Sound scattering by a compact circular pore

Abstract

The aim of this paper is to study the three-dimensional scattering of an oblique wave incident on a flanged circular compact pore of finite depth. The multipole structure with the scattering is resolved by the method of matched asymptotic expansion, where we assume smallness of ε = k * a * , the product of the incident wavenumber k * and the pore radius a * . Two distinguished cases are solved: the rigid boundary condition and the pressure-release boundary condition. The study presents by far the most complete solutions to these problems, with the outer solution up to O ( ε 5 ) and the inner solution up to O ( ε 2 ) . In particular, the sophisticated interplay between the pore depth and the incident angle is revealed in the different orders of solution. It is shown that the leading order of the outer wave field for both cases is O ( ε 3 ) . For the rigid boundaries, there is one dipole dependent on the incident angle and one monopole. Interestingly, the monopole arises from the second-order interaction of the pore volume and the small but non-negligible compressibility in the inner field. This is one of the few examples analytically solvable to demonstrate this property. On the other hand, only one dipole is found for the pressure-release boundary. The next order in the outer solutions for both types of the boundary conditions is of O ( ε 5 ) and is shown to contain quadrupoles and octupoles. The multipole structures for both types of boundaries are tabulated, explicitly with the effects of the incident angle and the pore depth.