The research undertaken in this thesis is in the broad area of diffraction theory. We consider three separate and distinct problems of acoustic scattering with rectangular geometries, which have a common underlying mathematical structure. The geometries are: the infinite wedge, the waveguide with a barrier, and the semi-infinite plate of finite thickness. It turns out that these problems may be formulated as matrix Wiener–Hopf problems with the special property that their matrix kernels $K$ may be formulated as $K=M-1JM$, where $J2=I$, the identity matrix. This special property makes the problems amenable to factorisation which enables an exact solution to be derived, at least in theory. In practice, in two of the cases, we end up with an infinite system of equations which must be truncated to allow for practical computation of coefficients. However, these coefficients are rapidly convergent aided by the use of a novel technique termed the `corner singularity method', in which the integration contour of an integral is shifted upwards in the complex plane to pick up a contribution from the infinite 'tail'. This work has applications in industrial and marine acoustics, and bears promise of fruitful extension to elastodynamics and other areas of wave theory.