  A set of semi-analytical techniques based on Fourier analysis is used to
solve wave scattering problems in variously shaped waveguides with varying
normal admittance boundary conditions. Key components are newly developed
conformal mapping methods, wave splitting, Fourier series expansions in
eigen-functions to non-normal operators, the building block method or the
cascade technique, Dirichlet-to-Neumann operators, and reformulation in terms
of stable differential equations for reflection and transmission matrices. For
an example the results show good correspondence with a finite element method
solution to the same problem in the low and medium frequency domain. The
Fourier method complements finite element analysis as a waveguide simulation
tool. For inverse engineering involving tuning of straight waveguide parts
joining complicated waveguide elements, the Fourier method is an attractive
alternative including time aspects. The prime motivation for the Fourier method
is its added physical understanding primarily at low frequencies.
