Creeping Waves in the Shadow of an Elliptic Cylinder

摘要

The geometrical theory of optics has been extended by Keller, whose idea of“creeping waves”propagating into the geometrical shadow region predicts a diffracted field in this region which is exponentially small for high frequencies.In this paper the exact solution is constructed for the velocity potential of an acoustic line source on the surface of a fixed elliptic cylinder, and the asymptotic behaviour of the field at high frequencies is examined. This canonical problem can be used as a check on Keller's theory.A standard technique is available for problems of this type. A formal series solution is constructed in terms of Mathieu functions and is transformed into integral form to improve convergence, by the Watson transformation. This integral can in turn be rewritten as a“residue series”, a sum of residues of the integrand.The first few terms of the residue series are exponentially small and successively decreasing when the frequency is large, and it is generally assumed in the literature that this decay continues throughout the series. This assumption is plausible, and is expected to hold for all points in the shadow, but is shown to be incorrect.An analysis of the residue series is carried out here, and shows that successive terms do not necessarily decrease. Further, it is found that some of the terms of the residue series are actually exponentially large in parts of the shadow, so that the full residue series is not a good representation of the solution.In spite of this surprising result, the leading term of Keller's theory is still verified. For the sum of all the terms of the residue series is small, even though individual terms may be large.Similar remarks are Found to hold for a source at any position outside the