Abstract
We consider a point source above a rough surface, and write the coherent response as the Sommerfeld–Weyl–Noether integral in terms of the coherent plane-wave reflection coefficient R for correlated distributions of bosses on rigid or free planes. The uniform asymptotic development of the integral (a generalization of the original Sommerfeld approximation based on the error function complement) is applied to near-grazing for a rigid base, and graphical results and simple analytical approximations are presented to exhibit the essentials for data inversion programs. The coefficient R and the associated angle-dependent impedance (determined by the ensemble-averaged multiple scattering amplitude for one fixed boss) are specified by the single scattered amplitude and the statistical-mechanics pair distribution function. Low-frequency illustrations for rigid bosses delineate multipole-coupling and packing effects on propagation and on attenuation arising from incoherent scattering, as functions of the packing fraction and of the shapes of the bosses and their exclusion regions.
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