On the derivation of boundary integral equations for scattering by an infinite two-dimensional rough surface

A plane acoustic wave is incident upon an infinite, rough, impenetrable surface S. The aim is to find the scattered held by deriving a boundary integral equation over S, using Green's theorem and the free-space Green's function. This requires careful consideration of certain integrals over a large hemisphere of radius r; it is known that these integrals vanish as r --> infinity if the scattered field satisfies the Sommerfeld radiation condition, but that is not the case here-reflected plane waves must be present. It is shown that the well-known Helmholtz integral equation is not valid in all circumstances. For example, it is not valid when the scattered field includes plane waves propagating away from S along the axis of the hemisphere. In particular, it is not valid for the simplest possible problem of a plane wave at normal incidence to an infinite flat plane. Some suggestions for modified integral equations are discussed. (C) 1998 American Institute of Physics. [S0022-2488(98)01302-4].