On inclusion of source in full-field approximation of acoustic wave equation and a limitation

Acoustic wave equation seeks to describe wavefield in terms of a volumetric radiation source, $s$, or a surface source. For the latter, an associated boundary-value problem yields a description for the wavefield in terms of a Kirchhoff-Helmholtz or Rayleigh-Sommerfeld integral. A rigid-baffle assumption for the surface source (aperture) gives an integral formula in terms of a monopole source $-\partial p /\partial \boldsymbol{n}$, or equivalently $\rho_0 \partial ( \boldsymbol{u} \cdot {\boldsymbol{n}}) / \partial t$. Here, $p$ is the pressure, $\rho_0$ is the ambient density, $\boldsymbol{u}$ is the velocity vector and $\boldsymbol{n}$ is a unit vector normal to the surface. Alternatively, a soft-baffle assumption yields an integral formula in terms of a dipole source $p$. One example is interior-field dipole formula, which describes back-projected wavefield in terms of measurements over a surface. It will be shown theoretically that the amplitude calculated by the dipole formula is a function of obliquity factor or equivalently solid angle, a parameter which has not been included in the state-of-the-art approaches for full-field approximation. It will be shown numerically for a specific case how inclusion of the obliquity factors in a full-field approximation of the dipole integral formula yields a solution which matches the associated analytic formula.