Abstract: The diffraction of waves by a wedge-shaped boundary separating two media has long been an open question in acoustics. In this paper, an exact solution is presented for such a physical system with arbitrary wedge angle and source placement. Kontorovich-Lebedev transforms are used to solve the equations of motion and to analyze the boundary conditions. This analysis results in a singular integral equation for the transformed potential. Function theoretic methods are used to reduce this equation to a Fredholm equation with a symmetric kernel. The exact solution is then given in terms of the eigenfunctions of the symmetric kernel. The principal contributions of this work are the production of a joint distribution for the radial eigenfunctions (with two distinct wave speeds) in cylindrical coordinates and the method used to reduce a nondegenerate singular integral to a Hilbert-Schmidt equation. These two techniques, however, not only allow the solution of the scalar problem reported here, but also show the way to solution of the vector problems for electromagnetic and elastic fields.