A Fourier Matching Method for Analyzing Resonances in a Sound-Hard Slab with Subwavelength Holes

Abstract

This paper presents a Fourier matching method to rigorously study resonances in a sound-hard slab with a finite number of narrow cylindrical holes. The cross sections of the holes, of diameters $\mathcal{O}\left(h\right)$ for $h\ll 1$, can be arbitrarily shaped. Outside the slab, a sound field can be represented in terms of its normal derivatives on the apertures of the holes. Inside each hole, the field can be represented in terms of a countable Fourier basis due to the zero Neumann boundary condition on the side surface. The countably infinite Fourier coefficients for all the holes constitute the unknowns. Matching the two field representatives leads to a countable-dimensional linear system governing the unknowns. Due to the invertibility of a principal submatrix of the infinite-dimensional coefficient matrix, we reduce the linear system to a finitedimensional one. Resonances are those when the finite-dimensional linear system becomes singular. We derive asymptotic formulae for the resonances in the subwavelength structure for $h\ll 1$. They reveal that a sound field with its real frequency close to a resonance frequency can be enhanced by a magnitude $\mathcal{O}\left({h}^{-2}\right)$. Numerical experiments are carried out to validate the proposed resonance formulae.

Keywords

Acoustic scattering problem / resonance frequency / subwavelength structure / Helmholtz equation / field enhancement