Scattering of low frequency sound by fluid and solid cylinders

Abstract

Wave scattering by objects is typically studied for plane incident waves. However, full Green's functions are necessary for problems where the separation of the scatterer from interfaces, sources, or other scatterers is comparable to the dimensions of the scatterer itself. In this paper, the two-dimensional problem of scattering of monochromatic cylindrical waves by an infinite cylinder embedded in a homogeneous fluid is considered. Fluid and solid cylinders are studied, and soft and hard cylinders are revisited. The exact solutions for the Green's functions are expressed as an infinite series of cylindrical functions with complex amplitudes determined by the acoustic boundary conditions at the surface of the cylinder. Here, we derive closed-form asymptotics for the scattered field in the Rayleigh scattering regime where radius of the cylinder is small compared to the wavelength. The scattered wave approximation is valid for arbitrary source and observation point positions outside the scatterer and is expressed as a sum of fields due to three linear image sources. When the source or receiver is located sufficiently far from the cylinder, the new uniform asymptotic solutions reduce to well-known results for plane-wave scattering. Image source solutions were anticipated due to classically studied electrostatic analog problems involving dielectric cylinders. Image source representation offers physical insights into the scattering physics and suggests simple analytic solutions for scattering by objects near interfaces and within waveguides.