Application of the Sommerfeld–Watson transformation to scattering of acoustic waves obliquely incident upon cylindrical shells

Abstract

The Sommerfeld–Watson transformation (SWT) is applied to the normal mode series for scattering of an obliquely incident plane acoustic wave from an infinite circularly cylindrical shell to express the scattered field in terms of its wave components. The ‘‘geometric’’ part of the scattered field is extracted and expressed as a line integral that is evaluated by shifting to a path of steepest descent (SDP). When deforming the path, one or more poles may be crossed and their residues must be accounted for, resulting in discontinuities in the computed field unless the effects of these poles are also included in the evaluation of the SDP integral. This correction, made in the analysis in this paper, is applied to the particular example of a thin steel shell in water, using thin-shell theory. Very good agreement for scattering into the backward half-space is obtained between the results of the normal mode series and the modified SWT calculation for frequency-angle combinations such that ka cos φ≥2, where φ is the angle of incidence of the plane wave relative to the shell normal. Inclusion of the corrections to the solution also improves the calculation even when no poles are crossed by the path deformation.