Scattering from a corrugated hard wall: Comparison of boundary conditions

The calculation of the scattered-wave amplitude by a hard-corrugated-wall potential is reduced to the determination of a source function, which is then calculated by application of an appropriate boundary condition. We discuss and compare the general features of the solution given by three different boundary conditions: (i) setting the wave function equal to zero on the corrugated surface, (ii) setting the wave function equal to zero on a plane beneath the surface, and (iii) the Rayleigh condition. It is argued that only the first of these three methods will always produce a solution. Detailed comparisons are made for the case of a triangular corrugation profile, and we show that the application of boundary conditions on the surface always gives a solution. However, it is argued that the other two methods cannot produce a convergent solution for this profile, and these conclusions are supported by numerical calculations.