Low-frequency scattering of acoustic waves by a bounded rough surface in ahalf-plane

The problem of the scattering of harmonic plane waves by a rough half-plane is studied here. The surface roughness is finite. The slope of the irregul arity is taken as arbitrary. Two boundary conditions are considered, those of Dirichlet and Neumann. An asymptotic solution is obtained, when the wave length lambda of the incident wave is much larger than the characteristic l ength of the roughness l, by means of the method of matched asymptotic expa nsions in terms of the small parameter epsilon =2 pil/lambda. For the Diric hlet problem, the solution of the near and far fields is obtained up to O(e psilon (2)). The far field solution is given in terms of a coefficient that have a simple explicit expression, which also appears in the corresponding solution to the Neumann problem, already solved. Also the scattering cross section is given by simple formulas to O(epsilon (3)). It is noted that, f or the Dirichlet problem, the leading term is of order epsilon (3) which, b y contrast, is different from that of the circular cylinder in full space, that is, of order epsilon (-1) (log epsilon)(-2). Some examples display the simplicity of the general results based on conformal mapping, which involv e arcs of circle, polygonal lines, surface cracks and the like. (C) 2001 Ac oustical Society of America.