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Register a new userA Fourier-matching Method for Analyzing Resonance Frequencies by a Sound-hard Slab with Arbitrarily Shaped Subwavelength Holes
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This paper presents a simple Fourier-matching method to rigorously study resonance frequencies of a sound-hard slab with a finite number of arbitrarily shaped cylindrical holes of diameter ${cal O}(h)$ for $hll1$. Outside the holes, a sound field can be expressed in terms of its normal derivatives on the apertures of holes. Inside each hole, since the vertical variable can be separated, the field can be expressed in terms of a countable set of Fourier basis functions. Matching the field on each aperture yields a linear system of countable equations in terms of a countable set of unknown Fourier coefficients. The linear system can be reduced to a finite-dimensional linear system based on the invertibility of its principal submatrix, which is proved by the well-posedness of a closely related boundary value problem for each hole in the limiting case $hto 0$, so that only the leading Fourier coefficient of each hole is preserved in the finite-dimensional system. The resonance frequencies are those making the resulting finite-dimensional linear system rank deficient. By regular asymptotic analysis for $h ll 1$, we get a systematic asymptotic formula for characterizing the resonance frequencies by the 3D subwavelength structure. The formula reveals an important fact that when all holes are of the same shape, the Q-factor for any resonance frequency asymptotically behaves as ${cal O}(h^{-2})$ for $hll1$ with its prefactor independent of shapes of holes.
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This paper proposes a novel, rigorous and simple Fourier-transformation approach to study resonances in a perfectly conducting slab with finite number of subwavelength slits of width $hll 1$. Since regions outside the slits are variable separated, by Fourier transforming the governing equation, we could express field in the outer regions in terms of field derivatives on the aperture. Next, in each slit where variable separation is still available, wave field could be expressed as a Fourier series in terms of a countable basis functions with unknown Fourier coefficients. Finally, by matching field on the aperture, we establish a linear system of infinite number of equations governing the countable Fourier coefficients. By carefully asymptotic analysis of each entry of the coefficient matrix, we rigorously show that, by removing only a finite number of rows and columns, the resulting principle sub-matrix is diagonally dominant so that the infinite dimensional linear system can be reduced to a finite dimensional linear system. Resonance frequencies are exactly those frequencies making the linear system rank-deficient. This in turn provides a simple, asymptotic formula describing resonance frequencies with accuracy ${cal O}(h^3log h)$. We emphasize that such a formula is more accurate than all existing results and is the first accurate result especially for slits of number more than two to our best knowledge. Moreover, this asymptotic formula rigorously confirms a fact that the imaginary part of resonance frequencies is always ${cal O}(h)$ no matter how we place the slits as long as they are spaced by distances independent of width $h$.
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