The scattering from quadratic residue diffusers has been predicted using methods based on the Helmholtz–Kirchhoff integral equation. The methods consisted of two boundary integral methods, and two more approximate solutions utilizing Kirchhoff’s boundary conditions. The predictions were compared to measurements made in both the near and far fields. An accurate boundary integral method which works below and above the well cutoff frequency has been found. The limitations of the more approximate methods have been defined. The approximations behind the quadratic residue diffuser’s design have been tested. The important assumption of a phase change local reacting admittance over each well face was found to be good over a wide range of frequencies. The frequency limit of this approximation has also been defined.