Acoustic impedance of a cylindrical orifice

Abstract

We use matched asymptotics to derive analytical formulae for the acoustic impedance of a subwavelength orifice consisting of a cylindrical perforation in a rigid plate. In the inviscid case, an end correction to the length of the orifice due to Rayleigh is shown to constitute an exponentially accurate approximation in the limit where the aspect ratio of the orifice is large; in the opposite limit, we derive an algebraically accurate correction, depending upon the logarithm of the aspect ratio, to the impedance of a circular aperture in a zero-thickness screen. Viscous effects are considered in the limit of thin Stokes boundary layers, where a boundary layer analysis in conjunction with a reciprocity argument provides the perturbation to the impedance as a quadrature of the basic inviscid flow. We show that for large aspect ratios the latter perturbation can be captured with exponential accuracy by introducing a second end correction whose value is calculated to be in between two guesses commonly used in the literature; we also derive an algebraically accurate approximation in the small-aspect-ratio limit. The viscous theory reveals that the resistance exhibits a minimum as a function of aspect ratio, with the orifice radius held fixed. It is evident that the resistance grows in the long-aspect-ratio limit; in the opposite limit, resistance is amplified owing to the large velocities close to the sharp edge of the orifice. The latter amplification arrests only when the the plate is as thin as the Stokes boundary layer. The analytical approximations derived in this paper could be used to improve circuit modelling of resonating acoustic devices.

Boundary-layer effects on electromagnetic and acoustic extraordinary transmission through narrow slits

“…Together these give expression (3.22) for the parameter C. We emphasise that the effective condition (3.56) relies on the one-dimensional nature of the wave propagation in the bulk slit region and the related independence of Φ −1 , Φ 0 and Φ 1 on the transverse coordinate X. When the bulk field is multi-dimensional, the viscous effect gives a contribution to the effective boundary condition proportional to the surface Laplacian of the bulk pressure, rather than the bulk pressure itself [29]. Given (3.17), in the present case Φ −1 and its surface Laplacian are in fact proportional.…”

Acoustic scattering by cascades with complex boundary conditions: compliance, porosity and impedance

“…The Rayleigh conductivity is a function of the aperture shape: K R = 2R for a circular aperture of radius R in a zero-thickness plate. Recently, Brandão & Schnitzer (2020) derived asymptotic expressions for the Rayleigh conductivity of a cylindrical orifice in a plate of finite thickness. These solutions also account for the viscous effects of a Stokes boundary later adjacent to the plate.…”

Boundary-layer effects on electromagnetic and acoustic extraordinary transmission through narrow slits

“…We emphasise that the effective condition (3.56) relies on the one-dimensional nature of the wave propagation in the bulk slit region and the related independence of Φ −1 , Φ 0 and Φ 1 on the transverse coordinate X. When the bulk field is multi-dimensional, the viscous effect gives a contribution to the effective boundary condition proportional to the surface Laplacian of the bulk pressure, rather than the bulk pressure itself [22]. Given (3.17), in the present case Φ −1 and its surface Laplacian are in fact proportional.…”