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. This paper discusses the question of extending solutions of the twodimensional Helmholtz equation across a curve. Although general results on extending solutions of elliptic systems are available [6], these results do not normally give useful information about the size of an extension. In most applications though, the size of the extension region matters. Our main result, stated in the following section, gives explicit criteria which can be used to determine a region in which an extension is guaranteed to exist. The criteria are easy to check and, often, the region that they determine is maximal in a natural sense. A classical application of extension results for the Helmholtz equation is to the socalled Rayleigh hypothesis. In 1907 Rayleigh solved the two-dimensional scattering problem for a sinusoidal surface illuminated by a plane wave in terms of a series of outgoing waves [8]. Assuming the validity of the expansion for points above the peaks of the sinusoid, there remains the question about whether it gives the correct results in the troughs of the surface. To resolve this issue it is necessary to extend the expansion to the region between the peaks and troughs of the scatterer. It was eventually shown that for a sinusoidal surface there is a relation between the amplitude and period of the surface in order for an extension to exist. Rayleigh's solution is correct only under this constraint on the surface geometry. A discussion may be found in [1]. Subsequently, numerous attempts were made to obtain similar results without using an explicit formula for the solution. The best known results were obtained by Millar [5]. This paper represents the solution of the Helmholtz equation using a form of Green's theorem in the complex plane and applies conformal mapping arguments based on the Schwartz function to locate the singularities. It seems that this approach may require

Extendability of solutions of Helmholtz’s equation to the interior of a two-dimensional scatterer