计算压力波场。从解析角度出发，本文旨在综述获得声波方程唯一解的典型方法：即通过体辐射源$s$，或通过压力法向导数$(\partial/\partial \boldsymbol{n})p$及其等价形式$\rho_0 u^{\boldsymbol{n}}$（其中$\rho_0$为环境密度）表示的单极子面源，这里$u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$为法向速度，$\boldsymbol{n}$为垂直于表面的单位外法向量。对于包括时间反演传播在内的某些情况，面源被定义为以压力$p$表示的偶极子源。随后将阐述推导所得公式的数值逼近方法。数值逼近推导的解析公式的关键步骤在于源项的引入，本文对此进行细致研究。结果表明，与采用格林函数的解析解或射线法相比，偶极子源声波方程的数值逼近在有效处理立体角方面存在局限性。