Acoustic wave equation seeks to represent wavefield in terms of a radiation source which possesses finite energy over space and time. The wavefield may be represented over a surface bounding the source, and calculated by solving an associated boundary-value problem via imposing conditions on the boundary of a chosen solution space. This manuscript aims to study approaches for obtaining unique solution to acoustic wave equation in terms of either a volumetric radiation source $s$, or surface source. For the latter, the wavefield is described using a Kirchhoff-Helmholtz or Rayleigh-Sommerfeld integral over the surface. Using a monopole version of these integral formulae, a singlet surface source is defined in terms of minus normal pressure derivative $-(\partial/\partial \boldsymbol{n})p$ or its equivalent $\rho_0 \partial u^{\boldsymbol{n}}/ \partial t$. Here, $p$ is the pressure, $\rho_0$ is the ambient density, and $u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$ is the normal velocity with $\boldsymbol{n}$ a unit vector outwardly normal to the surface. Using a dipole variant of these surface integral formulae, the surface source is defined as a doublet source in terms of pressure $p$. It will be shown that an interior-field dipole variant of these integral formulae represents the back-projected field from observations of the wavefield over a surface. The key step for numerically approximating all these derived analytical formulae is inclusion of source, and will be studied in this manuscript carefully. It will be shown that a numerical approximation of a dipole version of these surface integral formulae has a limitation regarding how to account for obliquity factors or their equivalent solid angles efficiently, especially for describing a back-projected field from observations over a measurement surface.

翻译：声波方程旨在根据一个在空间和时间上具有有限能量的辐射源来表示波场。波场可以在包围源的表面上表示，并通过在所选解空间的边界上施加条件求解相关的边值问题来计算。本文旨在研究基于体积辐射源$s$或面源获得声波方程唯一解的方法。对于后者，波场通过基尔霍夫-亥姆霍兹或瑞利-索末菲积分在表面上描述。使用这些积分公式的单极子形式，通过负的法向压力导数$-(\partial/\partial \boldsymbol{n})p$或其等效项$\rho_0 \partial u^{\boldsymbol{n}}/ \partial t$定义了一个单源面源。其中，$p$是压力，$\rho_0$是环境密度，$u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$是法向速度，$\boldsymbol{n}$是垂直于表面的单位外法向量。使用这些面积分公式的偶极子变体，面源被定义为以压力$p$表示的偶极子源。将证明，这些积分公式的内场偶极子变体表示从观测表面波场反投影的场。数值逼近所有这些推导的解析公式的关键步骤是包含源，本文将对此进行仔细研究。将证明，这些面积分公式的偶极子版本的数值近似有一个局限性，即如何有效考虑倾斜因子或其等效立体角，特别是用于描述来自测量表面观测的反投影场。