Sound scattering by cylinders of noncircular cross section: A conformal mapping approach

Abstract

A new method for predicting the far-field scattered pressure due to a plane wave incident upon an infinitely long cylinder of noncircular cross section is presented. The method, referred to herein as the Fourier matching method (FMM), involves conformally mapping the exterior and interior of a closed surface to a semi-infinite strip. This method is new in that the boundary conditions are satisfied using constraints described in the new angular variable. The resultant formulation is numerically efficient (much more so than the T-matrix method under certain conditions, for example) and works well for both small and large deviations from a circular cross section, as well as penetrable and impenetrable materials. Furthermore, the basis functions generated in the calculation can also improve the efficiency of other methods such as the T-matrix method. Example calculations are presented for elliptic, square, and three-leaf clover cross sections for several types of boundary conditions. In all cases, the results compare extremely well with exact or high-frequency asymptotic results.

Acoustic Scattering by Axisymmetric Finite-Length Bodies: An Extension of a 2-Dimensional Conformal Mapping Method

“…The derivation of the extended formulation for scattering by an axisymmetric finite-length body is conceptually very similar to the corresponding derivation of the two-dimensional solution described by DiPerna and Stanton (1994); in fact, some of the elements are identical. Both solutions begin with the wave equation in a known coordinate system and conformally map the coordinate variables to a new, orthogonal coordinate system in which the locus of all points where the new radial coordinate is a constant exactly coincides with the scatterer surface.…”

“…where (u, w, v) are the new coordinates, and F(u, w) is a function which depends on the specific transformation (Morse and Feshbach, 1953;DiPerna and Stanton, 1994). With the exception that the wave number is now a function of position, the new Helmholtz equation is formally identical to the Helmholtz equation in Cartesian coordinates.…”

“…Furthermore, the approach is inherently numerically efficient due to the nature of its formulation. For example, the FMM was shown by DiPerna and Stanton (1994) to be more efficient than the T-matrix method for the case of the high-aspect-ratio elliptic cylinder because fewer terms were needed for the numerical integrations. Even after incorporating the FMM basis functions into the T-matrix calculations, the FMM required 85% fewer integration points.…”

“…All of these approaches are limited in one or more of the following: frequency range, class of surfaces, types of boundary conditions, eccentricity of shape and/or computational implementation and numerical efficiency. DiPerna and Stanton (1994) introduced a conformal mapping approach to predicting farfield sound scattering by infinitely long cylinders of noncircular cross section. The approach, termed the Fourier Matching Method (FMM), involves a conformal mapping of variables to a new coordinate system in which the constant radial coordinate exactly matches the scatterer surface.…”

Three-dimensional visualization of fish morphometry and acoustic backscatter

“…The current approximation of body and swimbladder morphology using equivalent cylinders make KRM model predictions at insonifying angles near head-and tail-on less reliable. Expanding the KRM model and/or further investigation into other models (e.g., finite element boundary element [Foote, 1985] or conformal mapping [Diperna and Stanton, 1994]) may be required to achieve accurate model predictions at all orientations.…”