In this paper a mathematical model is given for the scattering of an incident wave from a surface covered with microscopic small Helmholtz resonators, which are cavities with small openings. More precisely, the surface is built upon a finite number of Helmholtz resonators in a unit cell and that unit cell is repeated periodically. To solve the scattering problem, the mathematical framework elaborated in [Ammari et al., Asympt. Anal., 2019] is used. The main result is an approximate formula for the scattered wave in terms of the lengths of the openings. Our framework provides analytic expressions for the scattering wave vector and angle and the phase-shift. It justifies the apparent absorption. Moreover, it shows that at specific lengths for the openings and a specific frequency there is an abrupt shift of the phase of the scattered wave due to the subwavelength resonances of the Helmholtz resonators. A numerically fast implementation is given to identify a region of those specific values of the openings and the frequencies.

中文翻译：

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基于周期性重复亥姆霍兹谐振器阵列的梯度超表面的数学和数值框架

在本文中，给出了入射波从覆盖有微小亥姆霍兹共振器的表面散射的数学模型，这些共振器是具有小开口的空腔。更准确地说，该表面建立在一个单位单元中有限数量的亥姆霍兹谐振器上，并且该单位单元周期性地重复。为了解决散射问题，[Ammari et al., Asympt. Anal., 2019] 使用。主要结果是散射波在开口长度方面的近似公式。我们的框架提供了散射波矢量和角度以及相移的解析表达式。它证明了表观吸收是合理的。而且，它表明，在开口的特定长度和特定频率处，由于亥姆霍兹谐振器的亚波长共振，散射波的相位会发生突然偏移。给出了数字快速实现来识别开口和频率的那些特定值的区域。