9.7. Possible profiles 
When it comes to considering various horn profiles, there are only a limited number with exact analytical solutions to the wave equation. For a start, an exact solution relies upon a profile that fits a coordinate system which leads to a separable wave equation. In other words, the coordinate system must be orthogonal, having coordinate surfaces that all meet at right angles. In Part V we presented solutions to the three-dimensional Helmholtz wave equation in three such coordinate systems, namely, rectangular, cylindrical, and spherical. Generally, cylindrical coordinates lead to parabolic
horns (with two parallel and two nonparallel side walls), while spherical coordinates lead to conical horns.
There are only ...