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

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.