On inclusion of time-varying source in numerical approximation to acoustic wave equation and a neglected limitation

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.