Creeping modes in a shadow

In the two-dimensional theory of diffraction by smooth curves there are certain canonical problems that can be solved explicitly in series form. The series converge slowly at short wavelengths but they can be transformed by Watson's transformation into another form (the residue series or creeping-mode expansion) which has been much used in shadow regions. It is found that throughout the shadow region the first few terms of the residue series are exponentially small and decrease rapidly, and these have often been used as an estimate of the wave potential without further justification. Leppington's recent work on the shadow of an ellipse has shown, however, that in part of the shadow some of the later terms of the residue series are exponentially large. In other words, the complete residue series in part of the shadow is even more slowly convergent than the original series.In the present paper the Watson transformation is re-examined in the light of this result. The original series is expressed as the sum of a finite number of terms of the residue series and of a remainder. It is shown that throughout the shadow the remainder is at short wavelengths asymptotically of smaller order than the last term retained in the finite residue series. It follows that the residue series (which is a convergent infinite series) is also an asymptotic series, and this fact is sufficient to justify most of the usual applications. The proof is given in detail for the circle and in outline for the ellipse; it makes use of the theory of conformal mapping.