Far-field approximation for an outgoing source ($\psi (R\mathbf{\theta })=R^{-1}{e}^{ikR}{f}_{\infty }(\mathbf{\theta })+O(R^{-2})$)

For a bounded source region and fixed positive wavenumber, expand $|R\mathbf{\theta }-{\mathbf{r}}^{\prime }|=R-\mathbf{\theta }\cdot {\mathbf{r}}^{\prime }+O(R^{-1})$. In the outgoing Green function for the three-dimensional Helmholtz equation, this gives $G=e^{ikR}{e}^{-ik\mathbf{\theta }\cdot {\mathbf{r}}^{\prime }}/(4\pi R)+O(R^{-2})$. Volume integration gives the outgoing field's far-field pattern with an $O(R^{-2})$ remainder. The estimate is an asymptotic statement for fixed source size and frequency.