On the Non-Specular Reflection of Plane Waves of Sound

The non-specular or scattered reflection of plane waves of sound by various rigid, non-absorbent non-porous surfaces composed of either semicylindrical or hemispherical bosses (protuberances) on an infinite plane is analyzed. Exact solutions for the problem of the single boss and a plane wave at an arbitrary angle of incidence are derived through consideration of a cylinder or sphere and two simultaneously incident “image waves.” Finite patterned distributions of such bosses are then treated and the far field solution obtained subject to the restriction that the secondary excitations of the various bosses be neglected. (The equivalent problems for cylinders and spheres are also considered as well as the second-order solution, for the cylindrical case, which takes into account the interaction of neighboring elements.) These solutions are found to contain the characteristic Fraunhofer terms for a grating or lattice. The asymptotic solutions for the single bosses (Kr≫1, Ka1) are then extended to consider both finite and infinite uniform random distributions. The solutions for the finite distributions are found to contain the characteristic Fraunhofer terms for similarly shaped apertures. The solutions for the infinite distributions (of semicylinders or hemispheres) are found to be remarkably similar when expressed in terms of the volumetric departure from the plane per cm2 of distribution. Some extensions and ramifications of the results are also considered.