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 derivative of pressure (or normal displacement) $-(\partial/\partial \boldsymbol{n})p$ 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：声波方程旨在通过具有有限时空能量的辐射源表征波场。该波场可在包围源表面的空间上表示，并通过在选定解空间边界上施加条件求解相应的边值问题进行计算。本文旨在研究通过体积辐射源$s$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-索末菲表面积分描述。采用这些积分公式的单极子形式，单重面源由压力（或法向位移）的法向导数负值$-(\partial/\partial \boldsymbol{n})p$或其等价形式$\rho_0 \partial u^{\boldsymbol{n}}/ \partial t$定义，其中$p$为压力，$\rho_0$为环境密度，$u^{\boldsymbol{n}} = \boldsymbol{u} \cdot \boldsymbol{n}$为法向速度（$\boldsymbol{n}$为垂直于表面向外单位向量）。利用这些表面积分公式的偶极子变体，面源被定义为以压力$p$表征的双重源。研究表明，这些积分公式的内域偶极子变体可表示从表面波场观测反投影的场。所有这些解析公式数值近似的关键步骤在于源的包含，本文对此进行细致研究。结果表明，偶极子表面积分公式的数值近似存在局限性：如何有效处理倾斜因子或其等效立体角，尤其是描述测量表面观测反投影场时。