Relation between Creeping Waves and Lateral Waves on a Curved Interface

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Abstract
Diffraction effects at a gently curved interface between two media are investigated. Particular attention is paid to the behavior of the field on the diffracted rays which propagate along the interface into the shadows. It is found that far from the launching point of such a ray the field comprises of a series of modes which decay exponentially, due to the continuous leakage of energy away from the interface. At moderate distances, in the penumbra region, this series is poorly convergent. It can be converted into an integral, which can be evaluated asymptotically there, yielding a field with an algebraic decay. The field is like that diffracted along a plane interface, the so‐called lateral wave, and reduces to it when the radius of curvature becomes infinite. The regions of transition from one representation to the other are determined, and uniform asymptotic expressions, valid across those regions, are given. All our results apply to a two‐dimensional scalar problem, but results for three dimensions and for vector problems can be derived in a similar way.

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