Scattering at a rough boundary—Extensions of the Kirchhoff approximation

A Helmholtz integral formula gives the acoustic field reflected from a rough surface, in terms of the values of the acoustic field and of its normal derivative at all points on the surface. For surface reflection problems in which one, or the other, of these surface field values are specified, we can make use of this formula to derive a boundary surface integral equation on the unspecified field values. Further, a series solution of this equation can be constructed by iteration, the zeroth‐order term of the series being the result predicted by a Kirchhoff approximation. The nth‐order iterate of this series requires an n‐fold integration over the region of the rough surface, an integration that does not in general converge absolutely for unboundedly large surfaces. For a pressure release surface, we consider the first two iterates in detail and using stationary phase approximations replace the required integration by summations. In this way the source of the convergence difficulty is made clear and is demo...