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The Half-Space Matching (HSM) method has recently been developed as a newmethod for the solution of 2D scattering problems with complex backgrounds,providing an alternative to Perfectly Matched Layers (PML) or other artificialboundary conditions. Based on half-plane representations for the solution, thescattering problem is rewritten as a system coupling (1) a standard finiteelement discretisation localised around the scatterer and (2) integralequations whose unknowns are traces of the solution on the boundaries of afinite number of overlapping half-planes contained in the domain. Whilesatisfactory numerical results have been obtained for real wavenumbers,well-posedness and equivalence of this HSM formulation to the originalscattering problem have been established only for complex wavenumbers. In thepresent paper we show, in the case of a homogeneous background, that the HSMformulation is equivalent to the original scattering problem also for realwavenumbers, and so is well-posed, provided the traces satisfy radiationconditions at infinity analogous to the standard Sommerfeld radiationcondition. As a key component of our argument we show that, if the trace on theboundary of a half-plane satisfies our new radiation condition, then thecorresponding solution to the half-plane Dirichlet problem satisfies theSommerfeld radiation condition in a slightly smaller half-plane. We expect thatthis last result will be of independent interest, in particular in studies ofrough surface scattering.

On the half-space matching method for real wavenumber