A three-dimensional Wiener–Hopf technique for general bodies of revolution, Part I: Symmetrization and factorization of the integral equation’s kernel.

Abstract

This is the first of a two-part paper on the development of analytic solutions of acoustic scattering and/or radiation from arbitrary bodies of revolution under heavy fluid loading. The approach followed is the construction of a 3-D Wiener–Hopf technique with Fourier transforms that operate on the finite object’s arclength variable (the object’s practical finiteness comes about, in a Wiener–Hopf sense, by formally bringing to zero the radius of its semi-infinite generator curve for points beyond a prescribed station). Unlike in the classical case of a planar semi-infinite geometry, the kernel of the integral equation is nontranslational and therefore with independent wavenumber spectra for its receiver and source arclengths. The solution procedure applies a symmetrizing spatial operator that reconciles the regions of (+) and (−) analyticity of the kernel’s two-wavenumber transform with those of the virtual sources. The spatially symmetrized integral equation is of the Fredholm second kind and thus with a strong unit “diagonal”–a feature that makes possible the Wiener–Hopf factorization of its transcendental doubly transformed kernel via secondary spectral manipulations. The second of this two-part paper describes how the theory addresses departures from axisymmetry through substructuring techniques and demonstrates the analysis for canonical problems of fluid-structure interaction.