Poisson’s equation is an important equation to postprocess the aerodynamic fields into linearized momentum modes and was recently found to be important for the computation of an isotropic pressure-like source for scalar aeroacoustic wave models, like the aeroacoustic wave equation based on Pierce’s operator (AWE-PO). Mathematically viable boundary conditions of the Poisson equation, which computes the AWE-PO source, are investigated. For the different source fields, the wave propagation is computed using the AWE-PO, and the details of the sound prediction results are compared to a reference direct numerical simulation of a mixing layer. The different boundary conditions of the Poisson equation were found to have a minor influence on the overall sound prediction characteristics of the AWE-PO equation. The AWE-PO is reformulated into a simplified version of the Phillips’s equation, which mitigates the intermediate step of computing an isotropic source potential. By doing so, a previously obtained interference radiation valley in the radiated acoustic intensity of the AWE-PO results is attributed to a missing shear-noise source term.