用傅里叶匹配法分析含异形亚波长孔声硬平板的共振频率
A Fourier-matching Method for Analyzing Resonance Frequencies by a Sound-hard Slab with Arbitrarily Shaped Subwavelength Holes
摘要：
本文提出了一种简单的傅里叶匹配方法，严格研究了有限数量的任意形状的圆柱孔直径${\cal O}(h)$的$h\ll1$的声硬板的共振频率。在孔的外部，声场可以用其在孔的孔径上的法向导数来表示。在每个孔内，由于垂直变量可以分离，因此场可以用可数的傅立叶基函数集表示。匹配每个孔径上的场产生根据未知傅里叶系数的可数集合的可数方程的线性系。利用主子阵的可逆性，线性系可化为有限维线性系，并通过在极限情形$h\to 0$下每个洞的一个密切相关的边值问题的适定性证明了这一点，使得在有限维系统中只保留每个洞的主导傅里叶系数.共振频率是使所得到的有限维线性系秩亏的那些。通过对$h \ll 1$的正则渐近分析，得到了表征三维亚波长结构谐振频率的系统渐近公式。该公式揭示了一个重要的事实，即当所有孔的形状相同时，任何谐振频率的Q因子渐近地表现为$h\ll1$的${\cal O}(h^{-2})$，其前因子与孔的形状无关。
英文摘要：
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 $h\ll1$. 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 $h\to 0$, 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 $h\ll1$ with its prefactor independent of shapes of holes.
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