## Summary

For certain wave scattering problems embedding formulae can be derived, which express the solution, or far-field behaviour of the solution, for arbitrary plane wave incident angle in terms of the corresponding quantities for a finite number of other related problems. Their scope has so far been limited to scattering in $$\mathbb{R}^2$$, and to a lesser extent $$\mathbb{R}^3$$; in this article we derive embedding formulae for wave scattering in a class of two-dimensional waveguide. The waveguide is straight and of uniform width outside a finite length region within which the boundaries are piecewise-linear and the waveguide can contain polygonal obstacles, a restriction being that all boundaries of the waveguide and obstacles must be inclined at a rational angle to the axis of the waveguide. Once solutions are determined for a finite set of incident propagating modes, the embedding formulae provide expressions for reflection and transmission coefficients for all remaining incident propagating modes. The precise number of solutions required is a function of the number and nature of the corners of the boundaries and obstacles. The formulae are illustrated for a particular waveguide geometry for which the problem can be formulated as an integral equation and approximate numerical solutions determined using the Galerkin method.