An extension to the short-wave asymptotics of the transmission coefficient for a semi-submerged circular cylinder

Abstract

The method of matched asymptotic expansions is used to extend the short-wave asymptotics of the transmission coefficient T by the addition of the terms of order 1/ N 5 , (log N ) 2 / N 6 and log N / N 6 as N → ∞ (where N = wavenumber times cylinder radius). The result is the formula \begin{eqnarray*} T &=& \frac{2{\rm i}}{\pi N^4}\exp (-2{\rm i}N)\left[1+\frac{4\log N}{\pi N}-\frac{4}{\pi N} \bigg(2-\gamma-\log 2+\frac{{\rm i}\pi}{8}\bigg)+\frac{8(\log N)^2}{\pi^2N^2}\right.\\ && \left.-\frac{8\log N}{\pi^2N^2}\bigg(5-2\gamma - \log 4+\frac{{\rm i}\pi}{4}\bigg)\right] + O\bigg(\frac{1}{N^6}\bigg)\quad {\rm as}\;N\rightarrow \infty \end{eqnarray*} (where γ = Euler's constant). The first term above is that derived rigorously by Ursell (1961) using an integral-equation method; the second term is that added by Leppington (1973) using matched asymptotic expansions; and the next three terms are those derived in this paper. Significant agreement between numerical values of T obtained from the completed fifth-order asymptotics and those obtained using Ursell's multipole expansions is demonstrated for 8 [les ] N [les ] 20 (table 2). The extensions of the perturbation expansions for the potential in the various fluid sub-domains (used in the method of matched expansions) provide some interesting cross-checks, between the solutions for potentials occurring later in the series and determined at advanced matching stages, with those for potentials occurring earlier on and determined independently at an earlier stage in the matching process. Some examples are given.